Maps on idempotent operators

Abstract

Abstract. The set of all bounded linear idempotent operators on a Banach space X is a poset with the partial order defined by P ≤ Q if P Q = QP = P . Another natural relation on the set of idempotent operators is the orthogonality relation defined by P ⊥ Q ⇔ P Q = QP = 0. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver problems will be explained. At the end we will prove a new result concerning bijective maps on idempotent operators preserving comparability.1. Introduction. Throughout this paper X will denote an infinite-dimensional real or complex Banach space and B(X) the algebra of all bounded linear operators on X. By X ′ we denote the dual of X. An operator P ∈ B(X) is called an idempotent operator if P 2 = P . If P is an idempotent, P ∈ {0, I}, then the underlying Banach space X can be decomposed into the direct sum X = Im P ⊕ Ker P , where Im P and Ker P denote the image and the kernel of P , respectively. Clearly, P acts like the identity on Im P , while the restriction of P to Ker P is the zero map. Thus, the matrix representation of P with respect to the above direct sum decomposition of X is

Maps completely preserving idempotents and maps completely preserving square-zero operators

“…In order to prove the main theorem in this section, we need a lemma which is slightly different from the main results in [22], [24] and [26]. The proof is similar and we omit it here.…”

“…For example, it is the case in solving the famous open problemKaplansky's problem, and additive maps preserving the elements vanished by a polynomial for standard operator algebras and von Neumann algebras (see [1,3,5,14,16,20] etc.). So the various idempotent preserver problems have been intensively studied (see [8,18,19,26]). …”

Characterizing sequential isomorphisms on Hilbert-space effect algebras

“…Extend f to the Hilbert space H by defining f (λx) = λf (x) for any complex scalars λ and any x ∈ S. Then, for any vectors x, y ∈ H , x, y = 0 ⇔ f (x), f (y) = 0. As dim H 3, we may apply Uhlhorn's version of Wigner's theorem [10], and by a similar argument as that in [11,12], it is easily checked that there is a unitary operator or an anti-unitary operator U on H such that (P ) = UP U † holds for all rank-1 projections P.…”

“…The following key lemma describes the structure of the zero product preservers (equivalently, orthogonal preservers) on P(H ), that is, the maps : P(H ) → P(H ) satisfying that P Q = 0 ⇔ (P ) (Q) = 0 for all P , Q ∈ P(H ). The idea of its proof is borrowed from [11,12]. Lemma 2.7.…”