Variation of Discrete Spectra for Non-Selfadjoint Perturbations of Selfadjoint Operators

Let B = A + K where A is a bounded selfadjoint operator and K is an element of the von Neumann–Schatten ideal ${\mathcal{S}_{p}}$ with p > 1. Let {λ _n } denote an enumeration of the discrete spectrum of B. We show that ${\sum_n {\rm dist}(\lambda_n, \sigma(A))^p}$ is bounded from above by a constant multiple of ${\|K\|_{p}^p}$ . We also derive a unitary analog of this estimate and apply it to obtain new estimates on zero-sets of Cauchy transforms.     

On Finite Rank Perturbations of Selfadjoint Operators in Krein Spaces and Eigenvalues in Spectral Gaps

It is shown that the finiteness of eigenvalues in a spectral gap of a definitizable or locally definitizable selfadjoint operator in a Krein space is preserved under finite rank perturbations. This results is applied to a class of singular Sturm–Liouville operators with an indefinite weight function.     

On Selfadjoint Homogeneous Operators

We study unbounded selfadjoint operators that are unitarily equivalent to their affine transformation. We investigate transformation properties of an operator-valued M-function associated with given affine-invariant operator as well as spectral properties of operators that occur in that investigation.     

Positive Decompositions of Selfadjoint Operators

Given a linear bounded selfadjoint operator a on a complex separable Hilbert space ${\mathcal{H}}Given a linear bounded selfadjoint operator a on a complex separable Hilbert space H{\mathcal{H}}, we study the decompositions of a as a difference of two positive operators whose ranges satisfy an angle condition. These decompositions are related to the canonical decompositions of the indefinite metric space (H,á ,  ?_a){(\mathcal{H},\langle\,, \,\rangle_a)}, associated to a. As an application, we characterize the orbit of congruence of a in terms of its positive decompositions.     

A Spectral Representation for Bounded Non-Selfadjoint Operators

An integral representation for an arbitrary bounded operator T defined on a Hilbert space ${\mathcal{H}}$ is given. The representing measure is in general defined on a Jordan curve surrounding the spectrum of T. It is obtained as a limit, in a certain weak sense, of a family (F _r ) of absolutely continuous measures the Radon?CNikodym derivative of which (with respect to the standard Lebesgue measure on the considered Jordan curve) are described explicitly in terms of the operator T and its adjoint T ^*.     

Inequalities for the Eigenvalues of Non-Selfadjoint Jacobi Operators

We prove Lieb-Thirring-type bounds on eigenvalues of non-selfadjoint Jacobi operators, which are nearly as strong as those proven previously for the case of selfadjoint operators by Hundertmark and Simon. We use a method based on determinants of operators and on complex function theory, extending and sharpening earlier work of Borichev, Golinskii and Kupin.     

Positivity for Perturbations of Polyharmonic Operators with Dirichlet Boundary Conditions in Two Dimensions

Higher order elliptic partial differential equations with Dirichlet boundary conditions in general do not satisfy a maximum principle. Polyharmonic operators on balls are an exception. Here it is shown that in IR² small perturbations of polyharmonic operators and of the domain preserve the maximum principle. Hence the Green function for the clamped plate equation on an ellipse with small eccentricity is positive.     

关于自伴算子的奇异连续谱

本文建立了自伴算子奇异连续谱的三个定理,它们推广了Barry Simon近期所获得的一些重要结果.     

On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space which can roughly be described as follows: (1) If is an open subset of and all spectral subspaces for A corresponding to compact subsets of have finite rank of negativity, the same is true for a selfadjoint operator B in for which the difference of the resolvents of A and B is compact. (2) The property that there exists some neighbourhood of such that the restriction of A to a spectral subspace for A corresponding to is a nonnegative operator in is preserved under relative perturbations in form sense if the resulting operator is again selfadjoint. The assertion (1) is proved for selfadjoint relations A and B. (1) and (2) generalize some known results.     

Some Selfadjoint 2$\times$2 Operator Matrices Associated with Closed Operators

We study a 22 operator matrix associated with a closed densely defined operator. Among others, the selfadjointness of a closed symmetric operator and the strong commutativity of two (unbounded) self-adjoint operators are characterized in terms of the related operator matrices. We propose a definition of strong commutativity for closed symmetric operators. Submitted: November 8, 2001     

Singular Perturbations of Self-Adjoint Operators

Singular relatively compact perturbations of self-adjoint operators are studied. The results obtained are applied to the Schrödinger operator with a singular potential.     

Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L ¹-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.     

Finite Rank Perturbations of Toeplitz Operators

We study finite rank perturbations of the Brown-Halmos type results involving products of Toeplitz operators acting on the Bergman space.      

About the Limit Spectra of Some Unbounded Selfadjoint Integral Operators

Siberian Mathematical Journal - We establish sufficient conditions for the zero to belong to the limit spectra of some unbounded selfadjoint integral operator.     

Coburn Type Operators and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is called Coburn operator if ker(T ? λ) = {0} or ker(T ? λ)^*= {0} for each λ ∈ C. In this paper, the authors define other Coburn type properties for Hilbert space operators and investigate the compact perturbations of operators with Coburn type properties. They characterize those operators for which has arbitrarily small compact perturbation to have some fixed Coburn property.Moreover, they study the stability of these properties under small compact perturbations.     

Invariance of Picard Dimensions Under Basic Perturbations

The Picard dimension \(\dim \mu\) of a signed Radon measure μ on the punctured closed unit ball 0?x|?≦?1 in the d-dimensional euclidean space with d?≧?2 is the cardinal number of the set of extremal rays of the cone of positive continuous distributional solutions u of the Schrödinger equation (???Δ?+?μ)u?=?0 on the punctured open unit ball 0?x|?x|?=?1. If the Green function of the above equation on 0?x|?Δ?+?μ)u?=?δ _y , the Dirac measure supported by the point y, exists for every y in 0?x|?μ is referred to as being hyperbolic on 0?x|?γ is a radial Radon measure which is both positive and absolutely continuous with respect to the d-dimensional Lebesgue measure dx whose Radon–Nikodym density dγ(x)/dx is bounded by a positive constant multiple of |x|^???2. The purpose of this paper is to show that the Picard dimensions of hyperbolic radial Radon measures μ are invariant under basic perturbations \(\gamma: \dim(\mu+\gamma)=\dim\mu\). Three applications of this invariance are also given.     

非线性多值算子扰动的映射定理

本文讨论非线性多值算子的非紧扰动的映射定理,并给出非线性泛函方程ｚ∈Ｔ（ｘ）＋Ｆ（ｘ）可解性的最新结果,其中Ｔ是多值算子且（Ｔ＋１/ｎＩ）^－１是１－集压缩,而Ｆ是１－集压缩或γ－凝聚．所得的结果改善了［５,８,１２］中的主要结果