On the fine spectrum of the second order difference operator over the sequence spaces \ell_p and bv_p, (1 < p < \infty)

TL;DR

This paper investigates the detailed spectral properties of a specific second order difference operator represented by symmetric tri-band matrices on sequence spaces _p and bv_p, extending known results from Hilbert spaces to Banach spaces.

Contribution

It determines the fine spectra of the symmetric tri-band matrix operator U(s, r, s) on _p and bv_p spaces, which was not previously established.

Findings

Spectral properties of U(s, r, s) are characterized on _p and bv_p.

Differences in behavior of tri-band matrices between Hilbert and Banach spaces are analyzed.

The work extends spectral theory of difference operators to broader sequence spaces.

Abstract

In general, it is well known the behaviors of the symmetric tri-band matrices on the Hilbert spaces. But the symmetric tri-band matrices have different the behavior on the Banach spaces. The main purpose of this work is to determine the fine spectra of the operator U(s, r, s) defined by symmetric tri-band matrix over the sequence spaces {\ell}_p and bvp.