Multiplicative maps on \({\mathcal B}(X)\)
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)].
scientific article; zbMATH DE number 1329982