Similarity-preserving linear maps on $B(X)$

Fangyan Lu ¹   ; Chaoran Peng ¹

¹ Department of Mathematics Soochow University Suzhou 215006, P.R. China

Studia Mathematica, Tome 209 (2012) no. 1, pp. 1-10

Résumé

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 $A\in B(X)$.

Détail

DOI : 10.4064/sm209-1-1

Keywords: infinite dimensional banach space algebra bounded linear operators phi bijective similarity preserving linear map only following holds there exist nonzero complex number invertible bounded operator similarity invariant linear functional c phi ctat there exist nonzero complex number invertible bounded linear operator * similarity invariant linear functional c phi cta *t