Linear maps preserving $A$-unitary operators

Abstract

Let $\mathcal{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $\mathscr{B}(\mathcal{H})$ and $\mathscr{B}_A(\mathcal{H})$ the sub-algebra of $\mathscr{B}(\mathcal{H})$ of all $A$-self-adjoint operators. Assume $\phi \mathscr{B}_A(\mathcal{H})$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi(I)=P$ then $\psi$ defined by $\psi(T)=P\phi(PT)$ is a homomorphism or an anti-homomorphism and $\psi(T^{\sharp})=\psi(T)^{\sharp}$ for all $T \in\mathscr{B}_A(\mathcal{H})$, where $P=A^+A$ and $A^+$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there exists an operator $T$ satisfying $P\phi(T)=P$.

Keywords

• linear preserver problem
• semi-inner product