Lipschitz-type bounds for the map \(A \to|A|\) on \({\mathcal L}({\mathcal H})\)
type: Item
instance of: scholarly article
title: Lipschitz-type bounds for the map \(A \to|A|\) on \({\mathcal L}({\mathcal H})\)
zbMATH Open document ID: 1027.47007
DOI: 10.1016/S0024-3795(02)00453-6
author: Laura Smithies
author: Alfred S. jun. Cavaretta
published in: Linear Algebra and its Applications
publication date: 2003-03-11T00:00:00Z
review text: Let \(L(H)\) be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space \(H\). For a positive real number \(R\), let \(B_R= \{A\in L(H):\|A\|> R\}\), where \(\|\cdot\|\) denotes the usual operator norm. Denote by \(B^+_R\) the subset of \(B_R\) consisting of all nonnegative operators. In this paper, the authors show the following: (1) If \(0< R< {1\over 16}\), then the square root map \(A\to A^{1/2}\) is Lipschitz continuous on the set \(B^+_R\), that is \(\|A^{1/2}- B^{1/2}\|\leq {\sqrt{8}\over R} \|A-B\|\) for all \(A\), \(B\in B^+_R\). (2) If \(0< R< {1\over 4}\), then the absolute value map \(A\to|A|\) is Lipschitz continuous on the set \(B_R\), that is \(\||A|-|B|\|\leq {4\sqrt{2}\over R} \|A- B\|\) for all \(A\), \(B\in B_R\).
reviewed by: Omar Hirzallah
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 1880038
zbMATH Keywords: algebra of bounded linear operators
zbMATH Keywords: Hilbert space
zbMATH Keywords: absolute value map
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880038