Maps on real Hilbert spaces preserving the area of parallelograms and a preserver problem on self-adjoint operators
type: Item
instance of: scholarly article
title: Maps on real Hilbert spaces preserving the area of parallelograms and a preserver problem on self-adjoint operators
zbMATH Open document ID: 1302.47056
author: György Pál Gehér
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1407.1868
reviewed by: Abdellatif Bourhim
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 39B42
zbMATH DE Number: 6362883
zbMATH Keywords: preserver problems
zbMATH Keywords: area of parallelogram
zbMATH Keywords: commutativity of self-adjoint operators
zbMATH Keywords: unitarily invariant norm
MaRDI profile type: Publication
OpenAlex ID: W2008689415
arXiv ID: 1407.1868
cites work: Q4306891
cites work: Q5427871
cites work: Maps preserving matrix pairs with zero Jordan product
cites work: Q4850048
cites work: An elementary proof for the non-bijective version of Wigner's theorem
cites work: Maps on classes of Hilbert space operators preserving measure of commutativity
cites work: A new proof of Wigner's theorem
cites work: Martin's theorem for Euclidean n-space and a generalization to the perimeter case
cites work: Wigner's theorem revisited
cites work: Q4893588
cites work: Q4244353
cites work: Nonlinear commutativity preserving maps on self-adjoint operators
cites work: Transformations on bounded observables preserving measure of compatibility
cites work: Commutativity preserving maps on quantum states
cites work: On Wigner's theorem: Remarks, complements, comments, and corollaries
cites work: Volume preserving mappings in the spirit of the Mazur-Ulam theorem
cites work: Nonlinear commutativity-preserving maps on Hermitian matrices
cites work: Orthogonality preserving transformations on the set of \(n\)-dimensional subspaces of a Hilbert space
cites work: Q3844299
DOI: 10.1016/J.JMAA.2014.09.071
review text: Let \(E\) be a real Hilbert space and NEWLINE\[NEWLINE\blacklozenge(a,b):=\sqrt{|a|^2.|b|^2-\langle a, b\rangle^2}NEWLINE\]NEWLINE the area of the parallelogram spanned by two vectors \(a,~b\in E\). Here, \(\langle~.~,~.~\rangle\) denotes the scalar product of \(E\) and \(|.|\) its norm. In the paper under review, the author answers a question raised by \textit{T. M. Rassias} and \textit{P. Wagner} [Aequationes Math. 66, No. 1--2, 85--89 (2003; Zbl 1085.39022)] by describing all maps \(\phi\) from \(E\) into itself preserving the area of parallelograms spanned by two vectors in \(E\); i.e., NEWLINE\[NEWLINE\blacklozenge(\phi(a),\phi(b))=\blacklozenge(a,b)NEWLINE\]NEWLINE for all \(a,~b\in E\). He also characterizes all maps \(\phi\) on \(B_s({\mathcal H})\) which preserve the norm of the commutators, and solves a preserver problem of \textit{L. Molnár} and \textit{W. Timmermann} [Int. J. Theor. Phys. 50, No. 12, 3857--3863 (2011; Zbl 1243.81079)]. Here, \(B_s({\mathcal H})\) is the real vector-space of all linear bounded self-adjoint operators on a complex separable Hilbert space \({\mathcal H}\) and \(\|.\|\) is a unitarily invariant norm on \(B_s({\mathcal H})\).
description: scientific article; zbMATH DE number 6362883