On Bloch-to-Besov composition operators
type: Item
instance of: scholarly article
title: On Bloch-to-Besov composition operators
zbMATH Open document ID: 0873.30017
author: Shamil Makhmutov
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-09-29T00:00:00Z
review text: Let \(H(D)\) denote the space of analytic functions defined on the open unit disk \(D\). Every analytic self-map \(\phi: D \to D\) induces a linear operator \(C_{\phi}: H(D) \to H(D)\), defined by \(C_{\phi}f=f\circ \phi\), \(f \in H(D)\). Let \(\mathcal B\), \(B_p\) and \(B_p^h\), \(1<p<\infty\), denote the Bloch space, the Besov space and the hyperbolic Besov space, respectively. The two theorems proved in this paper are as follows. Theorem 1. Let \(\phi: D \to D\) be an analytic self-map and let \(1<p<\infty\). Then \(C_{\phi}\) is a Bloch-to-\(B_p\) composition operator if and only if \(\phi \in B_p^h\). Theorem 2. If \(\phi \in B_p^h\), \(1<p<\infty\), then \(\phi\) induces a compact composition operator \(C_{\phi}\) on \(\mathcal B\) into \(B_p\).
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 1022277
zbMATH Keywords: composition operator
zbMATH Keywords: Besov space
reviewed by: George Csordas
MaRDI profile type: Publication
full work available at URL: pjaa.72.232
OpenAlex ID: W2008876618
cites work: Q4853799
cites work: Bloch-to-BMOA pullbacks on the disk
cites work: Q5618011
cites work: Q5774138
cites work: Bounded mean oscillation of Bloch pull-backs
cites work: Smoothness of the Boundary Values of Functions Bounded and Holomorphic in the Disk
cites work: Composition operators from Bloch type spaces to Hardy and Besov spaces
cites work: Q3999007
DOI: 10.3792/PJAA.72.232
description: scientific article; zbMATH DE number 1022277