Operator balls
type: Item
instance of: scholarly article
title: Operator balls
zbMATH Open document ID: 0745.47034
DOI: 10.1007/BF01198922
author: S. Singh
published in: Integral Equations and Operator Theory
publication date: 1992-06-25T00:00:00Z
review text: Let \(H_ 1,H_ 2\) be Hilbert spaces and let \({\mathcal K}({\mathcal K}^ 0)\) denote the set of all bounded linear operators from \(H_ 1\) to \(H_ 2\) with norm \(\leq 1\) (resp. \(<1\)). For operators \(A,B,C\), the set \({\mathcal K}(C;A,B)=C+A{\mathcal K}B\) \(({\mathcal K}^ 0(C;A,B)=C+A{\mathcal K}^ 0B)\) is called the closed (resp. open) operator ball of the title. \(C\) is the `centre', \(A\), \(B\) are `radii'. Note that \({\mathcal K}(C;A,B)\) is the closure of \({\mathcal K}^ 0(C;A,B)\) in the weak, strong, and operator topologies but \({\mathcal K}^ 0(C;A,B)\) is not (relatively) open in any of the usual topologies. Many results are given including necessary and sufficient conditions for inclusions between operator balls to hold. For example, either an open ball \({\mathcal K}^ 0(C;A,B)\) is contained in \({\mathcal K}^ 0(C_ 1;A_ 1,B_ 1)\) or the entire closed ball \({\mathcal K}(C;A,B)\) lies on the boundary of \({\mathcal K}(C_ 1;A_ 1,B_ 1)\). Connections with the study of linear fractional transformations are promised for a later paper.
Mathematics Subject Classification ID: 47L07
zbMATH DE Number: 7069
zbMATH Keywords: operator ball
zbMATH Keywords: operator topologies
MaRDI profile type: Publication
cites work: Q3254057
cites work: Über monotone Matrixfunktionen
full work available at URL: https://doi.org/10.1007/bf01198922
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description: scientific article; zbMATH DE number 7069