Spectrum and Fine Spectrumof Generalized Second Order Difference Operator ∆2uv on Sequence Space c0

Bijaya Laxmi Panigrahi, Parmeshwary Dayal Srivastava

Abstract

The purpose of this paper is to determine spectrum and fine spectrumof newly introduced operator ∆2uv on sequence space c0. The operator ∆2uv on sequence space c0 is defined by ∆2uv x= (unxn−vn−1xn−1+un−2xn−2)∞n=0withx−1, x−2= 0,wherex= (xn)∈c0,u= (uk) is a either constant or strictlydecreasing sequence of positive real numbers withU= limk→∞uk6= 0,v= (vk)is a sequence of positive real numbers such thatvk6= 0 for eachk∈N0withV= limk→∞vk6= 0. In this paper we have obtained the results on spectrum andpoint spectrum for the operator ∆2uv over sequence space c0. We have also obtainedthe results on continuous spectrumσc(∆2uv, c0), residual spectrumσr(∆2uv, c0) andfine spectrum of the operator ∆2uv on sequence space c0.