Mappings preserving generalized and hyper-generalized projection operators

Abstract

Let $B\left(H\right)$ be the algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\ge 3$. For a fixed integer $k\ge 2$, an operator $A\in B\left(H\right)$ is called k-generalized projection if $A^k=A^{⁎}$, and k-hyper-generalized projection if $A^k=A^{†}$, where $A^{⁎}$ and $A^{†}$ denote the adjoint and the Moore–Penrose inverse of A, respectively. In this paper, we provide a complete characterization of surjective maps $\Phi :B\left(H\right)\to B\left(H\right)$ such that $A-\lambda B$ is k-generalized projection (resp. k-hyper-generalized projection) if and only if $\Phi \left(A\right)-\lambda \Phi \left(B\right)$ is k-generalized projection (resp. k-hyper-generalized projection), for any $A,B\in B\left(H\right)$ and $\lambda \in C$. We also study the non-linear preservers of k-potent operators.