Linear maps preserving quasi-commutativity

Let X and Y be Banach spaces and B(X) and B(Y) the algebras of all bounded linear operators on X and Y, respectively. We say that A, B is an element of B(X) quasi-commute if there exists a nonzero scalar omega such that AB = omega BA. We characterize bijective linear maps phi : B(X) -> B(Y) preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.