On the spectrum of fractional difference operator

Abstract

This work mainly explores the idea of the spectrum of the fractional difference operator (of order ) over the Banach space ℓ1. The difference operator via the sequence x = (x k ) is defined by where (α) i denotes the Pochhammer symbol, and ν = (ν k ) is either a constant or strictly decreasing sequence of positive real numbers with some conditions. Sets comprising of the point, the residual and the continuous spectrum of the operator on the Banach space ℓ1 are computed.