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Multiplication operators on Hardy and weighted Bergman spaces over planar regions

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Abstract

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.

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Acknowledgements

The author wishes to thank the referees for their careful review of the manuscript and many helpful suggestions. This work is part of the author’s Ph. D. dissertation under the guidance and encouragement of Professor Tyrone Duncan and Professor Albert Sheu.

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  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Yi Yan

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  1. Yi Yan
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Correspondence to Yi Yan.

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Communicated by Dechao Zheng.

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Yan, Y. Multiplication operators on Hardy and weighted Bergman spaces over planar regions. Banach J. Math. Anal. 14, 1443–1480 (2020). https://doi.org/10.1007/s43037-020-00070-1

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