Connected components in the space of composition operators on \(H^\infty\) functions of many variables
type: Item
instance of: scholarly article
title: Connected components in the space of composition operators on \(H^\infty\) functions of many variables
zbMATH Open document ID: 1029.46053
DOI: 10.1007/BF02789591
author: Pablo Galindo
author: Mikael Lindström
author: Richard M. Aron
published in: Integral Equations and Operator Theory
publication date: 2004-02-04T00:00:00Z
review text: Let \(H^\infty (B_E) \) be the space of all bounded analytic functions on the open unit ball \(B_E \) of a Banach space \(E . \) For an analytic mapping \( \phi :B_E \to B_F,\) the operator \( C_\phi : H^\infty (B_F)\to H^\infty (B_E) \) defined by \(C_\phi (f) =f\circ \phi \) is called the composition operator. For the case that \(E\) and \(F\) are the set of complex numbers, \textit{B. MacCluer, S. Ohno} and \textit{R. Zhao} have shown in [Integral Equ. Oper. Theory 40, 481-494 (2001; Zbl 1062.47511)] that (1) the identity mapping is an isolated point in the set \( C( H^\infty (B_F), H^\infty (B_E)) \) of all composition operators and (2) the path connected components in \( C( H^\infty (B_F), H^\infty (B_E))\) are open balls of radius 2 with respect to the operator norm. In the paper under review, the authors discuss the case of arbitrary Banach spaces \(E, F.\). In general, the set of all composition operators whose ranges lie strictly in \(B_E \) form a path connected component. For \(E=F=C_0 (X),\) the identity operator is an isolated point. It is conjectured that this should be true for general Banach spaces. If \(F\) is a Hilbert space or a \(C_0(X)\) space, two composition operators \( C_\phi , C_\psi \) are in the same path connected component if and only if \( ||C_\phi -C_\psi ||<2 \) with respect to the operator norm.
reviewed by: Hermann Render
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 1890553
zbMATH Keywords: composition operator
zbMATH Keywords: spaces of analytic functions
zbMATH Keywords: peak point
zbMATH Keywords: pseudohyperbolic distance
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1890553