Maps preserving certain orthogonality of operators on \(\mathcal {B(H)}\)

Abstract

Let \({\mathcal {H}}\) be a complex Hilbert space of dimension at least 3 and \(\mathcal {B(H)}\) the algebra of all bounded linear operators on \({\mathcal {H}}\). For any \(A, B\in {\mathcal {B}}({\mathcal {H}})\), A and B are said to be orthogonal if \(A^*B=0\). In this paper, we establish the general form of orthogonality preserving bijections on \(\mathcal {B(H)}\). Furthermore, we obtain a characterization of bijections \(\varphi :\mathcal {B(H)}\rightarrow \mathcal {B(H)}\) satisfying \(\varphi (A)\bot (\varphi (B)-\varphi (C))\) if and only if \(A\bot (B-C)\) for any \(A,B,C\in \mathcal {B(H)}\).