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| CHARACTERIZATIONS OF JORDAN $\dag$-SKEW MULTIPLICATIVE MAPS ON OPERATOR ALGEBRAS OF INDEFINITE INNER PRODUCT SPACES |
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Citation： |
AN Runling,HOU Jinchuan.CHARACTERIZATIONS OF JORDAN $\dag$-SKEW MULTIPLICATIVE MAPS ON OPERATOR ALGEBRAS OF INDEFINITE INNER PRODUCT SPACES[J].Chinese Annals of Mathematics B,2005,26(4):569~582 |
| Page view： 1377
Net amount： 1077 |
Authors： |
AN Runling; HOU Jinchuan |
Foundation： |
Project supported by the National Natural Science Foundation of China (No.10471082) and the Shanxi Provincial Natural Science Foundation of China (No.20021005). |
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| Abstract： |
Let $H$ and $K$ be indefinite inner product spaces. This paper shows that a bijective map $\Phi:{\mathcal B}(H)\rightarrow{\mathcal B}(K)$ satisfies $\Phi(AB^{\dag}+B^{\dag}A)=\Phi(A)\Phi(B)^\dag+\Phi(B)^\dag\Phi(A)$ for every pair $A,B\in{\mathcal B}(H)$ if and only if either $\Phi(A)=cUAU^\dag$ for all $A$ or $\Phi(A)=cUA^\dag U^\dag$ for all $A$; $\Phi$ satisfies $\Phi(AB^\dag A)=\Phi(A)\Phi(B)^\dag
\Phi(A)$ for every pair $A,B\in{\mathcal B}(H)$ if and only if either $\Phi(A)=UAV$ for all $A$ or $\Phi(A)=UA^\dag V$ for all $A$, where $A^\dag$ denotes the indefinite conjugate of $A$, $U$ and $V$ are bounded invertible linear or conjugate linear operators with $U^\dag U=c^{-1}I$ and $V^\dag V=cI$ for some nonzero real number $c$. |
Keywords： |
Indefinite inner product spaces, $\dag$-Automorphisms, Jordan product |
Classification： |
46C20, 47B49 |
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