单位双圆盘Hardy空间上乘法算子的一些性质

单位双圆盘Hardy空间上乘法算子的一些性质 Some Properties of Multiplication Operators on the Hardy Space over the Bidisk 1 first-author 韩凯凯，男，讲师，现从事泛函分析研究。E-mail：kkhan.math@whu.edu.cn 韩凯凯，男，讲师，现从事泛函分析研究。E-mail：kkhan.math@whu.edu.cn kkhan.math@whu.edu.cn 2 河北经贸大学 数学与统计学学院，河北 石家庄 050061 河北经贸大学 数学与统计学学院，河北 石家庄 050061 School of Mathematics and Statistics， Hebei University of Economics and Business， Shijiazhuang 050061， Hebei， China School of Mathematics and Statistics， Hebei University of Economics and Business， Shijiazhuang 050061， Hebei， China 武汉大学 数学与统计学院，湖北 武汉 430072 武汉大学 数学与统计学院，湖北 武汉 430072 School of Mathematics and Statistics， Wuhan University， Wuhan 430072， Hubei， China School of Mathematics and Statistics， Wuhan University， Wuhan 430072， Hubei， China 设 <math id="M1"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522931&type= 2.03200006 3.13266659 是单位双圆盘上的解析函数，研究了单位双圆盘Hardy空间上乘法算子 <math id="M2"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522939&type= 4.23333359 3.21733332 的一些性质。具体地，给出了单位双圆盘Hardy空间上有界线性算子是乘法算子的一种刻画。证明了有界乘法算子 <math id="M3"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522949&type= 4.23333359 3.21733332 的本性范数等于 <math id="M4"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522960&type= 2.03200006 3.13266659 的上确界范数，揭示了单位双圆盘Hardy空间上没有非平凡的紧乘法算子。给出了乘法算子是满射的充分必要条件，描述了乘法算子的谱。利用Poisson积分的方法刻画了闭值域的乘法算子。 This paper considers some properties of multiplication operators <math id="M5"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522968&type= 4.23333359 3.21733332 on the Hardy space over the bidisk, where <math id="M6"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522980&type= 2.03200006 3.13266659 is an analytic function on the bidisk. Concretely, we gave a characterization when a bounded linear operator on the Hardy space over the bidisk is a multiplication operator. The essential norm of a bounded multiplication operator <math id="M7"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522988&type= 4.23333359 3.21733332 was proved to be equal to the supremum norm of <math id="M8"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522997&type= 2.03200006 3.13266659 , which reveals that there exists no non-trivial compact multiplication operators on the Hardy space over the bidisk. Moreover, we gave a sufficient and necessary condition when a multiplication operator was surjective, and described the spectra of multiplication operators. Finally, by using Poisson integral, we characterized multiplication operators with closed range. 单位双圆盘Hardy空间 乘法算子 本性范数 谱 Hardy space over the bidisk multiplication operator essential norm spectrum 0　 引 言 1 设 <math id="M9"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523009&type= 2.79399991 2.79399991 为复平面 <math id="M10"><mi mathvariant="normal">ℂ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523018&type= 2.70933342 2.79399991 中的单位圆盘，即 <math id="M11"><mi>D</mi><mo>=</mo><mo stretchy="false">{</mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi><mo>:</mo><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><mo><</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523028&type= 30.56466866 3.55599999 。设 <math id="M12"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523036&type= 2.53999996 2.79399991 为单位圆周，即 <math id="M13"><mi>T</mi><mo>=</mo><mo stretchy="false">{</mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi><mo>:</mo><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523042&type= 30.56466866 3.55599999 。记单位双圆盘 <math id="M14"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523052&type= 3.97933340 3.04799986 表示 <math id="M15"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523060&type= 2.79399991 2.79399991 与自身的笛卡尔乘积集合， <math id="M16"><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523068&type= 4.06400013 3.04799986 表示 <math id="M17"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523076&type= 2.53999996 2.79399991 与自身的笛卡尔乘积集合。设 <math id="M18"><mi>H</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523083&type= 10.66799927 3.81000018 表示单位双圆盘 <math id="M19"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523092&type= 3.97933340 3.04799986 上解析函数的全体，单位双圆盘 <math id="M20"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523103&type= 3.97933340 3.04799986 上Hardy空间 <math id="M21"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523111&type= 12.02266693 3.81000018 的定义为 <math display="block" id="M22"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>f</mi><mo>∈</mo><mi>H</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>:</mo><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><munder><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><mn mathvariant="normal">0</mn><mo><</mo><mi>r</mi><mo><</mo><mn mathvariant="normal">1</mn></mrow></munder><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mi>ζ</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523117&type= 94.48799896 8.29733276 其中 <math id="M23"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523125&type= 3.97933340 3.72533321 表示 <math id="M24"><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523134&type= 4.06400013 3.04799986 上标准化的Haar测度。设 <math id="M25"><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523140&type= 17.61066628 3.97933340 ，则对几乎处处的 <math id="M26"><mi>ζ</mi><mo>∈</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523147&type= 9.65199947 3.89466667 ，径向极限 <math id="M27"><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>r</mi><mo>→</mo><msup><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo></mrow></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mi>ζ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523155&type= 28.36333275 5.50333357 存在 ［ 1 1 ］ ，而且 <math id="M28"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523165&type= 12.02266693 3.81000018 与 <math id="M29"><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523173&type= 11.51466751 3.81000018 <math id="M30"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523181&type= 5.24933338 4.48733330 的某个闭子空间等距同构。对任意 <math id="M31"><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523189&type= 17.61066628 3.97933340 ， <math display="block" id="M32"><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523198&type= 39.70866776 7.28133297 Hardy空间 <math id="M33"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523208&type= 12.02266693 3.81000018 的再生核函数与标准化再生核函数分别记为 <math display="block" id="M34"><msub><mrow><mi>K</mi></mrow><mrow><mi>w</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mover><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mo>¯</mo></mover><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mover><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mo>¯</mo></mover><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523216&type= 50.71533203 8.97466660 <math display="block" id="M35"><msub><mrow><mi>k</mi></mrow><mrow><mi>w</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">|</mo><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">|</mo><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mover><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mo>¯</mo></mover><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mover><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mo>¯</mo></mover><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523224&type= 52.83199692 9.39799976 其中， <math id="M36"><mi>w</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523233&type= 46.99000168 4.65666676 。设 <math id="M37"><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523241&type= 12.95399952 3.81000018 表示单位双圆盘 <math id="M38"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523249&type= 3.97933340 3.04799986 上有界解析函数的全体，即 <math display="block" id="M39"><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>f</mi><mo>∈</mo><mi>H</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>:</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub><mo>=</mo><munder><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></munder><mfenced open="|" close="|" separators="|"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523257&type= 76.45400238 9.48266697 设 <math id="M40"><mi>φ</mi><mo>∈</mo><mi>H</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523265&type= 16.59466553 3.97933340 ，由 <math id="M41"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523273&type= 2.37066674 3.64066648 诱导的乘法算子 <math id="M42"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523280&type= 4.82599974 3.72533321 的定义为： <math id="M43"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mi>f</mi><mo>=</mo><mi>φ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523288&type= 15.83266640 4.57200003 ， <math id="M44"><mi>f</mi><mo>∈</mo><mi>H</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523295&type= 16.25600052 3.97933340 。由文献［ 2 2 ］可知，乘法算子 <math id="M45"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523301&type= 4.82599974 3.72533321 在Hardy空间 <math id="M46"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523309&type= 12.02266693 3.81000018 上有界当且仅当 <math id="M47"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523317&type= 18.88066673 3.97933340 。进一步地，若 <math id="M48"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523324&type= 4.82599974 3.72533321 在 <math id="M49"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523332&type= 12.02266693 3.81000018 上有界，则 <math id="M50"><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523341&type= 22.69066620 5.84200001 。近些年，各种解析函数空间上的乘法算子得到了广泛研究。文献［ 3 3 ］研究了Hardy-Sobolev空间上乘法算子的谱理论。文献［ 4 4 ］运用Berezin变换刻画了Bergman空间上闭值域的乘法算子。文献［ 5 5 ］研究了平面区域Hardy空间和加权Bergman空间上的乘法算子。本文将研究单位双圆盘Hardy空间 <math id="M51"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523349&type= 12.02266693 3.81000018 上乘法算子的一些基本性质。 1　 主要结论及证明 1 设 <math id="M52"><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523359&type= 24.38399887 4.65666676 ， <math id="M53"><mi>ζ</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523370&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523366&type= 24.12999916 4.91066647 ， <math id="M54"><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523377&type= 13.63133430 4.31799984 ， <math id="M55"><msub><mrow><mi>ζ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>φ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523385&type= 11.34533310 5.16466665 ， <math id="M56"><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523391&type= 11.51466751 3.64066648 。Poisson核函数定义如下： <math id="M57"><mi>P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523398&type= 54.52533722 5.07999992 ，其中 <math id="M58"><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>r</mi></mrow><mrow><mi>j</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523405&type= 77.47000122 4.40266657 ， <math id="M59"><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523414&type= 11.51466751 3.64066648 。注意到 <math id="M60"><mi>P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523420&type= 18.11866570 3.72533321 ，以及 <math id="M61"><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523426&type= 35.05199814 7.28133297 ［ 1 1 ］ 。 引理1 设 <math id="M62"><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523433&type= 17.61066628 3.97933340 。若存在常数 <math id="M63"><mi>K</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523441&type= 9.65199947 2.79399991 ，满足对几乎处处的 <math id="M64"><mi>ζ</mi><mo>∈</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523447&type= 9.65199947 3.89466667 ， <math id="M65"><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523454&type= 17.77999878 5.16466665 成立，则对任意 <math id="M66"><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523460&type= 9.48266697 3.38666677 ， <math id="M67"><mfenced open="|" close="|" separators="|"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523469&type= 17.77999878 4.82599974 成立。 证 对任意 <math id="M68"><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523478&type= 9.48266697 3.38666677 ，由文献［ 1 1 ］可得， <math id="M69"><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523486&type= 49.53000259 7.28133297 。于是 <math display="block" id="M70"><mfenced open="|" close="|" separators="|"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo><mi>K</mi><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523493&type= 99.73733521 8.29733276 证毕。 定理1 设 <math id="M71"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523501&type= 2.20133328 2.79399991 是Hardy空间 <math id="M72"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523512&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523508&type= 12.02266693 3.81000018 上的有界线性算子，则 <math id="M73"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523520&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523516&type= 2.20133328 2.79399991 是乘法算子当且仅当 <math id="M74"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523528&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523524&type= 2.20133328 2.79399991 与 <math id="M75"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523532&type= 5.24933338 4.06400013 和 <math id="M76"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523541&type= 5.24933338 4.06400013 均交换，其中 <math id="M77"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523549&type= 40.72466660 5.07999992 ， <math id="M78"><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523556&type= 11.51466751 3.64066648 。 证 必要性是显然的，只需证充分性。设 <math id="M79"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523568&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523564&type= 2.20133328 2.79399991 与 <math id="M80"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523570&type= 5.24933338 4.06400013 和 <math id="M81"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523573&type= 5.24933338 4.06400013 均交换。令 <math id="M82"><mi>φ</mi><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523578&type= 14.30866623 3.72533321 ，下证 <math id="M83"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523582&type= 18.88066673 3.97933340 ，并且 <math id="M84"><mi>S</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523589&type= 11.76866722 3.72533321 。对任意 <math id="M85"><mi>n</mi><mo>,</mo><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523593&type= 13.03866673 3.21733332 ，直接计算可得 <math id="M86"><mi>S</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msubsup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msubsup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>⋅</mo><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523603&type= 27.94000053 4.91066647 。因此，对任意多项式 <math id="M87"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523611&type= 1.86266661 3.64066648 ， <math id="M88"><mi>S</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo>⋅</mo><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523618&type= 19.04999924 3.72533321 。 注意到 <math id="M89"><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523633&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523629&type= 34.62866592 4.74133348 。因为多项式在 <math id="M90"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523638&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523636&type= 12.02266693 3.81000018 中是稠密的，则存在一列多项式 <math id="M91"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523645&type= 5.58799982 4.91066647 依范数收敛到 <math id="M92"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523651&type= 2.37066674 3.64066648 。由文献［ 2 2 ］引理2可得， <math id="M93"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523658&type= 5.58799982 4.91066647 在 <math id="M94"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523666&type= 3.97933340 3.04799986 上点点收敛到 <math id="M95"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523678&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523674&type= 2.37066674 3.64066648 。因为 <math id="M96"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523682&type= 2.20133328 2.79399991 在 <math id="M97"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523690&type= 12.02266693 3.81000018 上有界，则 <math id="M98"><mfenced open="{" close="}" separators="|"><mrow><mi>S</mi><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523698&type= 7.70466709 4.91066647 依范数收敛到 <math id="M99"><mi>S</mi><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523706&type= 4.48733330 3.64066648 ，于是 <math id="M100"><mfenced open="{" close="}" separators="|"><mrow><mi>S</mi><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523714&type= 7.70466709 4.91066647 在 <math id="M101"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523721&type= 3.97933340 3.04799986 上点点收敛到 <math id="M102"><mi>S</mi><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523729&type= 4.48733330 3.64066648 。另一方面，因为 <math id="M103"><mi>S</mi><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⋅</mo><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523735&type= 17.77999878 4.57200003 ，则 <math id="M104"><mfenced open="{" close="}" separators="|"><mrow><mi>S</mi><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523743&type= 7.70466709 4.91066647 在 <math id="M105"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523750&type= 3.97933340 3.04799986 上点点收敛到 <math id="M106"><msup><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523762&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523759&type= 3.38666677 3.89466667 。由此可得， <math id="M107"><mi>S</mi><mi>φ</mi><mo>=</mo><msup><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523771&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523766&type= 12.44600010 3.89466667 ，以及 <math id="M108"><msup><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523774&type= 19.38866615 3.97933340 。归纳可得，对任意 <math id="M109"><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523786&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523781&type= 10.41399956 3.38666677 ， <math id="M110"><mi>S</mi><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523790&type= 17.35666656 3.89466667 成立，以及 <math id="M111"><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523798&type= 19.38866615 3.97933340 。于是， <math id="M112"><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>≤</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced><mo>⋅</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced><mo>≤</mo><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523806&type= 41.23266602 5.07999992 。归纳可得，对任意 <math id="M113"><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523814&type= 10.41399956 3.38666677 ， <math id="M114"><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced><mo>≤</mo><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523823&type= 20.31999969 5.07999992 成立。若 <math id="M115"><mi>S</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523832&type= 8.88999939 2.79399991 ，结论是显然的。所以设 <math id="M116"><mi>S</mi><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523840&type= 8.88999939 3.04799986 ，令 <math id="M117"><mi>ψ</mi><mo>=</mo><mi>φ</mi><mo>/</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523848&type= 17.52599907 3.97933340 。则对任意 <math id="M118"><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523858&type= 10.41399956 3.38666677 ， <math id="M119"><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523866&type= 14.39333344 4.74133348 成立。假设存在正数 <math id="M120"><mi>ε</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523874&type= 8.12800026 2.79399991 ，使得集合 <math id="M121"><mi>E</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>ζ</mi><mo>∈</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>:</mo><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>ψ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>ε</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523883&type= 45.04266739 5.50333357 具有正测度，即 <math id="M122"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523891&type= 17.18733406 4.48733330 。则对任意 <math id="M123"><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523898&type= 10.41399956 3.38666677 ，由（1）式可得 <math display="block" id="M124"><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><mo stretchy="false">|</mo><mover><mrow><mi>ψ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn><mi>n</mi></mrow></msup><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mo>≥</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn><mi>n</mi></mrow></msup><mo>⋅</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523906&type= 73.15200043 7.28133297 这与对任意 <math id="M125"><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523914&type= 10.41399956 3.38666677 ， <math id="M126"><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523922&type= 14.39333344 4.74133348 矛盾。因此，对几乎处处的 <math id="M127"><mi>ζ</mi><mo>∈</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523930&type= 9.65199947 3.89466667 ， <math id="M128"><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>ψ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523938&type= 17.27199936 4.82599974 。由引理1可得， <math id="M129"><mi>ψ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523946&type= 18.96533394 3.97933340 。于是， <math id="M130"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523952&type= 18.88066673 3.97933340 。 对任意 <math id="M131"><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523960&type= 17.61066628 3.97933340 ，存在一列多项式 <math id="M132"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523968&type= 5.58799982 4.91066647 依范数收敛到 <math id="M133"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523977&type= 1.10066664 3.64066648 。因为 <math id="M134"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523985&type= 2.20133328 2.79399991 在 <math id="M135"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523989&type= 12.02266693 3.81000018 上有界以及 <math id="M136"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523994&type= 18.88066673 3.97933340 ，则 <math id="M137"><mi>S</mi><mi>f</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>S</mi><mo stretchy="false">(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>φ</mi><mo>=</mo><mi>f</mi><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524001&type= 49.78400040 4.65666676 。因此 <math id="M138"><mi>S</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524005&type= 11.76866722 3.72533321 。证毕。 设 <math id="M139"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524011&type= 2.62466669 2.79399991 是复Banach空间， <math id="M140"><mi mathvariant="normal">ℬ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524017&type= 9.22866726 3.55599999 表示 <math id="M141"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524023&type= 2.62466669 2.79399991 上全体有界线性算子构成的代数， <math id="M142"><mi mathvariant="italic">𝒦</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524030&type= 10.32933331 3.55599999 表示 <math id="M143"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524037&type= 2.62466669 2.79399991 上紧算子的全体。设 <math id="M144"><mi>S</mi><mo>∈</mo><mi mathvariant="normal">ℬ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524044&type= 15.15533447 3.55599999 ， <math id="M145"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524050&type= 2.20133328 2.79399991 本性范数的定义为 <math id="M146"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mfenced open="{" close="}" separators="|"><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi><mo>-</mo><mi>R</mi></mrow></mfenced><mo>:</mo><mi>R</mi><mo>∈</mo><mi mathvariant="italic">𝒦</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524057&type= 51.64666748 5.41866684 。 <math id="M147"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524063&type= 2.20133328 2.79399991 是紧算子当且仅当 <math id="M148"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524069&type= 13.97000027 5.07999992 。对任意 <math id="M149"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524076&type= 8.29733276 3.13266659 ， <math id="M150"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524084&type= 2.20133328 2.79399991 第 <math id="M151"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524090&type= 1.86266661 2.79399991 个逼近数的定义为 <math display="block" id="M152"><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mfenced open="{" close="}" separators="|"><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi><mo>-</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced><mo>:</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℬ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>≤</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524096&type= 75.09933472 5.67266655 因为有限秩算子是紧算子，则对任意 <math id="M153"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524102&type= 8.29733276 3.13266659 ，成立 <math id="M154"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced></mrow><mrow><mi>e</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≤</mo><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>S</mi></mrow></mfenced></mrow><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">6</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524110&type= 35.81399918 5.84200001 。 定理2 设 <math id="M155"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524119&type= 18.88066673 3.97933340 ，则在Hardy空间 <math id="M156"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524127&type= 12.02266693 3.81000018 上下式成立 <math display="block" id="M157"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mo>∀</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524135&type= 66.80200195 5.92666674 因此， <math id="M158"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524144&type= 4.82599974 3.72533321 是紧算子当且仅当 <math id="M159"><mi>φ</mi><mo>≡</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524152&type= 8.88999939 3.64066648 。 证 因为 <math id="M160"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>e</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo stretchy="false">)</mo><mo>≤</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524160&type= 54.52533722 5.92666674 ，所以只需证明 <math id="M161"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>e</mi></mrow></msub><mo>≥</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524167&type= 23.79133415 5.92666674 。因为 <math id="M162"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524175&type= 17.10266685 5.75733376 ，则存在 <math id="M163"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524184&type= 3.97933340 3.04799986 中一个点列 <math id="M164"><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><msubsup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524191&type= 11.17599964 4.65666676 ，满足 <math id="M165"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced><mo>=</mo><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524200&type= 22.26733398 5.41866684 ，以及 <math id="M166"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfenced open="|" close="|" separators="|"><mrow><mi>φ</mi><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524208&type= 31.15733337 5.84200001 。由文献［ 2 2 ］中引理2可得，对任意 <math id="M167"><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524216&type= 10.41399956 3.38666677 ，有 <math id="M168"><mfenced open="|" close="|" separators="|"><mrow><mi>φ</mi><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524224&type= 40.80933380 6.26533318 。于是， <math id="M169"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524230&type= 29.97199821 6.26533318 。显然，对任意 <math id="M170"><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524237&type= 9.48266697 3.38666677 ， <math id="M171"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524245&type= 21.92866707 4.65666676 。根据文献［ 7 7 ］可得， <math id="M172"><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524253&type= 3.64066648 4.06400013 在 <math id="M173"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524261&type= 12.02266693 3.81000018 中弱收敛于 <math id="M174"><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524269&type= 1.86266661 2.79399991 。对 <math id="M175"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524276&type= 12.02266693 3.81000018 上任意紧算子 <math id="M176"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524284&type= 2.70933342 2.79399991 ，则 <math id="M177"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfenced open="‖" close="‖" separators="|"><mrow><mi>R</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524290&type= 23.28333473 5.67266655 。于是 <math display="block" id="M178"><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>-</mo><mi>R</mi></mrow></mfenced><mo>≥</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>-</mo><mi>R</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>≥</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfenced separators="|"><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>-</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi>R</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524297&type= 113.70732880 6.94266701 由此可得， <math id="M179"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>e</mi></mrow></msub><mo>≥</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524310&type= 23.79133415 5.92666674 。证毕。 定理3 设 <math id="M180"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524317&type= 18.88066673 3.97933340 ，下列结论成立。 （a） 若 <math id="M181"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524325&type= 2.37066674 3.64066648 不恒为零，则 <math id="M182"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524333&type= 4.82599974 3.72533321 在 <math id="M183"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524341&type= 12.02266693 3.81000018 上是满射当且仅当存在常数 <math id="M184"><mi>M</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524349&type= 10.41399956 2.87866688 ，使得对任意 <math id="M185"><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524357&type= 9.48266697 3.38666677 ， <math id="M186"><mfenced open="|" close="|" separators="|"><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≥</mo><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524368&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524364&type= 18.71133423 4.74133348 成立。 （b） <math id="M187"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524372&type= 4.82599974 3.72533321 在 <math id="M188"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524386&type= 12.02266693 3.81000018 上的谱 <math id="M189"><mi>σ</mi><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mover><mrow><mi>φ</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524394&type= 24.89200020 4.48733330 。 证 （a） 设 <math id="M190"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524403&type= 2.37066674 3.64066648 不恒为零，充分性是显然的，只需证明必要性。设 <math id="M191"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524412&type= 4.82599974 3.72533321 在 <math id="M192"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524420&type= 12.02266693 3.81000018 上是满射。若 <math id="M193"><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524428&type= 17.61066628 3.97933340 ，满足 <math id="M194"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mi>f</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524436&type= 14.30866623 4.57200003 ，则对任意 <math id="M195"><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524443&type= 9.48266697 3.38666677 ， <math id="M196"><mi>φ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524450&type= 22.69066620 3.81000018 。因为 <math id="M197"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524458&type= 2.37066674 3.64066648 在 <math id="M198"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524471&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524468&type= 3.97933340 3.04799986 上解析且不恒为零，则集合 <math id="M199"><mi>E</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>:</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≠</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524475&type= 36.15266418 4.74133348 是 <math id="M200"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524481&type= 3.97933340 3.04799986 中的非空开集。由此可得，对任意 <math id="M201"><mi>z</mi><mo>∈</mo><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524484&type= 8.04333305 3.13266659 ， <math id="M202"><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524491&type= 13.80066681 3.81000018 成立。于是 <math id="M203"><mi>f</mi><mo>≡</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524500&type= 8.55133343 3.64066648 ，这表明 <math id="M204"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524505&type= 4.82599974 3.72533321 在 <math id="M205"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524513&type= 12.02266693 3.81000018 上是单射，从而 <math id="M206"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524522&type= 4.82599974 3.72533321 在 <math id="M207"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524529&type= 12.02266693 3.81000018 上可逆。因此存在常数 <math id="M208"><mi>M</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524538&type= 10.41399956 2.87866688 ，使得对任意 <math id="M209"><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524546&type= 9.48266697 3.38666677 ，成立 <math id="M210"><mfenced open="|" close="|" separators="|"><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>⋅</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msub><mrow><mi>K</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfenced><mo>≥</mo><mi>M</mi><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524554&type= 56.38800049 4.91066647 。 （b） 设 <math id="M211"><mi>λ</mi><mo>∈</mo><mi>φ</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524563&type= 14.90133286 3.97933340 ，则存在 <math id="M212"><mi>w</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524571&type= 10.75266743 3.38666677 ，使得 <math id="M213"><mi>λ</mi><mo>=</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524579&type= 15.15533447 3.72533321 。对任意 <math id="M214"><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524587&type= 17.61066628 3.97933340 ， <math display="block" id="M215"><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>-</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>-</mo><mi>λ</mi><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524593&type= 68.41066742 4.65666676 由此可得， <math id="M216"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>-</mo><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524602&type= 11.34533310 3.72533321 不是满射，因此 <math id="M217"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>-</mo><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524610&type= 11.34533310 3.72533321 不可逆。于是， <math id="M218"><mi>φ</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>⊆</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524616&type= 24.29933357 4.91066647 ，从而 <math id="M219"><mover><mrow><mi>φ</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><mo>⊆</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524625&type= 25.06133270 4.48733330 。 对任意 <math id="M220"><mi>λ</mi><mo>∉</mo><mover><mrow><mi>φ</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524633&type= 15.49400043 4.40266657 ，令 <math id="M221"><mi>δ</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mover><mrow><mi>φ</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524641&type= 29.12533188 4.40266657 ，则 <math id="M222"><mi>δ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524649&type= 8.63599968 2.79399991 ，其中 <math id="M223"><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mover><mrow><mi>φ</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524655&type= 22.35199928 4.40266657 表示 <math id="M224"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524660&type= 1.77800000 2.79399991 到集合 <math id="M225"><mover><mrow><mi>φ</mi><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524666&type= 9.56733322 4.40266657 的距离。因为对任意 <math id="M226"><mi>z</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524676&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524673&type= 9.48266697 3.38666677 ， <math id="M227"><mfenced open="|" close="|" separators="|"><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>≥</mo><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524679&type= 23.36800003 4.74133348 ，则 <math id="M228"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>-</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524684&type= 20.65866661 4.23333359 在 <math id="M229"><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524693&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524690&type= 3.97933340 3.04799986 上解析，并且 <math id="M230"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>-</mo><mi>λ</mi></mrow></mfenced></mrow></mrow><mo>≤</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mi>δ</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524695&type= 32.25799942 5.24933338 。显然 <math id="M231"><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><msup><mrow><mo>-</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mi>φ</mi><mo>-</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524701&type= 34.54399872 5.24933338 ，于是 <math id="M232"><mi>λ</mi><mo>∉</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524707&type= 15.40933418 4.48733330 。证毕。 设 <math id="M233"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524713&type= 2.62466669 2.79399991 是复Banach空间， <math id="M234"><mi>S</mi><mo>∈</mo><mi mathvariant="normal">ℬ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524724&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524720&type= 15.15533447 3.55599999 。若 <math id="M235"><mi>S</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524728&type= 8.88999939 3.55599999 是 <math id="M236"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524734&type= 2.62466669 2.79399991 的闭子空间，则称 <math id="M237"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524740&type= 2.20133328 2.79399991 具有闭值域。设 <math id="M238"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524745&type= 18.88066673 3.97933340 且 <math id="M239"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524750&type= 2.37066674 3.64066648 不恒等于零，由定理3的证明可得， <math id="M240"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524755&type= 4.82599974 3.72533321 在 <math id="M241"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524761&type= 12.02266693 3.81000018 上是单射。根据文献［ 8 8 ］中定理1.2可知， <math id="M242"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524767&type= 4.82599974 3.72533321 具有闭值域当且仅当存在常数 <math id="M243"><mi>M</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524774&type= 10.41399956 2.87866688 ，使得对任意 <math id="M244"><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524779&type= 17.61066628 3.97933340 ，成立 <math id="M245"><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi><mi>f</mi></mrow></mfenced><mo>≥</mo><mi>M</mi><mfenced open="‖" close="‖" separators="|"><mrow><mi>f</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524786&type= 21.50533485 4.82599974 。 定理4 设 <math id="M246"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524793&type= 18.88066673 3.97933340 且 <math id="M247"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524801&type= 2.37066674 3.64066648 不恒等于零，则 <math id="M248"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524809&type= 4.82599974 3.72533321 在空间 <math id="M249"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524817&type= 12.02266693 3.81000018 上具有闭值域当且仅当存在常数 <math id="M250"><mi>M</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524839&type= 10.41399956 2.87866688 ，使得对几乎处处的 <math id="M251"><mi>ζ</mi><mo>∈</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524846&type= 9.65199947 3.89466667 ， <math id="M252"><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>φ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≥</mo><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524854&type= 18.62666702 4.82599974 。 证 充分性是显然的，只需证明必要性。设 <math id="M253"><msub><mrow><mi>M</mi></mrow><mrow><mi>φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524863&type= 4.82599974 3.72533321 在空间 <math id="M254"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524871&type= 12.02266693 3.81000018 上具有闭值域，则存在常数 <math id="M255"><mi>M</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524879&type= 10.41399956 2.87866688 ，使得对任意 <math id="M256"><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524886&type= 17.61066628 3.97933340 ，成立 <math id="M257"><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi><mi>f</mi></mrow></mfenced><mo>≥</mo><mi>M</mi><mfenced open="‖" close="‖" separators="|"><mrow><mi>f</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524893&type= 21.50533485 4.82599974 。因为 <math id="M258"><mi>φ</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524900&type= 18.88066673 3.97933340 ，则 <math id="M259"><mover><mrow><mi>φ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524908&type= 23.53733444 4.91066647 。由文献［ 1 1 ］中定理2.1.3和文献［ 9 9 ］中定理3.12可得，对几乎处处的 <math id="M260"><mi>ξ</mi><mo>∈</mo><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524916&type= 9.56733322 3.89466667 ， <math id="M261"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>φ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>φ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>ξ</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524924&type= 66.12466431 7.28133297 ，其中 <math id="M262"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524932&type= 16.25600052 4.23333359 ， <math id="M263"><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msup><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524941&type= 11.59933376 3.97933340 。令 <math id="M264"><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524949&type= 12.61533356 4.57200003 ，则 <math id="M265"><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524957&type= 11.09133339 4.23333359 并且 <math display="block" id="M266"><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>φ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>ξ</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mo>∫</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>φ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524964&type= 107.52666473 7.28133297 于是， <math id="M267"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>φ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>φ</mi><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><msup><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524971&type= 59.60533142 6.34999990 <math id="M268"><mo>=</mo><msup><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524979&type= 9.14400005 3.04799986 。证毕。 2　 结 语 1 本文给出了单位双圆盘Hardy空间上有界线性算子是乘法算子的一种刻画。研究了单位双圆盘Hardy空间上乘法算子的一些基本性质，包括乘法算子的本性范数、逼近数、谱与闭值域性质。Vukotić ［ 6 6 ］ 研究了单位圆盘Hardy空间 <math id="M269"><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524987&type= 4.40266657 3.04799986 上解析Toeplitz算子的一些性质，本文的相关结论是在单位双圆盘Hardy空间上得到的，是其工作的推广。 参考文献 RUDIN W . 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DOI： 10.1007/s11785-019-00949-4 http://dx.doi.org/10.1007/s11785-019-00949-4 . Closed range type properties of Toeplitz operators on the Bergman space and the Berezin transform YAN Y . Multiplication operators on Hardy and weighted Bergman spaces over planar regions ［J］. Banach Journal of Mathematical Analysis ， 2020 ， 14 （ 4 ）： 1443 - 1480 . DOI： 10.1007/s43037-020-00070-1 http://dx.doi.org/10.1007/s43037-020-00070-1 . Multiplication operators on Hardy and weighted Bergman spaces over planar regions VUKOTIĆ D . Analytic Toeplitz operators on the Hardy space <math id="M270"><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524993&type= 3.72533321 2.53999996 ： A survey ［J］. Bulletin of the Belgian Mathematical Society Simon Stevin ， 2003 ， 10 （ 1 ）： 101 - 113 . 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