Nonlinear preservers of pseudospectral radius on Banach spaces
Let \({\mathcal L}(X)\) be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space \(X\). For \(\epsilon>0\), the \(\epsilon\)--pseudospectrum of an operator \(T\in{\mathcal L}(X)\) is \(\sigma_\epsilon(T)=\left\{\lambda\in\mathbb C:\|(\lambda-T)^{-1}\|\geq\frac{1}{\epsilon}\right\},\) with the convention that \(\|(\lambda-T)^{-1}\|=\infty\) if \(\lambda-T\) is not invertible. This set is compact and the \(\epsilon\)-pseudospectral radius of \(T\) is \(r_\epsilon(T):=\max\big\{|\lambda|:\lambda\in\sigma_\epsilon(T)\big\}.\) The author proves that a surjective map \(\varphi\) on \({\mathcal L}(X)\) satisfies \[ r_\epsilon(\varphi(T)\varphi(S))=r_\epsilon(TS),\ T,S\in{\mathcal L}(X), \] if and only if there exist a complex-valued functional \(\xi\) on \({\mathcal L}(X)\) and a bijective linear or conjugate linear isometry \(U\) on \(X\) such that \(|\xi(T)|=1\) and \(\varphi(T)=\xi(T)UTU^{-1}\) for all \(T\in{\mathcal L}(X)\). When \(X=\mathbb{C}^n\) is a finite-dimensional space with any norm and \(n\geq3\), he obtains a similar result but without the surjectivity assumption on \(\varphi\). The proof of these results uses several auxiliary lemmas, some of them are new. Among the new ones, the author proves that an operator \(T\in{\mathcal L}(X)\) is zero if and only if \(r_\epsilon(T)=\epsilon\).
scientific article; zbMATH DE number 7138410
scientific article