On the fine spectrum of the generalized difference operator $B(r,s)$ over the sequence space $\ell_1$ and $bv$

Abstract

The main purpose of this paper is to determine the fine spectrum of the operator $B(r,s)$ defined by a band matrix over the sequence spaces $\ell_1$ and $bv$. Since the operator $B(r,s)$ is reduced to the right-shift, difference and Zweier matrices in special cases $(r,s)=(0,1),(r,s)=(1,-1) $ and $(r,s)=(r,1-r)$, respectively, our results are more general than the corresponding results in the existing literature.