Spectrum-preserving linear maps (Q1074078)

Let X and Y be Banach spaces. A linear map \(\phi\) : \({\mathcal B}(X)\to {\mathcal B}(Y)\) is called spectrum-preserving if for every operator \(T\in {\mathcal B}(X)\) we have \(\sigma (\phi (T))=\sigma (T)\), where \(\sigma\) denotes the spectrum. We show that a spectrum-preserving surjective linear map \(\phi\) is either of the form \(\phi (T)=ATA^{-1}\) for an isomorphism A of X onto Y or of the form \(\phi (T)=BT^*B^{-1}\) for an isomorphism B of \(X^*\) onto Y. This is an extension of a related finite-dimensional result of \textit{M. Marcus} and \textit{B. N. Moyls} [Can. J. Math. 11, 61-66 (1959; Zbl 0086.017)].