Involution similarity preserving linear maps
Volume 249 / 2019
Studia Mathematica 249 (2019), 319-328
MSC: Primary 47B49.
DOI: 10.4064/sm180311-5-7
Published online: 23 April 2019
Abstract
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.
