Spectra of the lower triangular matrix $\mathbb{B}(r_1,\dots , r_l; s_1, \dots, s_{l'})$ over $c_0$

TL;DR

This paper determines the spectra and fine spectra of a specific lower triangular matrix with oscillatory sequences over the sequence space c0, providing detailed spectral analysis for particular cases.

Contribution

It introduces a detailed spectral analysis of a class of lower triangular matrices with oscillatory sequences over c0, including explicit spectra for specific parameter choices.

Findings

Spectra and fine spectra are explicitly characterized.

Spectral properties depend on the oscillatory sequences.

Results include specific cases with detailed spectra.

Abstract

The spectra and fine spectra of the lower triangular matrix $\mathbb{B}$ $(r_1,\dots , r_l;$ $ s_1, \dots, s_{l'})$ over the sequence space $c_0$ are determined. The diagonal and sub-diagonal entries of the matrix consist of two oscillatory sequences $r=(r_{k (\text{mod} \ l)+1})$ and $s= (s_{k(\text{mod} \ l')+1})$ respectively, whereas the rest of the entries of the matrix are zero. In particular, the spectra and fine spectra of the lower triangular matrix $\mathbb{B}(r_1,\dots , r_4; s_1, \dots, s_{6})$ over $c_0$ are discussed.