Error Bounds for Low-Rank Approximations of the First Exponential Integral Kernel

A hierarchical matrix is an efficient data-sparse representation of a matrix, especially useful for large dimensional problems. It consists of low-rank subblocks leading to low memory requirements as well as inexpensive computational costs. In this work, we discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and, hence, it is weakly singular. We develop analytical expressions for the approximate degenerate kernels and deduce error upper bounds for these approximations. Some computational results illustrating the efficiency and robustness of the approach are presented.     

On a Continuous Third-Order Method for Monotone Operator Equations in Hilbert Space

A continuous third-order method has been constructed for a nonlinear operator equation in a Hilbert space with a strongly monotone operator satisfying the Lipshitz condition. Conditions sufficient for the strong convergence of the method have been derived.     

On the Spectrum of the Product of Closed Operators

In this note we study the connection between the spectra of the products AB and BA of unbounded closed operators A and B acting in Banach spaces. Under the condition that the resolvent sets of these products are not empty we show that the spectra of AB and BA coincide away from zero and prove the commutation relation 𝕀 . Further, we prove statements concerning the relationship between the spectra of the operator AB and the block operator matrix .     

Double Operator Integrals in a Hilbert Space

Double operator integrals are a convenient tool in many problems arising in the theory of self-adjoint operators, especially in the perturbation theory. They allow to give a precise meaning to operations with functions of two ordered operator-valued non-commuting arguments. In a different language, the theory of double operator integrals turns into the problem of scalarvalued multipliers for operator-valued kernels of integral operators.The paper gives a short survey of the main ideas, technical tools and results of the theory. Proofs are given only in the rare occasions, usually they are replaced by references to the original papers. Various applications are discussed.     

Hilbert空间上线性算子广义逆A_(T,S)~((2))的存在性及其表示式

设H1和H2是两个Hilbert空间,B(H1,H2)表示从H1到H2的所有有界线性算子的集合,T和S分别是H1和H2的两个闭子空间．如果存在线性算子X∈B(H2,H1)满足XAX=X,R(X)=T,N(X)=S,则称X为线性算子A的具有指定像空间T和零空间S的外逆,记为AT,S(2)．该文进一步研究了线性算子广义逆AT,S(2)存在的若干等价条件及其性质,建立了算子广义逆AT,S(2)的表示形式．     

Factorization of Operator Functions in a Hilbert Space

Let be an integer. A Fermat-Euler dynamical system acts on the set of mod- residues coprime to by multiplication by a constant (which is also coprime to ). We study the dependence of the period and the number of orbits of this dynamical system on . Theorems generalizing Fermat's little theorem, as well as empirical conjectures, are given.     

Hilbert空间L2[a,b]上m个微分算子积的自伴性

考虑 [a ,b]( -∞   

Hilbert空间L^2[α，6]上m个微分算子积的自伴性

考虑[a,b](-∞＜a＜b＜∞)上n阶复值系数正则对称微分算式ly＝∑n j＝0 aj(t)y(j).本文首先给出由lmy(m∈N且m≥2)生成的微分算子T(lm)自伴边条件一种新的描述,其次研究了由微分算式ly生成的m个微分算子Tk(l)(k=1,…,m)的积Tm(l)…T2(l)T1(l)的自伴性并获得其自伴的充分必要条件.     

On the fine spectra of triangular Toeplitz operators

The fine spectra of triangular double-band and triple-band matrices were examined by several authors. Here we determine the fine spectra of Toeplitz operators, which are represented by upper and lower triangular n-band infinite matrices, over the sequence spaces c₀ and c. Also some spectral results over ?_∞ are given.     

一类再生Hilbert空间上偏微分算子的有界性(英文)

研究了一类再生Hilbert空间上偏微分算子的有界性,得到了偏微分算子有界的一个充分必要条件,推广了文献[1]中的结果.     

Inner bounds for the spectrum of quasinormal operators

A linear operator in a separable Hilbert space is called a quasinormal one if it is a sum of a normal operator and a compact one. In the paper, bounds for the spectrum of quasinormal operators are established. In addition, the lower estimate for the spectral radius is derived. Under some restrictions, that estimate improves the well-known results. Applications to integral operators and matrices are discussed. Our results are new even in the finite-dimensional case.

     

一类再生Hilbert空间上偏微分算子的有界性

研究了一类再生Hilbert空间上偏微分算子的有界性,得到了偏微分算子有界的一个充分必要条件,推广了文献中的结果．     

Hilbert空间上两个正交射影的乘积

该文研究Hilbert空间H上正则射影对(P,Q)的性质和结构,给出H上有界线性算子A表示为两个正交射影乘积的充分必要条件.     

缺项算子矩阵的本性谱

Let =(A C X B)be a 2×2 operator matrix acting on the Hilbert space н( )κ.For given A ∈B (H),B ∈B(K)and C ∈B(K,H)the set Ux∈B(H,к)σe(Mx)is determined,where σe(T)denotes the essential spectrum.     

On the Convergence of an Interpolation Process to an Entire Operator in a Hilbert Space

For an entire operator in a Hilbert space, we consider an interpolation operator-valued polynomial that has a fairly simple structure. A theorem on pointwise convergence of the iteration process to the entire operator is proved.     

On Fixed Point Index of a Gradient Operator and Applications

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On lifting of operators to Hilbert spaces induced by positive selfadjoint operators

We introduce the notion of induced Hilbert spaces for positive unbounded operators and show that the energy spaces associated to several classical boundary value problems for partial differential operators are relevant examples of this type. The main result is a generalization of the Krein-Reid lifting theorem to this unbounded case and we indicate how it provides estimates of the spectra of operators with respect to energy spaces.