Maps on idempotents

Peter Šemrl¹

¹ Department of Mathematics University of Ljubljana Jadranska 19 SI-1000 Ljubljana, Slovenia

Studia Mathematica, Tome 169 (2005) no. 1, pp. 21-44

Résumé

Let $X$ be an infinite-dimensional real or complex Banach space, $B(X)$ the algebra of all bounded linear operators on $X$, and $P(X)\subset B(X)$ the subset of all idempotents. We characterize bijective maps on $P(X)$ preserving commutativity in both directions. This unifies and extends the characterizations of two types of automorphisms of $P(X)$, with respect to the orthogonality relation and with respect to the usual partial order; the latter have been previously characterized by Ovchinnikov. We also describe bijective orthogonality preserving maps on the set of idempotents of a fixed finite rank. As an application we present a nonlinear extension of the structural result for bijective linear biseparating maps on $B(X)$.

Détail

DOI : 10.4064/sm169-1-2

Keywords: infinite dimensional real complex banach space algebra bounded linear operators subset subset idempotents characterize bijective maps preserving commutativity directions unifies extends characterizations types automorphisms respect orthogonality relation respect usual partial order latter have previously characterized ovchinnikov describe bijective orthogonality preserving maps set idempotents fixed finite rank application present nonlinear extension structural result bijective linear biseparating maps