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

Tom 209 / 2012

Fangyan Lu, Chaoran Peng Studia Mathematica 209 (2012), 1-10 MSC: Primary 47B49. DOI: 10.4064/sm209-1-1

Streszczenie

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 $A\in B(X)$.

Autorzy

  • Fangyan LuDepartment of Mathematics
    Soochow University
    Suzhou 215006, P.R. China
    e-mail
  • Chaoran PengDepartment of Mathematics
    Soochow University
    Suzhou 215006, P.R. China

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