共查询到20条相似文献,搜索用时 31 毫秒
1.
Let X be a real finite-dimensional normed space with unit sphere SX and let L(X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A,C∈L(X)
2.
When A∈B(H) and B∈B(K) are given, we denote by MC the operator matrix acting on the infinite-dimensional separable Hilbert space H⊕K of the form In this paper, for given A and B, the sets and ?C∈Inv(K,H)σl(MC) are determined, where σl(T),Bl(K,H) and Inv(K,H) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H. 相似文献
3.
Khakim D. Ikramov 《Linear algebra and its applications》2010,433(4):838-842
Every square complex matrix is known to be consimilar to a real matrix. Unitary congruence is a particular type of consimilarity. We prove that a matrix A∈Mn(C) is unitarily congruent to a real matrix if and only if A is unitarily congruent to via a symmetric unitary matrix. It is shown by an example that there exist matrices that are congruent, but not unitarily congruent, to real matrices. 相似文献
4.
Roland Hildebrand 《Linear algebra and its applications》2008,429(4):901-932
Let K⊂E, K′⊂E′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E⊗E′ is K⊗K′-separable if it can be represented as finite sum , where xl∈K and for all l. Let S(n), H(n), Q(n) be the spaces of n×n real symmetric, complex Hermitian and quaternionic Hermitian matrices, respectively. Let further S+(n), H+(n), Q+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix A∈H(mn)=H(m)⊗H(n) is H+(m)⊗H+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m)⊗S(n), H(m)⊗S(n), and for m?2 in the space Q(m)⊗S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n)⊗S(2) is Q+(n)⊗S+(2)- separable if and only if it is positive semidefinite. 相似文献
5.
Lajos Molnár 《Journal of Mathematical Analysis and Applications》2007,327(1):302-309
Let H be a Hilbert space and let A and B be standard ∗-operator algebras on H. Denote by As and Bs the set of all self-adjoint operators in A and B, respectively. Assume that and are surjective maps such that M(AM∗(B)A)=M(A)BM(A) and M∗(BM(A)B)=M∗(B)AM∗(B) for every pair A∈As, B∈Bs. Then there exist an invertible bounded linear or conjugate-linear operator and a constant c∈{−1,1} such that M(A)=cTAT∗, A∈As, and M∗(B)=cT∗BT, B∈Bs. 相似文献
6.
Joseph A. Ball Vladimir Bolotnikov 《Journal of Mathematical Analysis and Applications》2008,341(1):519-539
An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert space H(kd) on the unit ball Bd⊂Cd, where kd is the positive kernel kd(λ,ζ)=1/(1−〈λ,ζ〉) on Bd. The reproducing kernel space H(KS) associated with the positive kernel KS(λ,ζ)=(I−S(λ)S∗(ζ))⋅kd(λ,ζ) is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints of the multiplication operators on H(kd). We show that invariance of H(KS) under for each j=1,…,d is equivalent to the existence of a realization for S(λ) of the form S(λ)=D+C−1(I−λ1A1−?−λdAd)(λ1B1+?+λdBd) such that connecting operator has adjoint U∗ which is isometric on a certain natural subspace (U is “weakly coisometric”) and has the additional property that the state operators A1,…,Ad pairwise commute; in this case one can take the state space to be the functional-model space H(KS) and the state operators A1,…,Ad to be given by (a de Branges-Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A1,…,Ad satisfy an additional stability property. 相似文献
7.
8.
B.P. Duggal 《Journal of Mathematical Analysis and Applications》2010,370(2):584-587
Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A○∈B(K) denote the Berberian extension of an operator A∈B(H). It is proved that the set theoretic function σ, the spectrum, is continuous on the set C(i)⊂B(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and at every non-zero λ∈σp(A○) for some operators X and B such that λ∉σp(B) and σ(A○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where Aii∈C(i) and Aii has SVEP for all 1?i?m, then σ is continuous on CS(m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP. 相似文献
9.
Let A(λ) be a complex regular matrix polynomial of degree ? with g elementary divisors corresponding to the finite eigenvalue λ0. We show that for most complex matrix polynomials B(λ) with degree at most ? satisfying rank the perturbed polynomial (A+B)(λ) has exactly elementary divisors corresponding to λ0, and we determine their degrees. If does not exceed the number of λ0-elementary divisors of A(λ) with degree greater than 1, then the λ0-elementary divisors of (A+B)(λ) are the elementary divisors of A(λ) corresponding to λ0 with smallest degree, together with rank(B(λ)-B(λ0)) linear λ0-elementary divisors. Otherwise, the degree of all the λ0-elementary divisors of (A+B)(λ) is one. This behavior happens for any matrix polynomial B(λ) except those in a proper algebraic submanifold in the set of matrix polynomials of degree at most ?. If A(λ) has an infinite eigenvalue, the corresponding result follows from considering the zero eigenvalue of the perturbed dual polynomial. 相似文献
10.
Marek Niezgoda 《Linear algebra and its applications》2010,433(1):136-640
Let a,b>0 and let Z∈Mn(R) such that Z lies into the operator ball of diameter [aI,bI]. Then for all positive definite A∈Mn(R),
11.
Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices
Chi-Kwong Li 《Linear algebra and its applications》2009,430(7):1739-1398
Let be the set of entrywise nonnegative n×n matrices. Denote by r(A) the spectral radius (Perron root) of . Characterization is obtained for maps such that r(f(A)+f(B))=r(A+B) for all . In particular, it is shown that such a map has the form
12.
Hwa-Long Gau 《Linear algebra and its applications》2010,432(5):1310-1321
In this paper, we show that a reducible companion matrix is completely determined by its numerical range, that is, if two reducible companion matrices have the same numerical range, then they must equal to each other. We also obtain a criterion for a reducible companion matrix to have an elliptic numerical range, put more precisely, we show that the numerical range of an n-by-n reducible companion matrix C is an elliptic disc if and only if C is unitarily equivalent to A⊕B, where A∈Mn-2, B∈M2 with σ(B)={aω1,aω2}, , ω1≠ω2, and . 相似文献
13.
In a recent paper, Neumann and Sze considered for an n × n nonnegative matrix A, the minimization and maximization of ρ(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n × n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n × n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A + X) over all n × n real matrices X of Frobenius norm was sought. 相似文献
14.
15.
The paper studies the eigenvalue distribution of some special matrices. Tong in Theorem 1.2 of [Wen-ting Tong, On the distribution of eigenvalues of some matrices, Acta Math. Sinica (China), 20 (4) (1977) 273-275] gives conditions for an n × n matrix A ∈ SDn ∪ IDn to have |JR+(A)| eigenvalues with positive real part, and |JR-(A)| eigenvalues with negative real part. A counter-example is given in this paper to show that the conditions of the theorem are not true. A corrected condition is then proposed under which the conclusion of the theorem holds. Then the corrected condition is applied to establish some results about the eigenvalue distribution of the Schur complements of H-matrices with complex diagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, … , n} are presented such that the Schur complement matrix A/α of the matrix A has eigenvalues with positive real part and eigenvalues with negative real part. 相似文献
16.
Let H be the real quaternion algebra and Hn×m denote the set of all n×m matrices over H. Let P∈Hn×n and Q∈Hm×m be involutions, i.e., P2=I,Q2=I. A matrix A∈Hn×m is said to be (P,Q)-symmetric if A=PAQ. This paper studies the system of linear real quaternion matrix equations
17.
Fritz Gesztesy Yuri Latushkin Konstantin A. Makarov Fedor Sukochev Yuri Tomilov 《Advances in Mathematics》2011,(1):145
We compute the Fredholm index, index(DA), of the operator DA=(d/dt)+A on L2(R;H) associated with the operator path , where (Af)(t)=A(t)f(t) for a.e. t∈R, and appropriate f∈L2(R;H), via the spectral shift function ξ(⋅;A+,A−) associated with the pair (A+,A−) of asymptotic operators A±=A(±∞) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A−.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A+,A−) for the pair (A+,A−), and the corresponding spectral shift function ξ(⋅;H2,H1) for the pair of operators in this relative trace class context,This formula is then used to identify the Fredholm index of DA with ξ(0;A+,A−). In addition, we prove that index(DA) coincides with the spectral flow of the family {A(t)}t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A+,A−): with the choice of the branch of ln(detH(⋅)) on C+ such thatWe also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant. 相似文献
18.
Juan-Miguel Gracia 《Linear algebra and its applications》2009,430(4):1196-1215
Given four complex matrices A,B,C and D, where A∈Cn×n and D∈Cm×m, and given a complex number z0: What is the (spectral norm) distance from D to the set of matrices X∈Cm×m such that z0 is a multiple eigenvalue of the matrix
19.
When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite dimensional separable Hilbert space H⊕K of the form . In this paper, it is shown that a 2×2 operator matrix MC is upper semi-Fredholm and ind(MC)?0 for some C∈B(K,H) if and only if A is upper semi-Fredholm and
20.
A.B. Aleksandrov 《Journal of Functional Analysis》2010,258(11):3675-5251
This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α‖ in terms of ‖A−BSp‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K). 相似文献