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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