NOTES ON THE SPECTRAL PROPERTIES OF THE WEIGHTED MEAN DIFFERENCE OPERATOR G(u,v;?) OVER THE SEQUENCE SPACE ?_1

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; ?) over the sequence space ?_1. The product operator G(u, v; ?) over ?_1 is defined by(G(u,v; ?) x)_k=k∑i=1u_kv_i(x_i-x_(i-1)) with x_k = 0 for all k 0, where x =(x_k) ∈ ?_1,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; ?) on the sequence space ?_1.