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Commutants and complex symmetry of finite Blaschke product multiplication operator in $L^2(\mathbb{T})$

Part of: General theory of linear operators Spaces and algebras of analytic functions Special classes of linear operators

Published online by Cambridge University Press:  11 January 2024

Arup Chattopadhyay  and
Soma Das
Arup Chattopadhyay
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India (arupchatt@iitg.ac.in; 2003arupchattopadhyay@gmail.com)
Soma Das
Affiliation:
Statistics and Mathematics Unit, Indian Statistical Institute Bangalore, Bangalore 560059, India (dsoma994@gmail.com; somadas_pd@isibang.ac.in; soma18@iitg.ac.in)
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Abstract

Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.

Keywords

commutantconjugationHardy spacemodel spacecomplex symmetric operatorBlaschke product

MSC classification

Primary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A15: Invariant subspaces 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators 30H10: Hardy spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 1 , February 2024 , pp. 261 - 286
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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