On the Spectral Properties of the Weighted Mean Difference Operator over the Sequence Space

In the present work the generalized weighted mean difference operator has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences and . For any two sequences and of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator is defined by with for all . Furthermore, we compute the spectrum and the fine spectrum of the operator over the sequence space . In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space . 1. Introduction, Preliminaries, and Definitions Let and be two bounded sequences of either constant or strictly decreasing positive real numbers such that and for all , and By and , we denote the spaces of all absolutely summable and p-bounded variation series, respectively. Also, by , , and , we denote the spaces of all bounded, convergent, and null sequences, respectively. The main perception of this paper is to introduce the weighted mean difference operator as follows. Let be any sequence in , and we define the weighted mean difference transform of by where denotes the set of nonnegative integers and we assume throughout that any term with negative subscript is zero. Instead of writing (3), the operator can be expressed as a lower triangular matrix , where Equivalently, in componentwise the triangle can be represented by The main objective of this paper is to determine the spectrum of the operator over the basic sequence space . The operator has been studied by Polat et al. [1] in detail by introducing the difference sequence spaces , , and . In the existing literature several researchers have been actively engaged in finding the spectrum and fine spectrum of different bounded linear operators over various sequence spaces. The spectrum of weighted mean operator has been studied by Rhoades [2], whereas that of the difference operator over the sequence spaces for and , has been studied by Altay and Ba？ar [3, 4]. Kayaduman and Furkan [5] have determined the fine spectrum of the difference operator over the sequence spaces and and on generalizing these results, Srivastava and Kumar [6, 7] have determined the fine spectrum of the operator over the sequence spaces and , where is a sequence of either constant or strictly deceasing sequence of reals satisfying certain conditions. Dutta and Baliarsingh [8–10] have computed the spectrum of the operator ( ) and over the sequence spaces , , and , respectively. The fine spectrum of the generalized