# Spectrum and fine spectrum of generalised lower triangular triple band matrices over the sequence space $l_p$

## TL;DR

This paper investigates the spectrum and fine spectrum of a new generalized difference operator represented by a lower triangular triple band matrix with periodic sequences over the sequence space l_p, extending previous studies on band matrices.

## Contribution

It introduces a more general class of lower triangular triple band matrices with periodic sequences and analyzes their spectral properties on l_p spaces.

## Key findings

- Spectrum and fine spectrum characterized for the new operator
- Includes approximate point, defect, and compression spectra
- Provides examples and discusses special cases

## Abstract

The spectrum of triangular band matrices defined on the sequence spaces where the entries of each band is a constant or convergent sequence is well studied. In this article, the spectrum and fine spectrum of a new generalised difference operator defined by a lower triangular triple band matrix on the sequence space $l_p (1 \leq p < \infty)$ are obtained where the bands are considered as periodic sequences. The approximate point spectrum, defect spectrum, compression spectrum and the Goldberg classification of the spectrum are also discussed. Suitable examples are given in order to supplement the results. Several special cases of our findings are discussed which confirm that our study is more general and extensive.