Compact differences of composition operators on Bloch and Lipschitz spaces
type: Item
instance of: scholarly article
title: Compact differences of composition operators on Bloch and Lipschitz spaces
zbMATH Open document ID: 1146.47016
DOI: 10.1007/BF03321648
author: Pekka J. Nieminen
published in: Computational Methods and Function Theory
publication date: 2008-05-30T00:00:00Z
review text: For \(0<\alpha \leq 1\), let \({\mathcal B}^\alpha\) denote the space of analytic functions on the unit disc \({\mathbb D}\) such that \(\sup_{| z| <1}(1-| z| ^2)^\alpha | f'(z)| <\infty\), that is, the Bloch space for \(\alpha=1\) and the analytic Lipschitz classes for \(0<\alpha<1\). Using the notation \({\mathcal D}^\alpha\phi(z)=(\frac{1-| z| ^2}{1-| \phi(z)| ^2})^\alpha \phi'(z)\), \({\mathcal D}^1= {\mathcal D}\) and \({\mathcal B}^1= {\mathcal B}\), the author shows that, in the case when \(\alpha=1\), the difference of composition operators \(C_\phi -C_\psi\) is (weakly) compact on \(\mathcal B\) if and only if \(\lim_{| \phi(z)| \to 1}{\mathcal D}\phi(z)\rho(\phi(z),\psi(z))=0\) and \(\lim_{| \psi(z)| \to 1}{\mathcal D}\psi(z)\rho(\phi(z),\psi(z))=0\), where \(\rho(w_1,w_2)\) stands for the pseudo-hyperbolic metric, improving the result by \textit{T.\,Hosokawa} and \textit{S.\,Ohno} [J.~Oper.\ Theory 57, No.\,2, 229--242 (2007; Zbl 1174.47019)] where an extra condition was assumed. In the case when \(0<\alpha<1\) and assuming that \(\| \mathcal D\phi\| _\infty<\infty\) and \(\| \mathcal D\psi\| _\infty<\infty\), the (weak) compactness of \(C_\phi- C_\psi\) in \({\mathcal B}^\alpha\) is equivalent to \(\lim_{| \phi(z)| \to 1}{\mathcal D}^\alpha \phi(z)\rho(\phi(z),\psi(z))=0\), \(\lim_{| \psi(z)| \to 1}{\mathcal D}^\alpha \psi(z)\rho(\phi(z),\psi(z))=0\) and \(\lim_{\min\{| \phi(z)| , | \psi(z)| \} \to 1}{\mathcal D}^\alpha \phi(z)-{\mathcal D}^\alpha \psi(z)=0\). It is shown that the extra condition is necessary in the case \(0<\alpha<1\). The author uses a unified approach based on the difference of weighted composition operators acting between two weighted \(H^\infty\)-spaces. Some complementary remarks on the compactness of a single composition operator on Lipschitz spaces are also provided.
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 47B07
zbMATH DE Number: 5281672
zbMATH Keywords: composition operators
zbMATH Keywords: compactness
zbMATH Keywords: Bloch space
MaRDI profile type: Publication
cites work: Composition Operators Isolated in the Uniform Operator Topology
cites work: DIFFERENCES OF COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF HOLOMORPHIC FUNCTIONS
cites work: Q4515420
cites work: Q4888816
cites work: Q3688898
cites work: Q5618011
cites work: Q3922907
cites work: Hyperbolic derivatives and generalized Schwarz-Pick estimates
cites work: Q4462315
cites work: Q4490700
cites work: Essential norms of differences of composition operators on \(H^{\infty}\)
cites work: Topological structure of the space of weighted composition operators on \(H^\infty\)
cites work: Isolated points and essential components of composition operators on \(H^\infty\)
cites work: Q5308901
cites work: Topological structures of the sets of composition operators on the Bloch spaces
cites work: Linear relations in the Calkin algebra for composition operators
cites work: Components in the space of composition operators
cites work: Topological structure of the space of composition operators on \(H^\infty\).
cites work: Generalized Schwarz-Pick estimates
cites work: Composition Operators on Analytic Lipschitz Spaces
cites work: Compact Composition Operators on the Bloch Space
cites work: Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions
cites work: Compact differences of composition operators
cites work: On compactness of the difference of composition operators
cites work: Composition operators on a space of Lipschitz functions
cites work: Q5484160
cites work: Q4529295
cites work: Compact Composition Operators on Spaces of Boundary-Regular Holomorphic Functions
cites work: Isolation amongst the composition operators
cites work: Topological components of the set of composition operators on \(H^\infty(B_N)\)
cites work: Composition operators associated with bloch-type spaces
cites work: Bloch type spaces of analytic functions
full work available at URL: https://doi.org/10.1007/bf03321648
OpenAlex ID: W2089616253
Wikidata QID: Q125366386
Recommended article: On compactness of the difference of composition operators
Recommended article: Q5196159
Recommended article: Compact composition operators on Bloch type spaces
Recommended article: COMPACT DIFFERENCES OF COMPOSITION OPERATORS
Recommended article: Compact differences of composition operators
Recommended article: Compact composition operators on generalized Lipschitz spaces
Recommended article: Differences of composition operators on Bloch type spaces
Recommended article: Differences of Composition Operators on Bloch Type Spaces
Recommended article: Q5319488
Recommended article: Compact composition operators on certain analytic Lipschitz spaces
description: scientific article; zbMATH DE number 5281672