On compact perturbations of finite-zone Jacobi operators

2010 ◽  
Vol 165 (4) ◽  
pp. 473-482
Author(s):  
A. A. Kononova
Keyword(s):  
Jacobi Operators ◽  
Compact Perturbations

scholarly journals Spectra of Jacobi Operators via Connection Coefficient Matrices

2021 ◽  
Vol 382 (2) ◽  
pp. 657-707
Author(s):  
Marcus Webb ◽  
Sheehan Olver
Keyword(s):  
Error Control ◽  
Jacobi Operator ◽  
Asymptotic Properties ◽  
Hausdorff Metric ◽  
Trace Class ◽  
Coefficient Matrix ◽  
Jacobi Operators ◽  
Root Finding ◽  
Compact Perturbations ◽  
Connection Coefficient

AbstractWe address the computational spectral theory of Jacobi operators that are compact perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In particular, for Jacobi operators that are finite-rank perturbations we show that the computation of the spectrum can be reduced to a polynomial root finding problem, from a polynomial that is derived explicitly from the entries of a connection coefficient matrix. A formula for the spectral measure of the operator is also derived explicitly from these entries. The analysis is extended to trace-class perturbations. We address issues of computability in the framework of the Solvability Complexity Index, proving that the spectrum of compact perturbations of the free Jacobi operator is computable in finite time with guaranteed error control in the Hausdorff metric on sets.

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

2003 ◽  
Vol 46 (3) ◽  
pp. 575-595 ◽  
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

scholarly journals On a question of Olsen concerning compact perturbations of operators.

1977 ◽  
Vol 24 (1) ◽  
pp. 119-127 ◽  
Author(s):  
C. K. Chui ◽  
D. A. Legg ◽  
P. W. Smith ◽  
J. D. Ward
Keyword(s):  
Compact Perturbations

Coburn Type Operators and Compact Perturbations

2021 ◽  
Vol 42 (5) ◽  
pp. 677-692
Author(s):  
Tingting Zhou ◽  
Bin Liang ◽  
Chaoyue Wang
Keyword(s):  
Compact Perturbations

Variation of the Point Spectrum under Compact Perturbations

1988 ◽  
pp. 113-158 ◽  
Author(s):  
Domingo A. Herrero ◽  
Thomas J. Taylor ◽  
Zong Y. Wang
Keyword(s):  
Point Spectrum ◽  
Compact Perturbations
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