MAPS PRESERVING THE TRUNCATION OF OPERATORS ON POSITIVE CONES

Abstract

Let $\mathcal{\mathscr{H}}$ be a complex Hilbert space with $\text{dim }\mathcal{\mathscr{H}}\ge 2$ and $\mathcal{ℬ}{(\mathcal{\mathscr{H}})}^{+}$ the positive cone of the algebra of all bounded linear operators on $\mathcal{\mathscr{H}}$. For $A,B\in \mathcal{ℬ}{(\mathcal{\mathscr{H}})}^{+}$, $A$ is called a positive truncation of $B$ if $A=P_{A}B{P}_A$, where $P_A$ denotes the orthogonal projection onto the closure of $R(A)$. We determine structures of all bijections preserving the positive truncations of operators in both directions on the positive cone $\mathcal{ℬ}{(\mathcal{\mathscr{H}})}^{+}$.