{"title":"福克空间上豪斯多夫算子的有界性和紧凑性","authors":"Óscar Blasco, Antonio Galbis","doi":"10.1090/tran/9133","DOIUrl":null,"url":null,"abstract":"<p>We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F Subscript alpha Superscript p\"> <mml:semantics> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>α</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">F^p_\\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and taking its values into a larger one <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F Subscript alpha Superscript q Baseline comma 0 greater-than p less-than-or-equal-to q less-than-or-equal-to normal infinity\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>α</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>q</mml:mi> <mml:mo>≤</mml:mo> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">F^q_\\alpha ,\\ 0 > p \\leq q \\leq \\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F Subscript alpha Superscript normal infinity\"> <mml:semantics> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>α</mml:mi> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">F^\\infty _\\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-summing are also included.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":"44 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Boundedness and compactness of Hausdorff operators on Fock spaces\",\"authors\":\"Óscar Blasco, Antonio Galbis\",\"doi\":\"10.1090/tran/9133\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper F Subscript alpha Superscript p\\\"> <mml:semantics> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>α</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:annotation encoding=\\\"application/x-tex\\\">F^p_\\\\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and taking its values into a larger one <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper F Subscript alpha Superscript q Baseline comma 0 greater-than p less-than-or-equal-to q less-than-or-equal-to normal infinity\\\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>α</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>q</mml:mi> <mml:mo>≤</mml:mo> <mml:mi mathvariant=\\\"normal\\\">∞</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">F^q_\\\\alpha ,\\\\ 0 > p \\\\leq q \\\\leq \\\\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. 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引用次数: 0
摘要
我们得到了作用于Fock空间F α p F^p_\alpha 并将其值转化为更大的F α q , 0 > p ≤ q ≤ ∞ F^q_\alpha ,\ 0 > p \leq q \leq \infty 的有界Hausdorff算子的完整描述，以及Hausdorff算子将Fock空间转化为更小的Fock空间的一些必要或充分条件。一些结果是在混合规范 Fock 空间的背景下写出的。此外，还描述了福克空间上豪斯多夫算子的紧凑性。福克空间 F α ∞ F^\infty _\alpha 上豪斯多夫算子的紧凑性结果被扩展到更一般的全函数巴纳赫空间，其加权超规范是以径向权重定义的，豪斯多夫算子成为 p p - 和的条件也包括在内。
Boundedness and compactness of Hausdorff operators on Fock spaces
We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space FαpF^p_\alpha and taking its values into a larger one Fαq,0>p≤q≤∞F^q_\alpha ,\ 0 > p \leq q \leq \infty, as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space Fα∞F^\infty _\alpha is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become pp-summing are also included.
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