Some general techniques on linear preserver problems

Guterman, Alexander; Li, Chi-Kwong; Šemrl, Peter

doi:10.1016/s0024-3795(00)00119-1

Cited by 97 publications

(34 citation statements)

References 34 publications

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“…Questions of this type have a long history, and fall under the broader purview of linear preserver problems (two useful surveys are [5,6]). Recently there has been work done on finding and classifying linear preservers of various properties or sets related to quantum information theory.…”

Section: Introduction and Notationmentioning

confidence: 99%

Preservers of maximally entangled states

Poon

2015

Linear Algebra and its Applications

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The linear structure of the real space spanned by maximally entangled states is investigated, and used to completely characterize those linear maps preserving the set of maximally entangled states on M m ⊗ M m , where M m denotes the space of m × m complex matrices. Aside from a degenerate rank one map, such preservers are generated by a change of orthonormal basis in each tensor factor, interchanging the two tensor factors, and the transpose operator.

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Section: Introduction and Notationmentioning

confidence: 99%

Preservers of maximally entangled states

Poon

2015

Linear Algebra and its Applications

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“…For some recent surveys on preserver problems and papers with general introduction to this subject, see e.g. [13,24,28].…”

Section: Discussionmentioning

confidence: 99%

Adjacency preservers, symmetric matrices, and cores

Orel

2011

J Algebr Comb

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“…So, the characterization of linear maps on B(X) preserving idempotents in both directions can be easily deduced from the results on order or orthogonality preservers on I(X). Recently, many linear preserver problems were solved by reducing them to the problem of characterizing linear maps preserving idempotent operators [7]. Perhaps the most important linear preserver result obtained in this way is the Aupetit's solution [2] of Kaplansky's problem of characterizing linear invertibility preserving maps for von Neumann algebras.…”

Section: Motivationmentioning

confidence: 99%

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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Some general techniques on linear preserver problems

Cited by 97 publications

References 34 publications

Preservers of maximally entangled states

Preservers of maximally entangled states

Adjacency preservers, symmetric matrices, and cores

Maps on idempotent operators

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