Invertibility preserving linear maps and algebraic reflexivity of elementary operators of length one (Q2759026)

Let \(X\) and \(Y\) be real or complex Banach spaces and let \({\mathcal B}(X)\) (\({\mathcal B}(Y)\)) be the algebra of all bounded linear operators on \(X\) (\(Y\)). The author shows that a surjective linear map \(\Phi :{\mathcal B}(X)\rightarrow {\mathcal B}(Y)\) preserves the invertibility of operators in both directions if and only if there exist bounded invertible linear operators \(A:X\rightarrow Y\), \(B:Y\rightarrow X\), \(C:X^\prime \rightarrow Y\) and \(D:Y\rightarrow X^\prime \) such that \(\Phi (T)=ATB\) for all \(T\in {\mathcal B}(X)\) or \(\Phi (T)=CT^\prime D\) for all \(T\in {\mathcal B}(X)\). As an application, the author improves a result of Larson and Sourour on algebraic reflexivity of elementary operators of length one.