Automorphisms of the Poset of Skew Projections

Ovchinnikov, Peter G.

doi:10.1006/jfan.1993.1086

Cited by 57 publications

(51 citation statements)

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“…In particular, it can be proved that a bijective map on I(X) preserves order in both directions if and only if it preserves orthogonality in both directions [25, Proposition 1.1]. Therefore, Ovchinnikov's result [19] follows directly from the characterization of ⊥-automorphisms. Corollary 3.3.…”

Section: Results and Open Problemsmentioning

confidence: 99%

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Maps on idempotent operators

Šemrl

2007

Perspectives in Operator Theory

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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

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Section: Results and Open Problemsmentioning

confidence: 99%

“…We will conclude the paper with a new result. Ovchinnikov [19] characterized bijective maps on I(X) preserving order in both directions. He posed the problem whether we can get a nice structural result under the weaker assumption that the comparability of idempotents is preserved in both directions.…”

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confidence: 99%

Maps on idempotent operators

Šemrl

2007

Perspectives in Operator Theory

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“…In this case we write P ⊥ Q. Motivated by some problems in quantum mechanics (see the review MR 95a:46093) Ovchinnikov [16] characterized automorphisms of the poset P (X) in the case that X is a Hilbert space of dimension at least 3. Recall that an automorphism φ of the poset P (X) is a bijective map preserving order in both directions, that is, P ≤ Q if and only if φ(P ) ≤ φ(Q), P, Q ∈ P (X).…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Let us just mention that in the finite-dimensional complex case the results are slightly more complicated as the map T need not be linear or conjugate-linear, but merely semilinear (see [16]). On the other hand, in the finite-dimensional case the last two corollaries can be proved under weaker assumptions (see [21,22]).…”

Section: Bijective Map φ : P (X) → P (X) Preserves Commutativity In Bmentioning

confidence: 99%

Maps on idempotents

Šemrl

2005

Studia Math.

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Abstract. 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) ⊂ 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). Introduction and statement of main results.Let X be a real or complex Banach space and B(X) the algebra of all bounded linear operators on X. Denote by X * the dual of X and by P (X) ⊂ B(X) the subset of all idempotents. Recall that P (X) is a poset with the partial order defined by P ≤ Q ⇔ P Q = QP = P , P, Q ∈ P (X). Orthogonality is another important relation on P (X). Two idempotents P, Q ∈ P (X) are said to be orthogonal if P Q = QP = 0. In this case we write P ⊥ Q. Motivated by some problems in quantum mechanics (see the review MR 95a:46093) Ovchinnikov [16] characterized automorphisms of the poset P (X) in the case that X is a Hilbert space of dimension at least 3. Recall that an automorphism φ of the poset P (X) is a bijective map preserving order in both directions, that is, P ≤ Q if and only if φ(P ) ≤ φ(Q), P, Q ∈ P (X).As far as we know, the automorphisms of P (X) with respect to the orthogonality relation have not been treated in the literature. In fact, the structural result for bijective maps on P (X) preserving orthogonality in both directions follows almost directly from the result of Ovchinnikov. Namely, for a subset S ⊂ P (X) denote by S ⊥ the set of all idempotents from P (X) that are orthogonal to every member of S. In the case S = {P } we simply write P ⊥ = {P } ⊥ . Then, as we shall see, it is easy to verify that for an arbitrary pair of idempotents P, Q ∈ P (X) we have P ≤ Q if and only if

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“…So, we obtain that ψ preserves the partial ordering ≤ between the idempotents in M 3 (C) in both directions. We now apply a nice result of Ovchinnikov [10] describing the automorphisms of the poset of all idempotents on a Hilbert space of dimension at least 3. It is a trivial corollary of his result that our transformation ψ is orthoadditive on the set of all idempotents in M 3 (C), that is, if P, Q are mutually orthogonal idempotents in M 3 (C), then we have ψ(P + Q) = ψ(P ) + ψ(Q).…”

Section: Then We Infermentioning

confidence: 99%