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

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 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.