A new characterization of Bergman-Schatten spaces and a duality result
type: Item
instance of: scholarly article
title: A new characterization of Bergman-Schatten spaces and a duality result
zbMATH Open document ID: 1192.46016
author: Lars-Erik Persson
author: I. Popa
author: L. G. Marcoci
author: Nicolae Popa
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-09-10T00:00:00Z
review text: The authors consider some problems on duality of spaces of analytic matrices of the Bergman and Bloch type and study some characterizations by means of Taylor coefficients. If \(A=(a_{jk})\) and \(B=(b_{jk})\) are matrices of the same size (finite or infinite) the Schur (or Hadamard) product is defined by \(A*B=(a_{jk}b_{jk})\) and the space of Schur multipliers between spaces of matrices \(X\) and \(Y\), \(M(X;Y)\), is defined to be the set of matrices \(M\) such that \(M*A\in Y\) for any \(A\in X\). The authors consider functions \(r\to A(r)\) which are matrix-valued and use the correspondence with functions defined on the unit disc \(f_A(re^{it})=\sum_{k\in \mathbb Z} A_k(r)e^{ikt}\) where \(A_k(r)\) stands for the \(kth\)-diagonal matrix. In the particular case where \(A_k(r)=A_k r^k\), \(k\in \mathbb Z\), for a given upper triangular matrix \(A\) the matrix is called analytic matrix. The spaces of matrices considered in the paper are the Bergman-Schatten classes for \(1\leq p<\infty\), to be denoted \(L^p(D,\ell^2)\), given by \(r\to A(r)\) such that \(A(r)\) belongs to the the Schatten class \(C_p\) and \(\int_0^1\|A(r)\|^p_{C_p} dr<\infty\) (which with the classical notation of Bochner integrable functions corresponds to \(L^p([0,1), C_p)\)) and its subspace of analytic matrices \(\tilde L_a^p(D,\ell^2)\) where \(A(r)=A*C(r)\) for some upper triangular matrix \(A\) and \(C(r)\) the Toepliz matrix associated to the Cauchy kernel \(\frac{1}{1-r}\). The obvious modification for \(p=\infty\) where the authors denote separately the spaces \(L^\infty(D,\ell^2)\) and \(\tilde L^\infty(D,\ell^2)\) whenever the functions \(r\to A(r)\) are assumed to be either \(w^*\)-measurable or strong measurable with values in \(B(\ell^2)\). The Bloch space \({\mathcal B}(D,\ell^2)\) is defined to be the space of analytic matrices \(A(r)\) such that \(\sup_{0\leq r<1}(1-r^2) \|A'(r)\|_{B(\ell^2)} + \|A_0\|_{B(\ell^2)}<\infty\) where \(A'(r)=\sum_{k=0}^\infty A_kkr^{k-1}\). Also the…
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46B10
zbMATH DE Number: 5602698
zbMATH Keywords: Schur multipliers
zbMATH Keywords: Bergman-Schatten class
zbMATH Keywords: Bloch spaces
zbMATH Keywords: duality
reviewed by: Oscar Blasco
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.06.027
OpenAlex ID: W1967871145
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DOI: 10.1016/J.JMAA.2009.06.027
description: scientific article; zbMATH DE number 5602698