Nonlinear maps preserving numerical radius of indefinite skew products of operators
Let \(H\) be a complex Hilbert space with \(\dim H \geq 3\) and let \(B(H)\) be the algebra of all bounded linear operators. For \(A \in B(H)\), let \(w(A)\) denote its numerical radius. Given an invertible selfadjoint operator \(J\), the indefinite conjugate of \(A \in B(H)\) with respect to \(J\) is \(A^\dagger= J^{-1}A^* J\). Let \({\mathcal V}, {\mathcal W}\) be subsets of \(B(H)\) containing all rank one operators. In the paper under review, the authors give a complete characterization of surjective maps \(\varphi: {\mathcal W} \to {\mathcal V}\) with the property that \( w (A^\dagger B)= w(\varphi (A)^\dagger \varphi ( B) )\) and \(w(A B^\dagger)= w(\varphi (A) \varphi (B)^\dagger)\) for all \(A, B \in {\mathcal W}\). They also describe maps \(\varphi: {\mathcal W} \to {\mathcal V} \) satisfying \( w( A B^\dagger A)= w( \varphi (A) \varphi (B)^\dagger \varphi (A))\) for all \(A, B \in {\mathcal W}\). Let \(\mathcal A\) be a standard algebra on \(H\). Recall that surjective maps \(\varphi: \mathcal A \to \mathcal A\) preserving the numerical radius of operator products (\(w(AB)= w( \varphi (A) \varphi (B)\) for all \(A, B \in \mathcal A\)) were studied by \textit{J.--L.\thinspace Cui} and \textit{J.--C.\thinspace Hou} [Linear Algebra Appl.\ 428, No.\,7, 1649--1663 (2008; Zbl 1146.47023)]. Now, let \(B_s (H)\) denote the subspace of all selfadjoint operators in \(B(H)\). It should be pointed out that surjective maps \(\varphi: B_s (H) \to B_s (H)\) satisfying \(w(AB)= w( \varphi (A) \varphi (B))\) (respectively, \(w(ABA)= w( \varphi (A) \varphi (B) \varphi (A))\)) for all \(A, B \in B_s (H)\) have been previously determined by the authors in a preprint.
scientific article; zbMATH DE number 5549278
scientific article