On the Spectral Properties of the Weighted Mean Difference OperatorGu,v;Δover the Sequence Spaceℓ1

摘要

In the present work the generalized weighted mean difference operatorGu,v;Δhas been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequencesu=(uk)andv=(vk). For any two sequencesuandvof either constant or strictly decreasing real numbers satisfying certain conditions the difference operatorGu,v;Δis defined by(G(u,v;Δ)x)k=∑i=0k‍ukvi(xi-xi-1)withxk=0for allk<0. Furthermore, we compute the spectrum and the fine spectrum of the operatorGu,v;Δover the sequence spacel1. In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence spacel1.