Variation of the Point Spectrum under Compact Perturbations

Topics in Operator Theory - Operator Theory: Advances and Applications ◽

10.1007/978-3-0348-5475-7_9 ◽

1988 ◽

pp. 113-158 ◽

Cited By ~ 9

Author(s):

Domingo A. Herrero ◽

Thomas J. Taylor ◽

Zong Y. Wang

Keyword(s):

Point Spectrum ◽

Compact Perturbations

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Approximate point spectrum and commuting compact perturbations

Glasgow Mathematical Journal ◽

10.1017/s0017089500006509 ◽

1986 ◽

Vol 28 (2) ◽

pp. 193-198 ◽

Cited By ~ 32

Author(s):

Vladimir Rakočević

Keyword(s):

Banach Space ◽

Essential Spectrum ◽

Point Spectrum ◽

Complex Banach Space ◽

Linear Operators ◽

Complex Numbers ◽

Approximate Point Spectrum ◽

Infinite Dimensional ◽

Compact Perturbations ◽

Image Position

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B (X) (K(X)). Let σ(A) and σa(A) denote, respectively, the spectrum and approximate point spectrum of an element A of B(X). Setσem(A)and σeb(A) are respectively Schechter's and Browder's essential spectrum of A ([16], [9]). σea (A) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([13], [14]). In this note we characterize σab(A) and show that if f is a function analytic in a neighborhood of σ(A), then σab(f(A)) = f(σab(A)). The relation between σa(A) and σeb(A), that is exhibited in this paper, resembles the relation between the σ(A) and the σeb(A), and it is reasonable to call σab(A) Browder's essential approximate point spectrum of A.

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SIMILARITY AND THE POINT SPECTRUM OF SOME NON-SELFADJOINT JACOBI MATRICES

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091502000925 ◽

2003 ◽

Vol 46 (3) ◽

pp. 575-595 ◽

Cited By ~ 1

Author(s):

Jan Janas ◽

Maria Malejki ◽

Yaroslav Mykytyuk

Keyword(s):

Point Spectrum ◽

Trace Class ◽

Jacobi Matrices ◽

Secondary 47B37 ◽

Unilateral Shift ◽

Jacobi Operators ◽

Rank One Perturbation ◽

Compact Perturbations ◽

Primary 47B36 ◽

Rank One

AbstractIn this paper spectral properties of non-selfadjoint Jacobi operators $J$ which are compact perturbations of the operator $J_0=S+\rho S^*$, where $\rho\in(0,1)$ and $S$ is the unilateral shift operator in $\ell^2$, are studied. In the case where $J-J_0$ is in the trace class, Friedrichs’s ideas are used to prove similarity of $J$ to the rank one perturbation $T$ of $J_0$, i.e. $T=J_0+(\cdot,p)e_1$. Moreover, it is shown that the perturbation is of ‘smooth type’, i.e. $p\in\ell^2$ and$$ \varlimsup_{n\rightarrow\infty}|p(n)|^{1/n}\le\rho^{1/2}. $$When $J-J_0$ is not in the trace class, the Friedrichs method does not work and the transfer matrix approach is used. Finally, the point spectrum of a special class of Jacobi matrices (introduced by Atzmon and Sodin) is investigated.AMS 2000 Mathematics subject classification: Primary 47B36. Secondary 47B37

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