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Multiplication operators on Hardy and weighted Bergman spaces over planar regions
Banach Journal of Mathematical Analysis ( IF 1.2 ) Pub Date : 2020-05-19 , DOI: 10.1007/s43037-020-00070-1
Yi Yan

This paper studies some aspects of commutant theory and functional calculus for analytic multiplication operators on Hardy and weighted Bergman spaces over bounded planar regions. Multiplication operators defined by univalent functions are shown to commute only with multiplication operators. This result is generalized to a tuple of operators, and a sufficient condition is given for irreducibility of that induced by finite Blaschke products. Operators defined by fairly general ancestral functions are shown to commute with no nonzero compact operators, and these include the ones by monomial functions over annuli. For such operators over annuli, we characterize a certain dense subalgebra of the commutant. Norm and sequential weak closures of the analytic functional calculus algebra generated by a multiplication operator are characterized and essential spectral mapping properties obtained. Generalizing the similarity for finite Blaschke products to a larger class of weighted Bergman spaces, the commutant classification of these operators is obtained and seen strictly finer than the similarity classification, among other related results.

中文翻译:

平面区域上 Hardy 和加权 Bergman 空间的乘法算子

本文研究了有界平面区域上哈代空间和加权伯格曼空间上的解析乘法算子的交换理论和泛函演算的一些方面。单价函数定义的乘法运算符仅与乘法运算符交换。这个结果被推广到一个算子元组,并且给出了由有限 Blaschke 积引起的不可约性的充分条件。由相当一般的祖先函数定义的算子被证明可以与非零紧凑算子交换,其中包括环上的单项式函数的算子。对于环上的这些算子,我们刻画了换向式的某个稠密子代数。由乘法算子生成的解析泛函微积分代数的范数和顺序弱闭包被表征,并获得了基本的谱映射特性。将有限 Blaschke 乘积的相似性推广到更大的加权 Bergman 空间类别,获得了这些算子的交换分类,并且可以看到比相似性分类严格得多,以及其他相关结果。
更新日期:2020-05-19
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