On the commutant of certain multiplication operators on spaces of analytic functions
type: Item
instance of: scholarly article
title: On the commutant of certain multiplication operators on spaces of analytic functions
zbMATH Open document ID: 1065.47030
DOI: 10.1007/BF02904268
publication date: 2003-03-18T00:00:00Z
review text: Let \(\mathcal{M}(B)\) be the space of multipliers in the Banach space \(B\) of analytic functions defined on the open unit disk \(D\) and let \(M_\varphi (f)=\varphi f\) denote the multiplication operator on \(B\), where \(\varphi\) is a complex valued function on \(D\). The author characterizes the commutant of \(M_{z^n}, n>1\). He shows that if the operator \(S\) commutes with the operator \(M_{z^2}\) and \(SM_{z^{2k+1}}-M_{z^{2k+1}}S\) is compact for some \(k\in N\) then there exists some \(\varphi \in \mathcal{M}(B)\) such that \(S=M_\varphi\). The analogous assertion \(S=M_\varphi\) under some additional assumptions also holds if \(S\) commutes with the operator \(M_{z^n}, n>1\) on a functional Hilbert spaces of analytic functions and \(SM_{z}-M_{z}S\) is a compact operator.
reviewed by: N. Karapetiants
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 1884296
zbMATH Keywords: commutant
zbMATH Keywords: multiplication operator
zbMATH Keywords: Banach space of analytic functions
zbMATH Keywords: functional Hilbert spaces
zbMATH Keywords: bounded point evaluation
author: B. Khani Robati
MaRDI profile type: Publication
cites work: Commuting Toeplitz operators with harmonic symbols
cites work: Commutants of Toeplitz operators on the Bergman space
cites work: Invariant Subspaces in Banach Spaces of Analytic Functions
cites work: Commutants of certain multiplication operators on Hilbert spaces of analytic functions
cites work: Q5607489
cites work: Intersections of Commutants of Analytic Toeplitz Operators
cites work: The Commutants of Certain Analytic Toeplitz Operators
full work available at URL: https://doi.org/10.1007/bf02904268
OpenAlex ID: W1999654580
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1884296