Similarity-preserving linear maps on $B(X)$
Volume 209 / 2012
Studia Mathematica 209 (2012), 1-10
MSC: Primary 47B49.
DOI: 10.4064/sm209-1-1
Abstract
Let $X$ be an infinite-dimensional Banach space, and $B(X)$ the algebra of all bounded linear operators on $X$. Then $\phi: B(X)\to B(X)$ is a bijective similarity-preserving linear map if and only if one of the following holds:
(1) There exist a nonzero complex number $c$, an invertible bounded operator $T$ in $B(X)$ and a similarity-invariant linear functional $h$ on $B(X)$ with $h(I)\ne -c$ such that $\phi(A)=cTAT^{-1}+h(A)I$ for all $A\in B(X)$.
(2) There exist a nonzero complex number $c$, an invertible bounded linear operator $T: X^*\to X$ and a similarity-invariant linear functional $h$ on $B(X)$ with $h(I)\ne -c$ such that $\phi(A)=cTA^*T^{-1}+h(A)I$ for all $A\in B(X)$.
