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

Abstract

Let \(\mathcal {H}\) be a complex Hilbert space with \(\dim {\mathcal {H}}\ge 2\) and \(\mathcal {B}(\mathcal {H})\) be the algebra of all bounded linear operators on \(\mathcal {H}\). For \(A, B \in \mathcal {B}(\mathcal {H})\), B is called a truncation of A, denoted by \(B\prec A\), if \(B=PAQ\) for some projections \(P,Q\in {\mathcal {B}}({\mathcal {H}})\). And B is called a maximal truncation of A if \(B\not =A\) and there is no other truncation C of A such that \(B\prec C\). 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 \(\mathcal {B}(\mathcal {H})\).