Instability of Dense Point Spectrum Under Finite Rank Perturbations

1997 ◽  
Vol 187 (3) ◽  
pp. 583-595 ◽  
Author(s):  
A.Y. Gordon
Keyword(s):  
Finite Rank ◽  
Point Spectrum ◽  
Dense Point

scholarly journals A Note on Property(gb)and Perturbations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Qingping Zeng ◽  
Huaijie Zhong
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Point Spectrum ◽  
Bounded Linear ◽  
Approximate Point Spectrum ◽  
Weyl Spectrum ◽  
Nilpotent Operators ◽  
The Stability ◽  
Quasinilpotent Operators ◽  
Study Property

An operatorT∈ℬ(X)defined on a Banach spaceXsatisfies property(gb)if the complement in the approximate point spectrumσa(T)of the upper semi-B-Weyl spectrumσSBF+-(T)coincides with the setΠ(T)of all poles of the resolvent ofT. In this paper, we continue to study property(gb)and the stability of it, for a bounded linear operatorTacting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting withT. Two counterexamples show that property(gb)in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.

SINGULARLY FINITE RANK NONSYMMETRIC PERTURBATIONS ${\mathcal H}_{-2}$-CLASS OF A SELF-ADJOINT OPERATOR

2021 ◽  
Vol 9 (1) ◽  
pp. 140-151
Author(s):  
O. Dyuzhenkova ◽  
M. Dudkin
Keyword(s):  
Finite Rank ◽  
Scalar Product ◽  
Separable Hilbert Space ◽  
Point Spectrum ◽  
Adjoint Operator ◽  
Formal Expression ◽  
Rank One Perturbation ◽  
Rank One ◽  
Negative Space ◽  
First Time

The singular nonsymmetric rank one perturbation of a self-adjoint operator from classes ${\mathcal H}_{-1}$ and ${\mathcal H}_{-2}$ was considered for the first time in works by Dudkin M.E. and Vdovenko T.I. \cite{k8,k9}. In the mentioned papers, some properties of the point spectrum are described, which occur during such perturbations. This paper proposes generalizations of the results presented in \cite{k8,k9} and \cite{k2} in the case of nonsymmetric class ${\mathcal H}_{-2}$ perturbations of finite rank. That is, the formal expression of the following is considered \begin{equation*} \tilde A=A+\sum \limits_{j=1}^{n}\alpha_j\langle\cdot,\omega_j\rangle\delta_j, \end{equation*} where $A$ is an unperturbed self-adjoint operator on a separable Hilbert space ${\mathcal H}$, $\alpha_j\in{\mathbb C}$, $\omega_j$, $\delta_j$, $j=1,2, ..., n<\infty$ are vectors from the negative space ${\mathcal H}_{-2}$ constructed by the operator $A$, $\langle\cdot,\cdot\rangle$ is the dual scalar product between positive and negative spaces.

Singular perturbations of finite rank. Point spectrum

1997 ◽  
Vol 49 (9) ◽  
pp. 1335-1344 ◽  
Author(s):  
V. D. Koshmanenko ◽  
O. V. Samoilenko
Keyword(s):  
Singular Perturbations ◽  
Finite Rank ◽  
Point Spectrum

On the Embedded Eigenvalues and Dense Point Spectrum of the Stark-Like Hamiltonians

1997 ◽  
Vol 183 (1) ◽  
pp. 185-200 ◽  
Author(s):  
Sergei N. Naboko ◽  
Alexander B. Pushnitski
Keyword(s):  
Point Spectrum ◽  
Dense Point ◽  
Embedded Eigenvalues

Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential

1996 ◽  
Vol 126 (5) ◽  
pp. 1087-1096 ◽  
Author(s):  
Karl Michael Schmidt
Keyword(s):  
Dirac Operator ◽  
Essential Spectrum ◽  
Electrostatic Potential ◽  
Simple Proof ◽  
Point Spectrum ◽  
Electrostatic Potentials ◽  
One Dimensional ◽  
Decay Behaviour ◽  
Dense Point ◽  
The One

For the one-dimensional Dirac operator, examples of electrostatic potentials with decay behaviour arbitrarily close to Coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum. A simple proof that the essential spectrum of one-dimensional Dirac operators with electrostatic potentials is never empty is given in the appendix.

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