Spectrum and Fine Spectrum of Band Matrices Generated by Oscillatory Sequences

TL;DR

This paper investigates the spectral properties of a new class of band matrices with entries forming oscillatory sequences, providing detailed spectral subdivisions and conditions for the absence of point spectrum.

Contribution

It introduces a novel class of band matrices with oscillatory sequence entries and analyzes their spectrum using compact perturbation techniques.

Findings

Spectral subdivisions such as fine, discrete, and essential spectrum are characterized.

Conditions for the absence of point spectrum over the essential spectrum are established.

The spectrum is studied over the  sequence space for these matrices.

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.