On the fine spectra of the generalized r th difference operator Δνr on the sequence space ℓ1ℓ1

The main purpose of this paper is to determine the spectrum and the fine spectrum of the r  th difference operator Δνr on sequence space ℓ1ℓ1. The operator Δνr over ℓ1ℓ1 is defined by Δνr(x)=(Δνrxk)k=0∞=(Δνr-1xk-Δνr-1xk-1)k=0∞ with xk=νk=0xk=νk=0 for k<0k<0, where x=(xk)∈ℓ1,r∈N and ν=(νk)ν=(νk) is either constant or strictly decreasing sequence of positive real numbers satisfying certain conditions.