On the fine spectra of the generalized rth difference operator Δνr on the sequence space ℓ1

摘要

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