Linear Preserver Problems

Li, Chi-Kwong; Pierce, S. Gareth

doi:10.2307/2695268

Cited by 136 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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“…The study of the Schur product is related to many pure and applied areas; see [9]. There has been considerable interest in studying linear maps, additive maps, and multiplicative maps f on matrices with some special properties such as f (S) ⊆ S for a certain subset of matrices, or ( f (A)) = (A) for a given function on matrices; for example, see [3,8,11,12,15] and the references therein. In this paper, we study Schur multiplicative maps on matrices with some of these special properties.…”

Section: Introductionmentioning

confidence: 99%

Schur Multiplicative Maps on Matrices

Clark

Rastogi

2008

Bulletin of the Australian Mathematical Society

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The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying f (S) ⊆ S for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that ( f (A)) = (A) for various functions including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.2000 Mathematics subject classification: 15A03, 15A15.

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“…Besides Hua and Siegel's original motivation, related respectively to complex analysis and symplectic geometry, this result has many other applications, say in the theory of linear preservers [15] and in the geometry of Grassmann spaces. We refer an interested reader to [23] for a detailed explanation.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings

Pazzis

Šemrl

2015

Journal of Algebra

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The fundamental theorem of geometry of rectangular matrices describes the general form of bijective maps on the space of all m×n matrices over a division ring D which preserve adjacency in both directions. This result proved by Hua in the nineteen forties has been recently improved in several directions. One can study such maps without the bijectivity assumption or one can try to get the same conclusion under the weaker assumption that adjacency is preserved in one direction only. And the last possibility is to study maps acting between matrix spaces of different sizes. The optimal result would describe maps preserving adjacency in one direction only acting between spaces of rectangular matrices of different sizes in the absence of any regularity condition (injectivity or surjectivity).A division ring is said to be EAS if it is not isomorphic to any proper subring. It has been known before that it is possible to construct adjacency preserving maps with wild behaviour on matrices over division rings that are not EAS. For matrices over EAS division rings it has been recently proved that adjacency preserving maps acting between matrix spaces of different sizes satisfying a certain weak surjectivity condition are either degenerate or of the expected simple standard form. We will remove this weak surjectivity assumption, thus solving completely the long standing open problem of the optimal version of Hua's theorem.

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Linear Preserver Problems

Cited by 136 publications

References 0 publications

Preservers of maximally entangled states

Preservers of maximally entangled states

Schur Multiplicative Maps on Matrices

Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings

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