Linear maps preserving elements annihilated by the polynomial$XY-YX^{\dagger}$

Jianlian Cui¹ ; Jinchuan Hou²

¹ Department of Mathematical Science University of Tsinghua Beijing, 100084, P.R. China
² Department of Applied Mathematics Taiyuan University of Technology Taiyuan 030024, P.R. China and Department of Mathematics Shanxi Teachers University Linfen, 041004, P.R. China

Studia Mathematica, Tome 174 (2006) no. 2, pp. 183-199

Résumé

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 $A\in {\mathcal B}(H)$.

Détail

DOI : 10.4064/sm174-2-5

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