Spectral inclusion for unbounded diagonally dominant $n\times n$ operator matrices

This paper deals with the spectral inclusion properties of 2 x 2 operator matrices with unbounded entries in Hilbert space. The conditions for spectral inclusion by the quadratic numerical range are described. In addition, some examples are given to illustrate the main results.

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Spectral inclusion properties of some unbounded block operator matrices

Scientia Sinica Mathematica ◽

10.1360/n012013-00104 ◽

2014 ◽

Vol 44 (10) ◽

pp. 1099-1110

Author(s):

JunJie HUANG ◽

YaRu QI ◽

atancang Al

Keyword(s):

Operator Matrices ◽

Inclusion Properties ◽

Spectral Inclusion ◽

Block Operator ◽

Block Operator Matrices

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Local Spectral Properties Under Conjugations

Mediterranean Journal of Mathematics ◽

10.1007/s00009-021-01731-7 ◽

2021 ◽

Vol 18 (3) ◽

Author(s):

Pietro Aiena ◽

Fabio Burderi ◽

Salvatore Triolo

Keyword(s):

Hilbert Space ◽

Spectral Properties ◽

Operator Matrices ◽

Weyl Type Theorems ◽

Triangular Operator ◽

Analytic Core ◽

Complex Symmetric ◽

Symmetric Operators ◽

The Relationship

AbstractIn this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and $$T\in L(H)$$ T ∈ L ( H ) . We also study the relationship between the quasi-nilpotent part of the adjoint $$T^*$$ T ∗ and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form $$JT^*J$$ J T ∗ J . The theory is exemplified in some concrete cases.

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Polyhedral approximations of the semidefinite cone and their application

Computational Optimization and Applications ◽

10.1007/s10589-020-00255-2 ◽

2021 ◽

Author(s):

Yuzhu Wang ◽

Akihiro Tanaka ◽

Akiko Yoshise

Keyword(s):

Initial Approximation ◽

Semidefinite Relaxation ◽

Stable Set ◽

Cutting Plane Methods ◽

Diagonally Dominant Matrices ◽

Maximum Stable Set ◽

Simple Expansion ◽

Diagonally Dominant ◽

Polyhedral Approximations ◽

Semidefinite Cone

AbstractWe develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

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Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

Boundary Value Problems ◽

10.1186/s13661-019-01316-0 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Maozhu Zhang ◽

Kun Li ◽

Hongxiang Song

Keyword(s):

Boundary Conditions ◽

Operator Theory ◽

Resolvent Convergence ◽

Spectral Inclusion ◽

Regular Approximation ◽

Special Cases ◽

Abstract Operator ◽

Norm Resolvent Convergence ◽

Sturm Liouville ◽

Spectral Exactness

AbstractIn this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

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