Communications of the
Korean Mathematical Society CKMS

Let ${\mathcal B}(H)$ be the algebra of all bounded linear operators on a Hilbert space $H$ with $\operatorname{dim} (H)>2$. Let ${\mathcal{G} \mathcal{P}}$ be the subset of ${\mathcal B}(H)$ of all generalized projection operators. In this paper, we give a complete characterization of surjective maps $\Phi: {\mathcal B}(H) \rightarrow {\mathcal B}(H)$ satisfying $A-\lambda B \in {\mathcal{G} \mathcal{P}} \Leftrightarrow \Phi(A)-\lambda \Phi(B) \in {\mathcal{G} \mathcal{P}}$ for any $A, B \in {\mathcal B}(H)$ and $\lambda \in \mathbb{C}$.

Keywords: Hilbert space, generalized projection, preserver

MSC numbers: 47B49, 47B48