Linear maps preserving quasi-commutativity

Heydar Radjavi ¹   ; Peter Šemrl ²

¹ Department of Pure Mathematics University of Waterloo 200 University Avenue West Waterloo, ON, Canada N2L 3G1
² Department of Mathematics University of Ljubljana Jadranska 19 SI-1000 Ljubljana, Slovenia

Studia Mathematica, Tome 184 (2008) no. 2, pp. 191-204

Résumé

Let $X$ and $Y$ be Banach spaces and ${\cal B}(X)$ and ${\cal B}(Y)$ the algebras of all bounded linear operators on $X$ and $Y$, respectively. We say that $A,B \in {\cal B}(X)$ quasi-commute if there exists a nonzero scalar $\omega $ such that $AB = \omega BA$. We characterize bijective linear maps $\phi : {\cal B}(X) \to {\cal B}(Y)$ preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.

Détail

DOI : 10.4064/sm184-2-7

Keywords: banach spaces cal cal algebras bounded linear operators respectively say cal quasi commute there exists nonzero scalar omega omega characterize bijective linear maps phi cal cal preserving quasi commutativity characterization proved much general algebras finite dimensional result obtained without bijectivity assumption