Linear bijective maps preserving spectral functions on pairs of similar operators (Q6063626)

Let \({\mathcal B}(X)\) be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space \(X\). For any \(T\in{\mathcal B}(X)\), let \(\sigma(T)\) be its spectrum and \(r(T)\) be its spectral radius. Recall that two operators \(A,~B\in{\mathcal B}(X)\) are said to be similar, denoted by \(A\sim B\), if there exists an invertible operator \(U\in{\mathcal B}(X)\) such that \(B=UAU^{-1}\). The author describes the form of all bijective linear maps \(\varphi\) on \({\mathcal B}(X)\) satisfying \[ A\sim B\Rightarrow r(\varphi(A))=r(\varphi(B)),~(A,~B\in{\mathcal B}(X)). \] As a consequence, he characterizes all bijective linear maps \(\varphi\) on \({\mathcal B}(X)\) satisfying \[ \sigma(A)=\sigma(B)\Rightarrow r(\varphi(A))=r(\varphi(B)),~(A,~B\in{\mathcal B}(X)). \]