scholarly journals Linear maps preserving rank 2 on the space of alternate matrices and their applications

2004 ◽  
Vol 2004 (63) ◽  
pp. 3409-3417 ◽  
Author(s):  
Chongguang Cao ◽  
Xiaomin Tang
Keyword(s):  
Linear Space ◽  
Linear Preservers ◽  
Linear Maps ◽  
Rank 2

Denote by𝒦n(F)the linear space of alln×nalternate matrices over a fieldF. We first characterize all linear bijective maps on𝒦n(F)(n≥4)preserving rank 2 whenFis any field, and thereby the characterization of all linear bijective maps on𝒦n(F)preserving the max-rank is done whenFis any field except for{0,1}. Furthermore, the linear preservers of the determinant (resp., adjoint) on𝒦n(F)are also characterized by reducing them to the linear preservers of the max-rank whennis even andFis any field except for{0,1}. This paper can be viewed as a supplement version of several related results.

scholarly journals Auerbach Bases and Minimal Volume Sufficient Enlargements

2011 ◽  
Vol 54 (4) ◽  
pp. 726-738
Author(s):  
M. I. Ostrovskii
Keyword(s):  
Linear Space ◽  
Isometric Embedding ◽  
Normed Linear Space ◽  
Convex Set ◽  
Linear Projection ◽  
Minimal Volume ◽  
Finite Dimensional ◽  
Dimensional Normed Space ◽  
Closed Convex Set

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

scholarly journals Characterization of linear maps onMnwhose multiplicity maps have maximal norm, with an application in quantum information

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 51
Author(s):  
Daniel Puzzuoli
Keyword(s):  
Quantum Information ◽  
Quantum Channel ◽  
Operator Algebras ◽  
Optimal Performance ◽  
Single Shot ◽  
Trace Norm ◽  
Linear Maps ◽  
The Family ◽  
Matrix Transposition

Given a linear mapΦ:Mn→Mm, its multiplicity maps are defined as the family of linear mapsΦ⊗idk:Mn⊗Mk→Mm⊗Mk, whereidkdenotes the identity onMk. Let‖⋅‖1denote the trace-norm on matrices, as well as the induced trace-norm on linear maps of matrices, i.e.‖Φ‖1=max{‖Φ(X)‖1:X∈Mn,‖X‖1=1}. A fact of fundamental importance in both operator algebras and quantum information is that‖Φ⊗idk‖1can grow withk. In general, the rate of growth is bounded by‖Φ⊗idk‖1≤k‖Φ‖1, and matrix transposition is the canonical example of a map achieving this bound. We prove that, up to an equivalence, the transpose is the unique map achieving this bound. The equivalence is given in terms of complete trace-norm isometries, and the proof relies on a particular characterization of complete trace-norm isometries regarding preservation of certain multiplication relations.We use this result to characterize the set of single-shot quantum channel discrimination games satisfying a norm relation that, operationally, implies that the game can be won with certainty using entanglement, but is hard to win without entanglement. Specifically, we show that the well-known example of such a game, involving the Werner-Holevo channels, is essentially the unique game satisfying this norm relation. This constitutes a step towards a characterization of single-shot quantum channel discrimination games with maximal gap between optimal performance of entangled and unentangled strategies.

Linear Maps Transforming the Unitary Group

2003 ◽  
Vol 46 (1) ◽  
pp. 54-58 ◽  
Author(s):  
Wai-Shun Cheung ◽  
Chi-Kwong Li
Keyword(s):  
Linear Transformation ◽  
Unitary Group ◽  
Linear Operators ◽  
Unitary Matrices ◽  
Linear Maps

AbstractLet U(n) be the group of n × n unitary matrices. We show that if ϕ is a linear transformation sending U(n) into U(m), then m is a multiple of n, and ϕ has the formfor some V, W ∈ U(m). From this result, one easily deduces the characterization of linear operators that map U(n) into itself obtained by Marcus. Further generalization of the main theorem is also discussed.

Linear preservers of Hadamard majorization

2016 ◽  
Vol 31 ◽  
pp. 593-609 ◽  
Author(s):  
Sara Motlaghian ◽  
Ali Armandnejad ◽  
Frank Hall
Keyword(s):  
Stochastic Matrix ◽  
Doubly Stochastic Matrix ◽  
Graph Theoretic ◽  
Doubly Stochastic ◽  
Linear Preservers ◽  
Linear Maps ◽  
Term Rank

Let $\textbf{M}_{n }$ be the set of all $n \times n $ realmatrices. A matrix $D=[d_{ij}]\in\textbf{M}_{n } $ with nonnegative entries is called doubly stochastic if $\sum_{k=1}^{n} d_{ik}=\sum_{k=1}^{n} d_{kj}=1$ for all $1\leq i,j\leq n$. For $ X,Y \in \textbf{M}_{n}$ we say that $X$ is Hadamard-majorized by $Y$, denoted by $ X\prec_{H} Y$, if there exists an $n \times n$ doubly stochastic matrix $D$ such that $X=D\circ Y$.In this paper, some properties of$\prec_{H}$ on $\textbf{M}_{n}$ are first obtained, and then the (strong) linear preservers of$\prec_{H}$ on $\textbf{M}_{n }$ are characterized. For $n\geq3$, it is shown that the strong linear preservers of Hadamard majorization on $\textbf{M}_{n}$ are precisely the invertible linear maps on $\textbf{M}_{n}$ which preserve the set of matrices of term rank 1.An interesting graph theoretic connection to the linear preservers of Hadamard majorization is exhibited. A number of examples are also provided in the paper.

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

scholarly journals A characterization of normed M-spaces

1983 ◽  
Vol 24 (1) ◽  
pp. 89-92 ◽  
Author(s):  
Garfield C. Schmidt
Keyword(s):  
Linear Space ◽  
Linear Structure ◽  
Topological Spaces ◽  
Intrinsic Properties ◽  
Small Step ◽  
Linear Lattice ◽  
Linear Spaces ◽  
And Topology ◽  
Necessary And Sufficient

Linear spaces on which both an order and a topology are defined and related in various ways have been studied for some time now. Given an order on a linear space it is sometimes possible to define a useful topology using the order and linear structure. In this note we focus on a special type of space called a linear lattice and determine those lattice properties which are both necessary and sufficient for the existence of a classical norm, called an M-norm, for the lattice. This result is a small step in a program to determine which intrinsic order properties of an ordered linear space are necessary and sufficient for the existence of various given types of topologies for the space. This study parallels, in a certain sense, the study of purely topological spaces to determine intrinsic properties of a topology which make it metrizable and the study of the relation between order and topology on spaces which have no algebraic structure, or. algebraic structures other than a linear one.

scholarly journals A characterization of cone-convex vector-valued functions

2016 ◽  
Vol 32 (1) ◽  
pp. 79-85
Author(s):  
DAISHI KUROIWA ◽  
◽  
NICOLAE POPOVICI ◽  
MATTEO ROCCA ◽  
◽  
...  
Keyword(s):  
Linear Space ◽  
Convex Analysis ◽  
Linear Functional ◽  
Partially Ordered ◽  
Vector Valued Functions ◽  
Vector Valued

An interesting result in convex analysis, established by J.-P. Crouzeix in 1977, states that a real-valued function defined on a linear space is convex if and only if each function obtained from it by adding a linear functional is quasiconvex. The aim of this paper is to extend this result for vector-valued functions taking values in a partially ordered linear space.

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