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## Involution similarity preserving linear maps

Studia Mathematica MSC: Primary 47B49. DOI: 10.4064/sm180311-5-7 Opublikowany online: 23 April 2019

#### Streszczenie

Let $X$ be a Banach space with dimension at least 3. Two operators $A$ and $B$ in $B(X)$ are said to be $p$-similar if there is a product $S$ of finitely many involutions such that $A=SBS^{-1}$. In this paper, we investigate linear bijections $\Phi : B(X) \to B(X)$ such that $\Phi (A)$ and $\Phi (B)$ are similar whenever $A$ and $B$ are $p$-similar. We show that such a map is either an isomorphism or an anti-isomorphism plus a $p$-similarity invariant functional. This result can be used to characterize Lie isomorphisms and Jordan isomorphisms.

#### Autorzy

• Zijie QinDepartment of Mathematics
Soochow University
Suzhou 215006, China
e-mail
• Fangyan LuDepartment of Mathematics
Soochow University
Suzhou 215006, China
e-mail

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