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1.
Variational principles for eigenvalues of certain functions whose values
are possibly unbounded self-adjoint operators T() are proved. A generalised
Rayleigh functional is used that assigns to a vector x a zero of the
function T()x, x), where it is assumed that there exists at most one zero.
Since there need not exist a zero for all x, an index shift may occur. Using
this variational principle, eigenvalues of linear and quadratic polynomials
and eigenvalues of block operator matrices in a gap of the essential spectrum
are characterised. Moreover, applications are given to an elliptic eigenvalue
problem with degenerate weight, Dirac operators, strings in a medium with a
viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue
parameter. 相似文献
2.
Victor I. Lomonosov Heydar Radjavi Vladimir G. Troitsky 《Integral Equations and Operator Theory》2008,60(3):405-418
An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of
conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions
of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x
n
) in B there exists a subsequence and a bounded sequence (A
k
) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive
rings.
The second and the third authors were supported by NSERC. 相似文献
3.
4.
Hristo S. Sendov 《Linear algebra and its applications》2007,424(1):240-281
We are interested in higher-order derivatives of functions of the eigenvalues of real symmetric matrices with respect to the matrix argument. We describe a formula for the k-th derivative of such functions in two general cases.The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues.The second case describes the derivatives of the composition of a k-times differentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula significantly simplifies when the separable symmetric function is k-times continuously differentiable.As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one.To make the exposition as self contained as possible, we have included the necessary background results and definitions. Proofs of the intermediate technical results are collected in the appendices. 相似文献
5.
Double variational principles are established for eigenvalues of a (norm) continuous self-adjoint operator valued functionL defined on a real interval [, [.L() is not required to be definite for any . Applications are made to linear, quadratic and rational functionsL.This author acknowledges support from NSERC of Canada and the I.W. Killam Foundation.This author was supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 12176-MAT. 相似文献
6.
A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional
spaces, finite rank operators, Jordan blocks, and C0 contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of An, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A* has trivial kernel. 相似文献
7.
Let A be a self-adjoint operator in a Krein space
Under certain natural assumptions, it is shown precisely which real eigenvalues of A can be given a max-inf characterization generalizing the usual one in Hilbert space. This result unifies several approaches in the recent literature. 相似文献
8.
Heinz Langer Matthias Langer Alexander Markus Christiane Tretter 《Complex Analysis and Operator Theory》2008,2(1):99-134
We establish sufficient conditions for the so-called Virozub–Matsaev condition for twice continuously differentiable self-adjoint
operator functions. This condition is closely related to the existence of a local spectral function and to the notion of positive
type spectrum. Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given.
Received: September 22, 2007. Accepted: September 29, 2007. 相似文献
9.
V. A. Khatskevich M. I. Ostrovskii V. S. Shulman 《Integral Equations and Operator Theory》2005,51(1):109-119
Inspired by some problems on fractional linear transformations the authors introduce and study the class of operators satisfying the condition
where stands for the spectral radius; and the class of Banach spaces in which all operators satisfy this condition, the authors call such spaces V-spaces. It is shown that many well-known reflexive spaces, in particular, such spaces as Lp(0,1) and Cp, are non-V-spaces if p 2; and that the spaces lp are V-spaces if and only if 1 < p < . The authors pose and discuss some related open problems. 相似文献
10.
Mohammad Sal Moslehian 《Linear algebra and its applications》2011,434(8):1981-1987
We establish several operator versions of the classical Aczél inequality. One of operator versions deals with the weighted operator geometric mean and another is related to the positive sesquilinear forms. Some applications including the unital positive linear maps on C*-algebras and the unitarily invariant norms on matrices are presented. 相似文献
11.
E.B. Davies 《Mathematische Zeitschrift》2003,243(4):719-743
We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that
the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions
need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations
of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon,
and that the spectrum does not determine the operator up to similarity. Two elliptic systems which only differ in their boundary
conditions may have entirely different spectral asymptotics. While our study makes no claim to generality, the results obtained
will have to be incorporated into any future general theory.
Received: 15 August 2001 / in final form: 11 February 2002 / Published online: 24 February 2003 相似文献
12.
Pier Domenico Lamberti Massimo Lanza de Cristoforis 《Mediterranean Journal of Mathematics》2007,4(4):435-449
Let Ω be an open connected subset of for which the imbedding of the Sobolev space W
1,2(Ω) into the space L
2(Ω) is compact. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset (Ω) of , where is a Lipschitz continuous homeomorphism of Ω onto (Ω). Then we prove a result of real analytic dependence for symmetric functions of the eigenvalues upon variation of .
This paper represents an extension of a part of the work performed by P.D. Lamberti in his PhD Thesis at the University of
Padova under the guidance of M. Lanza de Cristoforis. 相似文献
13.
A. Böttcher 《Linear algebra and its applications》2011,435(8):1823-1836
This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)m=(QP)m and (PQ)m-1≠(QP)m-1. The main result is the classification of all these algebras, implying that for each m?2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra has a nondegenerate leading term, then it is group invertible, and a formula for the explicit computation of the group inverse is given. 相似文献
14.
LetP() be ann×n analytic matrix function andW(P) be its numerical range. In this paper classical results on the normality of matrix eigenvalues on W(P) are generalized to the context of such matrix functions. Special attention is paid to corners of W(P) and to the special case of matrix polynomialsP(). 相似文献
15.
S. Sallam 《Periodica Mathematica Hungarica》1985,16(1):1-5
In this paper, we continue the study [4] of the stability question of the quasidouble step spline function approximations,s(x) C
m–2
, to the initial value problemy
(n) = f(x, y, , y
(n–1)
). It will be shown that the method is unstable and hence divergent form n + 4. 相似文献
16.
For any symmetric function f:Rn?Rn, one can define a corresponding function on the space of n×n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability. 相似文献
17.
Let the functionQ be holomorphic in he upper half plane + and such that ImQ(z 0 and ImzQ(z) 0 ifz +. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf +
fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in + and such that the kernel
has negative squares of + and ImzQ(z) 0 ifz + is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f
dm +
2
fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832 相似文献
18.
In the present paper we consider a selfadjoint and nonsmooth operator-valued function on (c, d)R
1. We suppose that the equation (L()x, x)=0,x0, has exactly one rootp(x) (c, d) and the functionf()=(L()x, x) is increasing at the pointp(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved. 相似文献
19.
M.S. Moslehian 《Linear algebra and its applications》2009,430(4):1131-1987
We give an extension of Hua’s inequality in pre-Hilbert C∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is equivalent to operator convexity of given continuous real function. 相似文献
20.
J. Sprekels 《Aequationes Mathematicae》1977,16(3):283-295
Some existence theorems for fixed points of nonlinear integral operators expanding a cone in a Banach space in the sense of Krasnoselskii [5] are given. Using special cones we find pointwise inclusions for the fixed points. The question of finding the best bounds for the fixed points leads to a nonlinear optimization problem with infinitely many restrictions. 相似文献