Linear maps preserving elements annihilated by the polynomial$XY-YX^{\dagger}$
Volume 174 / 2006
Studia Mathematica 174 (2006), 183-199
MSC: Primary 47B48, 47B50.
DOI: 10.4064/sm174-2-5
Abstract
Let $H$ and $K$ be complex complete indefinite inner product spaces, and ${\mathcal B}(H,K)$ (${\mathcal B}(H)$ if $K=H$) the set of all bounded linear operators from $H$ into $K$. For every $T\in {\mathcal B}(H,K)$, denote by $T^\dagger$ the indefinite conjugate of $T$. Suppose that ${\mit\Phi} :{\mathcal B}(H)\rightarrow {\mathcal B}(K)$ is a bijective linear map. We prove that ${\mit\Phi} $ satisfies ${\mit\Phi} (A){\mit\Phi} (B)={\mit\Phi} (B){\mit\Phi} (A)^\dagger$ for all $A, B\in {\mathcal B}(H)$ with $AB=BA^\dagger $ if and only if there exist a nonzero real number $c$ and a generalized indefinite unitary operator $U\in {\mathcal B}(H, K)$ such that ${\mit\Phi} (A)=cUAU^{\dagger}$ for all $A\in {\mathcal B}(H)$.
