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.

scholarly journals Linear Maps Which are Anti-derivable at Zero

2020 ◽  
Vol 43 (6) ◽  
pp. 4315-4334
Author(s):  
Doha Adel Abulhamil ◽  
Fatmah B. Jamjoom ◽  
Antonio M. Peralta
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complete Characterization ◽  
Linear Operators ◽  
Continuous Linear ◽  
Bounded Linear ◽  
Linear Maps ◽  
Continuous Linear Operators

Abstract Let $$T:A\rightarrow X$$ T : A → X be a bounded linear operator, where A is a $$\hbox {C}^*$$ C ∗ -algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent: (a) T is anti-derivable at zero (i.e., $$ab =0$$ a b = 0 in A implies $$T(b) a + b T(a)=0$$ T ( b ) a + b T ( a ) = 0 ); (b) There exist an anti-derivation $$d:A\rightarrow X^{**}$$ d : A → X ∗ ∗ and an element $$\xi \in X^{**}$$ ξ ∈ X ∗ ∗ satisfying $$\xi a = a \xi ,$$ ξ a = a ξ , $$\xi [a,b]=0,$$ ξ [ a , b ] = 0 , $$T(a b) = b T(a) + T(b) a - b \xi a,$$ T ( a b ) = b T ( a ) + T ( b ) a - b ξ a , and $$T(a) = d(a) + \xi a,$$ T ( a ) = d ( a ) + ξ a , for all $$a,b\in A$$ a , b ∈ A . We also prove a similar equivalence when X is replaced with $$A^{**}$$ A ∗ ∗ . This provides a complete characterization of those bounded linear maps from A into X or into $$A^{**}$$ A ∗ ∗ which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are $$^*$$ ∗ -anti-derivable at zero.

Linear Operators Preserving The Real Symplectic Group

1975 ◽  
Vol 27 (3) ◽  
pp. 715-724 ◽  
Author(s):  
Stephen Pierce
Keyword(s):  
Linear Operator ◽  
Unitary Group ◽  
Symplectic Group ◽  
Linear Operators ◽  
Complex Numbers ◽  
Unitary Matrices ◽  
The Real ◽  
Real Symplectic Group

1. Introduction and s t a t e m e n t of results. Let F be a field and Mn(F) the n X n matrices over F. Set GL(n, F) to be t he units in Mn(F). Suppose H is a subgroup of GL(n, F) and L is an L-linear operator on Mn(F) mapping H into itself. Can L be neatly characterized?Marcus [2] answered this question when F is the complex numbers C and H the unitary group. The set of such L is a group and L has the form L(A) = UAV or L(A) = UA'V where U, V are fixed unitary matrices and A’ is t he transpose of A.

scholarly journals Sets of periods for chaotic linear operators

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
J. A. Conejero ◽  
F. Martínez-Giménez ◽  
A. Peris ◽  
F. Rodenas
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Complete Characterization ◽  
Linear Operators

AbstractWe provide a complete characterization of the possible sets of periods for Devaney chaotic linear operators on Hilbert spaces. As a consequence, we also derive this characterization for linearizable maps on Banach spaces.

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.

scholarly journals Automatic discontinuity of intertwining operators

1982 ◽  
Vol 25 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Banach Spaces ◽  
Linear Transformation ◽  
Linear Operators ◽  
Intertwining Operators ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Converse Problem ◽  
Simple Modification ◽  
Continuous Linear Operators

Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

Linear Transformations On Matrices: The Invariance of Generalized Permutation Matrices, I

1976 ◽  
Vol 28 (3) ◽  
pp. 455-472 ◽  
Author(s):  
Hock Ong ◽  
E. P. Botta
Keyword(s):  
Vector Space ◽  
Unitary Group ◽  
Linear Transformations ◽  
Permutation Matrices ◽  
Linear Maps

Let F be a field, Mn(F) be the vector space of all w-square matrices with entries in F and a subset of Mn(F). It is of interest to determine the structure of linear maps T : Mn(F) →Mn(F) such that . For example: Let be GL(n, C), the group of all nonsingular n X n matrices over C [5]; the subset of all rank 1 matrices in MmXn(F) [4] (MmXn(F) is the vector space of all m X n matrices over F) ; the unitary group [2] ; or the set of all matrices X in Mn(F) such that det(X) = 0 [1]. Other results in this direction can be found in [3].

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

Supercuspidal representations and preservation principle of theta correspondence

2019 ◽  
Vol 2019 (750) ◽  
pp. 1-52
Author(s):  
Shu-Yen Pan
Keyword(s):  
Unitary Group ◽  
Orthogonal Group ◽  
Classical Groups ◽  
Theta Correspondence ◽  
Dual Pairs ◽  
Supercuspidal Representations ◽  
Explicit Characterization ◽  
A Chain

Abstract The preservation principle of the local theta correspondence predicts the existence of a chain of irreducible supercuspidal representations of p-adic classical groups. In this paper, we give an explicit characterization of the chain starting from an irreducible supercuspidal representations of a unitary group of one variable or an orthogonal group of two variables. In particular, we define the Lusztig-like correspondence of generic cuspidal data for p-adic groups and establish its relation with local theta correspondence of supercuspidal representations for p-adic dual pairs.

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.

Sign in / Sign up

Export Citation Format

Share Document