On the fine spectrum of the second order difference operator over the sequence spaces ℓpℓp and bvp,(1<p<∞)

In general, the behaviors of the symmetric tri-band matrices on the Hilbert spaces are well known. But the symmetric tri-band matrices have different behavior on the Banach spaces. The main purpose of this work is to determine the fine spectra of the operator U(s,r,s)U(s,r,s) defined by symmetric tri-band matrix over the sequence spaces ℓpℓp and bvpbvp.