Characterizing isomorphisms in terms of completely preserving invertibility or spectrum

Let A and B be standard operator algebras on complex Banach spaces X and Y, respectively. In this paper, we characterize the surjective maps completely preserving the invertibility in both directions and the surjective maps completely preserving the spectrum from A to B . We show that a surjective map from A to B is a ring isomorphism if and only if it is unital and completely preserves the invertibility of operators in both directions; is an isomorphism if and only if it completely preserves the spectrum of operators.