A joint similarity problem on vector-valued Bergman spaces
type: Item
instance of: scholarly article
title: A joint similarity problem on vector-valued Bergman spaces
zbMATH Open document ID: 1168.47017
author: Olivia Constantin
author: Frédéric Jaëck
published in: Journal of Functional Analysis
publication date: 2009-06-04T00:00:00Z
review text: Given a separable Hilbert space \({\mathcal H}\), let \({\mathcal B}({\mathcal H})\) denote the algebra of all bounded linear operators on \({\mathcal H}\). Let \({\mathbb D}\) be the open unit disk. A commuting pair of operators \((T_1,T_2)\) with \(T_j\in{\mathcal B}({\mathcal H})\) is said to be jointly polynomially bounded if there exists a positive constant \(k\) such that \(\|p(T_1,T_2)\|\leq k\sup_{z\in{\mathbb D}^2}\|p(z)\|\) for any analytic polynomial of two variables \(p\), and it is said to be jointly completely polynomially bounded if there exists a constant positive \(k\) such that \[ \Bigg|\sum_{1\leq i,j\leq n} \langle p_{ij}(T_1,T_2)x_i,y_j\rangle\Bigg| \leq k\sup_{z\in{\mathbb D}^2} \|(p_{ij}(z_1,z_2))\| \Bigg(\sum_{i=1}^n\|x_i\|^2\Bigg)^{1/2} \Bigg(\sum_{j=1}^n\|y_j\|^2\Bigg)^{1/2}, \] whenever \((p_{ij})_{1\leq i,j\leq n}\) is an \(n\times n\) matrix of analytic polynomials of two variables, \(n=1,2,\dots\), and \(\{x_i\}_{i=1}^n\), \(\{y_j\}_{j=1}^n\) are vectors in \({\mathcal H}\). Analogously, one can define the notions of a polynomially bounded operator and a completely polynomially bounded operator by substituting the pair of operators by one operator and the polynomials of two variables by polynomials of one variable in the above definitions. Consider the pair of operators \(R_T=\left(\begin{smallmatrix} M_z^* & \Gamma_T \\ 0 & M_z\end{smallmatrix}\right)\) and \(R_0=\left(\begin{smallmatrix} M_z^* & 0 \\ 0 & M_z\end{smallmatrix}\right)\) acting on the direct sum of weighted vector-valued Bergman spaces \(L_a^{2,\alpha}({\mathcal H})\oplus L_a^{2,\alpha}({\mathcal H})\), where \(M_z\) is the operator of multiplication by \(z\) on \(L_a^{2,\alpha}({\mathcal H})\) and \(\Gamma_T\) is a little Hankel operator. The main result of the paper is the following Theorem. Let \(T:{\mathbb D}\to{\mathcal B}({\mathcal H})\) be a holomorphic operator-valued function with \(\sup_{z\in{\mathbb D}}(1-|z|^2)\|T'(z)\|<\infty\). Then the following statements …
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 5561654
zbMATH Keywords: little Hankel operator
zbMATH Keywords: vector-valued Bergman space
zbMATH Keywords: joint polynomial boundedness
zbMATH Keywords: joint complete polynomial boundedness
zbMATH Keywords: similarity to a contraction
zbMATH Keywords: joint similarity to a pair of contractions
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.11.022
OpenAlex ID: W1990905326
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DOI: 10.1016/J.JFA.2008.11.022
description: scientific article; zbMATH DE number 5561654