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

[摘要] In this paper spectral properties of non-selfadjoint Jacobi operators $J$ which are compact perturbations of the operator $J_0=S+ho S^*$, where $hoin(0,1)$ 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. $pinell^2$ and$$ varlimsup_{nightarrowinfty}|p(n)|^{1/n}leho^{1/2}. $$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

[学科分类] 数学（综合）

[关键词] non-selfadjoint Jacobi matrix;one-dimensional perturbation;point spectrum;essential spectrum