Multiplicative maps on \({\mathcal B}(X)\)
type: Item
instance of: scholarly article
title: Multiplicative maps on \({\mathcal B}(X)\)
zbMATH Open document ID: 0948.47038
DOI: 10.1007/BF02879024
published in: Science in China. Series A
publication date: 2000-11-27T00:00:00Z
review text: Let \(X\) and \(Y\) be two Banach spaces over the same field \(\mathbb{K}\) (where \(\mathbb{K}= \mathbb{R}\) or \(\mathbb{C}\)). A mapping \(\Phi:{\mathcal B}(X)\to{\mathcal B}(Y)\) is said to be multiplicative provided \(\Phi(TS)= \Phi(T)\Phi(S)\) for all \(T,S\in{\mathcal B}(X)\). The paper under review contains a fine analysis of the multiplicative mappings \(\Phi\) which send some rank-1 operator to an operator of rank not greater than 1 and satisfy the condition \(\Phi(\alpha I)= \alpha\Phi(I)\) for every \(\alpha\in \mathbb{K}\). An immediate consequence is the following characterization of the automorphisms of \({\mathcal B}(X)\): Every multiplicative bijective mapping \(\Phi\) of \({\mathcal B}(X)\) into itself, with the property \(\Phi(\alpha I)= \alpha\Phi(I)\) for every \(\alpha\in\mathbb{K}\), is of the form \(\Phi(T)= ATA^{-1}\) for a suitable \(A\in{\mathcal B}(X)\). The paper includes also several nice characterizations of the spectrum preserving mappings, numerical range preserving mappings as well as of the positivity preserving mappings among the multiplicative mappings. For example, Theorem 5 states (assuming \(\mathbb{K}=\mathbb{C}\)) the equivalence of the following conditions for \(\Phi:{\mathcal B}(X)\to{\mathcal B}(Y)\) an onto multiplicative mapping: i) \(\Phi\) is spectrum preserving. ii) \(\Phi\) is point spectrum preserving. iii) \(\Phi\) preserves the spectrum of every rank-1 operator. iv) \(\Phi\) is spectral radius preserving and \(\Phi(\alpha I)= \alpha\Phi(I)\) for every \(\alpha\in \mathbb{C}\). v) There exists an invertible operator \(A\in{\mathcal B}(X,Y)\) such that \(\Phi(T)= ATA^{-1}\) for every \(T\in{\mathcal B}(X)\). This result answers in the affirmative a problem raised by \textit{S. H. Hochwald} [Lin. Alg. Appl. 212-213, 339-351 (1994; Zbl 0815.15009)].
Mathematics Subject Classification ID: 47B49
zbMATH DE Number: 1329982
zbMATH Keywords: multiplicative mappings
zbMATH Keywords: rank-1 operator
zbMATH Keywords: automorphisms
zbMATH Keywords: spectrum preserving mappings
zbMATH Keywords: numerical range preserving mappings
zbMATH Keywords: positivity preserving mappings
author: Jinchuan Hou
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1329982