Notes on the spectral properties of the weighted mean difference operator G (u, v; δ) over the sequence space ℓ1

Abstract

In the study by Baliarsingh and Dutta [Internat. J.Anal., Vol.2014(2014), Article ID 786437], the authors computed the spectrum and the fine spectrum of the product operator G (u, v; δ) over the sequence space ℓ1. The product operator G (u, v; δ) over ℓ1 is defined by (G(u,v; δ)x)k=∑i=0kukvi(xi-xi-1) with xk = 0 for all k < 0, where x = (xk) ∈ ℓ1, and u and v are either constant or strictly decreasing sequences of positive real numbers satisfying certain conditions. In this article we give some improvements of the computation of the spectrum of the operator G (u, v; δ) on the sequence space ℓ1. © 2016 Wuhan Institute of Physics and Mathematics.