On the fine spectrum of the operator B(r,s,t)B(r,s,t) over the sequence spaces ℓ1ℓ1 and bvbv

We determine the fine spectrum of the operator B(r,s,t)B(r,s,t) defined by a band matrix over the sequence spaces ℓ1ℓ1 and bvbv. This generalizes the spectrum of the second order difference operator Δ2Δ2 and includes other special cases such as the generalized difference operator B(r,s)B(r,s) in [H. Furkan, H. Bilgiç, K. Kayaduman, On the fine spectrum of the generalized difference operator B(r,s)B(r,s) over the sequence spaces ℓ1ℓ1 and bvbv, Hokkaido Math. J. (in press)], the difference operator ΔΔ in [K. Kayaduman, H. Furkan, The fine spectra of the difference operator ΔΔ over the sequence spaces ℓ1ℓ1 and bvbv, Int. Math. Forum 1 (21–24) (2006) 1153–1160], the right shift operator and Zweier matrices.