Subdivisions of the Spectra for Generalized Difference Operator over Certain Sequence Spaces

Feyzi Basar, Nuh Durna, Mustafa Yildirim

Abstract

The fine spectrum of the generalized difference operator B(r, s) represented by a double band matrix over the sequence spaces c0 and c was studied by Altay and Basar [B. Altay, F. Basar, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c0 and c, Int. J. Math. Math. Sci. 2005 (18) (2005) 3005–3013], over the sequence spaces ℓp and bvp was worked by Bilgic and Furkan [H. Bilgic, H. Furkan, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces ℓp and bvp, (1 < p < ∞), Nonlinear Anal. 68 (3) (2008) 499–506]; where bvp denotes the space of all sequences (xk) such that (xk − xk−1) in ℓp with 1 < p < ∞ and was studied by Basar and Altay [F. Basar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55 (1) (2003) 136–147]. In this paper, the approximate point spectrum, defect spectrum and compression spectrum of the generalized difference operator B(r, s) over the sequence spaces c0, c, ℓp and bvp have been determined.