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 point spectrum of ℋ–2-singular perturbations

2007 ◽  
Vol 280 (1-2) ◽  
pp. 20-27 ◽  
Author(s):  
Sergio Albeverio ◽  
Mykola Dudkin ◽  
Alexei Konstantinov ◽  
Volodymyr Koshmanenko
Keyword(s):  
Singular Perturbations ◽  
Point Spectrum

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.

scholarly journals On symmetries in the theory of finite rank singular perturbations

2009 ◽  
Vol 256 (3) ◽  
pp. 777-809 ◽  
Author(s):  
Seppo Hassi ◽  
Sergii Kuzhel
Keyword(s):  
Singular Perturbations ◽  
Finite Rank

PROPERTY (Bgw) AND PERTURBATIONS

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
M.H.M. Rashid ◽  
T. Prasad
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Point Spectrum ◽  
Space Operator ◽  
Approximate Point Spectrum ◽  
Finite Rank Operators ◽  
Riesz Operators ◽  
Nilpotent Operators ◽  
The Stability ◽  
Isolated Eigenvalues

AbstractA Banach space operator T satisfies property (Bgw) if the complement in the approximate point spectrum σa(T) of the semi-B-essential approximate point spectrum σSHF+-(T) coincides with the set of isolated eigenvalues of T of Unite multiplicity E°(T). We find conditions for Banach Space operator tosatfafy the property (Bgw). We also study the stability of property (Bgw) under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators and by Riesz operators commuting with T.

scholarly journals On some extensions of the A-model

2020 ◽  
Vol 40 (5) ◽  
pp. 569-597
Author(s):  
Rytis Juršėnas
Keyword(s):  
Singular Perturbations ◽  
Hilbert Spaces ◽  
Finite Rank ◽  
Orthogonal Decomposition

The A-model for finite rank singular perturbations of class \(\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}\), \(m \in \mathbb{N}\), is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces \((\mathfrak{H}_n)_{n\in\mathbb{Z}}\) admit an orthogonal decomposition \(\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n\), with the corresponding projections satisfying \(P^{\pm}_{n+1}\subseteq P^{\pm}_n\), nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.

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