On the spectral properties of the weighted mean difference operator \(G(u,v;\Delta)\) over the sequence space \(\ell_1\) (Q1637856)

Summary: In the present work, the generalized weighted mean difference operator \(G(u,v;\Delta)\) is introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences \(u=(u_k)\) and \(v=(v_k)\). For any two sequences \(u\) and \(v\) of either constant or strictly decreasing real numbers satisfying certain conditions, the difference operator \(G(u,v;\Delta)\) is defined by \[ (G(u,v;\Delta)x)_k = \sum^k_{i=0}u_kv_i(x_i -x_{i-1}) \] with \(x_k=0\) for all \(k<0\). Furthermore, we compute the spectrum and the fine spectrum of the operator \(G(u,v;\Delta)\) over the sequence space \(\ell_1\). In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space \(\ell_1\).