On the fine spectrum of the operator [formula omitted] over the sequence spaces [formula omitted] and [formula omitted

Abstract

We determine the fine spectrum of the operator B ( r , s , t ) defined by a band matrix over the sequence spaces ℓ 1 and b v . This generalizes the spectrum of the second order difference operator Δ 2 and includes other special cases such as the generalized difference operator B ( r , s ) in [H. Furkan, H. Bilgiç, K. Kayaduman, On the fine spectrum of the generalized difference operator B ( r , s ) over the sequence spaces ℓ 1 and b v , Hokkaido Math. J. (in press)], the difference operator Δ in [K. Kayaduman, H. Furkan, The fine spectra of the difference operator Δ over the sequence spaces ℓ 1 and b v , Int. Math. Forum 1 (21–24) (2006) 1153–1160], the right shift operator and Zweier matrices.