Truncations of operators in \({\mathcal {B}}({\mathcal {H}})\) and their preservers

Let \(\mathcal {H}\) $\mathcal{H}$ be a complex Hilbert space with \(\dim {\mathcal {H}}\ge 2\) $dim\mathcal{H}\ge 2$ and \(\mathcal {B}(\mathcal {H})\) $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on \(\mathcal {H}\) $\mathcal{H}$ . For \(A, B \in \mathcal {B}(\mathcal {H})\) $A,B\in \mathcal{B}(\mathcal{H})$ , B is called a truncation of A, denoted by \(B\prec A\) $B\prec A$ , if \(B=PAQ\) $B=PAQ$ for some projections \(P,Q\in {\mathcal {B}}({\mathcal {H}})\) $P,Q\in \mathcal{B}(\mathcal{H})$ . And B is called a maximal truncation of A if \(B\not =A\) $B\ne A$ and there is no other truncation C of A such that \(B\prec C\) $B\prec C$ . We give necessary and sufficient conditions for B to be a maximal truncation of A. Using these characterizations, we determine structures of all bijections preserving truncations of operators in both directions on \(\mathcal {B}(\mathcal {H})\) $\mathcal{B}(\mathcal{H})$ .