Composition operators belonging to operator ideals
type: Item
instance of: scholarly article
title: Composition operators belonging to operator ideals
zbMATH Open document ID: 0939.47026
author: Thomas Domenig
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-12-05T00:00:00Z
review text: Let \(D\) be the unit disc \(\{z\in\mathbb{C}:|z|< 1\}\) of the complex plane \(\mathbb{C}\), and let \(\mu\) be the normalized Lebesgue measure on \(D\). If \(1\leq p<\infty\), then the Bergman space \(A^p\) is the subspace of \(L^p(\mu)\) consisting of holomorphic functions on \(D\). The sequence space consisting of elements of the form \(a= (a_0,a_1,\dots, a_j,\dots)\) such that \(\|a\|^p_p= \sum^\infty_{j=0}|a_j|^p<\infty\) is denoted by \(\ell^p\). If \(\phi: D\to D\) is analytic (holomorphic), then the composition operator \(C_\phi\) is defined by \(C_\phi(f)(z)= f\circ\phi(z)= f(\phi(z))\), and the image measure \(\mu_\phi\) is defined by \(\mu_\phi(A)= \mu(\phi^{-1}(A))\). If \((D_k)\) is a partition of \(D\) consisting of polar rectangles defined in terms of a diadic lattice and \(v_k= \mu(Q_k)^{-1/p} \mu_\phi(Q_k)^{1/q}\), \(k= 0,1,2,\dots\), then the diagonal operator \(\underline D_v\) is defined by \(\underline D_v(a_k)= v_ka_k\), \(k= 0,1,2,\dots\)\ . In the main results of this paper, a quasi-Banach ideal is introduced as a method which assigns to each pair of Banach spaces \((X,Y)\) a bounded linear operator \({\mathcal A}(X,Y)\) from \(X\) into \(Y\), and it is shown in the main theorem that the composition operator \(C_\phi\) is in \({\mathcal A}(A^p,A^q)\), where \(1\leq p,q<\infty\), if and only if \(\underline D_v\) is in \({\mathcal A}(\ell^p, \ell^q)\). Some previously derived special cases of the general statements include results by \textit{W. W. Hastings} [Proc. Am. Math. Soc. 52, 237-241 (1975; Zbl 0304.31007)] and \textit{B. D. MacCluer} and \textit{J. H. Shapiro} [Can. J. Math. 38, 878-906 (1986; Zbl 0608.30050)], providing conditions for boundedness and compactness of \(C_\phi: A^p\to A^q\) in terms of finiteness or limits of sequences of the types defining \(v_k\), \(k= 0,1,2,\dots\)\ .
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 1348152
zbMATH Keywords: composition operator
zbMATH Keywords: diadic lattice
zbMATH Keywords: quasi-Banach ideal
zbMATH Keywords: boundedness
zbMATH Keywords: compactness
reviewed by: George O. Okikiolu
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6483
OpenAlex ID: W2025946409
cites work: Inégalités à poids pour le projecteur de Bergman dans la boule unité de $C^{n}$
cites work: Q4023934
cites work: Q4850055
cites work: Q4373535
cites work: Q4226431
cites work: Diagonal mappings between sequence spaces
cites work: A Carleson Measure Theorem for Bergman Spaces
cites work: Composition Operators which Improve Integrability
cites work: Q4015708
cites work: Q4283339
cites work: SOME FUNCTIONAL ANALYTIC PROPERTIES OF COMPOSITION OPERATORS
cites work: Q4249044
cites work: Factorization of composition operators through Bloch type spaces
cites work: Forward and Reverse Carleson Inequalities for Functions in Bergman Spaces and Their Derivatives
cites work: Q3324231
cites work: Multipliers of Bergman spaces into Lebesgue spaces
cites work: Trace ideal criteria for Toeplitz operators
cites work: Angular Derivatives and Compact Composition Operators on the Hardy and Bergman Spaces
cites work: Q4183861
cites work: Q4722695
cites work: Q4000499
cites work: Q3951096
cites work: Interpolation by functions in Bergman spaces
cites work: The essential norm of a composition operator
cites work: Q4402890
cites work: Q3999007
DOI: 10.1006/JMAA.1999.6483
Recommended article: Q4226431
Recommended article: Composition operators on weighted Bergman spaces of polydisk
Recommended article: Duality and multiplication operators
Recommended article: COMPOSITION OPERATORS BELONGING TO SCHATTEN CLASS 𝒮p
Recommended article: Erratum to: ``Closed-range composition operators on weighted Bergman spaces''
Recommended article: Q4578241
Recommended article: Composition operators induced by symbols defined on a polydisk
Recommended article: Weighted composition operators between different weighted Bergman spaces in polydiscs
Recommended article: Q4589721
Recommended article: Schatten class composition operators on weighted Bergman spaces of bounded symmetric domains
description: scientific article; zbMATH DE number 1348152