Transformations preserving norms of means of positive operators and nonnegative functions
type: Item
instance of: scholarly article
title: Transformations preserving norms of means of positive operators and nonnegative functions
zbMATH Open document ID: 1354.47026
author: Lajos Molnár
author: Patrícia Szokol
published in: Integral Equations and Operator Theory
publication date: 2015-10-28T00:00:00Z
full work available at URL: IEOT.pdf
review text: Let \(H\) be a complex Hilbert space and \(B(H)\) the algebra of all bounded linear operators on \(H\). Denote by \(\mathcal A_+\) the cone of all positive operators in a subalgebra \(\mathcal{A}\) of \(B(H)\) which contains all finite rank operators, i.e., \(\mathcal A\) is a standard operator algebra on \(H\). Let \(1\leq p<\infty\) be a given number and denote by Tr the usual trace functional. The symbol \(C_p(H)\) stands for the set of all operators \(A\in B(H)\) for which \(\operatorname{Tr}| A|^p<\infty\). As for the case where \(p\) is infinite, \(C_\infty(H)\) denotes \(B(H)\) equipped with the usual operator norm \(\| \cdot\|\). In the paper under review, the authors first describe the structure of all bijective transformations between the positive cones of standard operator algebras on \(H\) which preserve a given symmetric norm of a given (Kubo-Ando) mean of their elements. They present their result in two theorems, one of them is as follows. Let \(\dim H>1\) and \(1<p\leq\infty\). Consider standard operator algebras \(\mathcal A\) and \(\mathcal B\) on \(H\) which are contained in \(C_p(H)\). Let \(\phi:\mathcal A_+\rightarrow\mathcal B_+\) be a bijective transformation such that, for a given \(0<\lambda<1\), \[ \|\lambda\phi(A)+(1-\lambda)\phi(B)\|_p=\|\lambda A+(1-\lambda)B\|_p \] holds for all \(A,\) \(B\in\mathcal A_+\). Then there exists a unitary or antiunitary operator \(U:H\rightarrow H\) such that \[ \phi(A)=UAU^\ast,\quad A\in\mathcal A_+. \] After presenting their result concerning means on operator algebras, the authors turn to the discussion of the problem concerning means on algebras of continuous functions. With the third and final theorem, which is of a similar spirit as the above result, the authors describe the structure of bijective transformations between cones of nonnegative elements of certain algebras of continuous functions.
reviewed by: Janko Marovt
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A64
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 6499542
zbMATH Keywords: preservers
zbMATH Keywords: operator means
zbMATH Keywords: means of functions
zbMATH Keywords: symmetric norms
MaRDI profile type: Publication
OpenAlex ID: W430315086
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DOI: 10.1007/S00020-015-2241-6
description: scientific article; zbMATH DE number 6499542