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Journal of Operator Theory

Volume 60, Issue 1, Summer 2008  pp. 71-83.

Similarity preserving linear maps

Authors Peter Semrl
Author institution: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, SI-1000, Slovenia

Summary:  Let $H$ be an infinite-dimensional separable Hilbert space, $B(H)$ the algebra of all bounded linear operators on $H$, and $\phi : B(H) \to B(H)$ a bijective linear map such that $\phi (A)$ and $\phi (B)$ are similar for every pair of similar operators $A,B \in B(H)$. Then there exist a nonzero complex number $c$ and an invertible operator $T\in B(H)$ such that either $\phi (A) = cTAT^{-1}$, $A\in B(H)$, or $\phi (A) = cTA^{\mathrm t} T^{-1}$, $A\in B(H)$. Here, $A^{\mathrm t}$ denotes the transpose of $A$ with respect to some fixed orthonormal basis in $H$.

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