Spectrum and Fine Spectrum of Band Matrices Generated by Oscillatory Sequences

TL;DR

This work analyzes the spectrum and fine spectrum of a class of oscillatory, non-constant-band matrices acting on $\ell_p$ spaces via a compact perturbation framework. The limit operator $T_0$ is shown to have a two-interval spectrum $\sigma(T_0,\ell_p) = [r_1-2s_1, r_1+2s_1] \cup [r_2-2s_2, r_2+2s_2]$ which equals its essential and continuous spectra, while $T = T_0 + K$ with compact $K$ may introduce a finite (or countable) set of eigenvalues $S_1$ outside this union. The paper provides sufficient conditions, based on the rate of convergence of the generating sequences and transfer-matrix criteria, for the absence of point spectrum on the essential spectrum, and shows that under exponential convergence every eigenvalue is of finite type and lies in $S_1$. Consequently, the full spectrum of $T$ is the union of the two intervals and $S_1$, with a complete description of the point, residual, continuous, and essential spectra, clarifying how non-constant bands affect spectral properties on $\ell_p$ spaces.

Abstract

In this paper, a new class of band matrices is considered where the entries of each non-zero band form a sequence with two limit points. The compact perturbation technique is used to study the spectrum over the $\ell_{p}, (1<p<\infty)$ sequence space. Several spectral subdivisions such as fine spectrum, discrete spectrum, essential spectrum, etc. are obtained. In addition, a few sufficient conditions on the absence of point spectrum over the essential spectrum are also discussed.

Theorems & Definitions (39)

• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Corollary 3.4