Linear maps preserving $A$-unitary operators

• Main Authors: Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi
• Format: Article
• Language: English
• Published: Institute of Mathematics of the Czech Academy of Science 2016-04-01
• Series: Mathematica Bohemica
• Subjects: linear preserver problem; semi-inner product
• Online Access: http://mb.math.cas.cz/full/141/1/mb141_1_4.pdf
• ISSN: 0862-7959; 2464-7136

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 $P\phi(T)=P$.