On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space . The perturbed operators are defined by the Krein resolvent formula , Im z 0, where B _z are finite-rank operators such that dom B _z dom A = |0}. For an arbitrary system of orthonormal vectors satisfying the condition span | _i } dom A = |0} and an arbitrary collection of real numbers , we construct an operator that solves the eigenvalue problem . We prove the uniqueness of under the condition that rank B _z = n.     

Spectrum of a Self-Adjoint Operator and Palais-Smale Conditions

The spectrum and essential spectrum of a self-adjoint operatorin a real Hilbert space are characterized in terms of Palais–Smaleconditions on its quadratic form and Rayleigh quotient respectively.     

Scalar Operators Representable as a Sum of Projectors

We study sets there exist n projectors P₁,...,P_n such that . We prove that if n 6, then .     

Point Spectrum of Singularly Perturbed Self-Adjoint Operators

We study the inverse spectral problem for the point spectrum of singularly perturbed self-adjoint operators. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 654–658, May, 2005.     

Continuity of the Spectrum of a Field of Self-Adjoint Operators

Given a family of self-adjoint operators \({(A_t)_{t \in T}}\) indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum \({\sigma(A_t)}\) to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T, d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.     

Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators

The Keldysh theorem is generalized to an arbitrary closed operator that is not necessarily close to self-adjoint operators and has a resolvent of Schatten–von Neumann class S _p . Based on this theorem, conditions of spectrum localization are obtained for certain classes of non-self-adjoint differential operators.     

A Class of Singular Self-Adjoint Ordinary Differential Operators with Maximal Spectrum

We present a characterization of the almost everywhere convergence of the partial Fourier series of functions in L^p(T), 1 < p < ∞, in terms of a discrete weak-type inequality.     

On the Decomposition of an Operator into a Sum of Four Idempotents

We prove that operators of the form (2 ± 2/n)I + K are decomposable into a sum of four idempotents for integer n > 1 if there exists the decomposition K = K ₁ K ₂ ... K _n, , of a compact operator K. We show that the decomposition of the compact operator 4I + K or the operator K into a sum of four idempotents can exist if K is finite-dimensional. If n trK is a sufficiently large (or sufficiently small) integer and K is finite-dimensional, then the operator (2 – 2/n)I + K [or (2 + 2/n)I + K] is a sum of four idempotents.     

On Some Product of Two Unbounded Self-Adjoint Operators

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).     

每个算子都是两个强不可约算子的和

本文证明了可分无穷维 Hilbert空间上每个有界线性算子均可写成两个强不可约算子之和 .这回答了文献 [9]中提出一个公开问题     

On the Defect Index of Quadratic Self-Adjoint Operator Pencils

Mathematical Notes -     

一类无穷维Hamilton算子的谱

研究了一类无穷维Hamilton算子的近似点谱及本质谱.进而通过无穷维Hamilton算子内部元素的乘积的谱对整体谱进行了刻画,最后证明了结论的正确性.     

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.     

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.     

Pseudodifferential Operators on Spaces of Distributions Associated with Non-negative Self-Adjoint Operators

We consider Hörmander type symbols on a family of spaces associated with non-negative self-adjoint operators, and we prove boundedness of the corresponding pseudodifferential operators on both classical and non-classical Besov and Triebel–Lizorkin spaces. Consequently, this also covers the case of Sobolev spaces. As an application, we obtain boundedness of spectral multipliers on the mentioned spaces.