On the Spectral Properties of the Weighted Mean Difference Operator Gu,v;Δ over the Sequence Space ℓ1

Abstract

In the present work the generalized weighted mean difference operator Gu,v;Δ has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences u=(uk) and v=(vk). For any two sequences u and v of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator Gu,v;Δ is defined by (G(u,v;Δ)x)k=∑i=0k‍ukvi(xi-xi-1) with xk=0 for all k<0. Furthermore, we compute the spectrum and the fine spectrum of the operator Gu,v;Δ over the sequence space l1. In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space l1.