Some Spectral Aspects of the Operator Δνr over the Sequence Spaces ℓp and bvp (1

Abstract EN

The main idea of the present paper is to compute the spectrum and the fine spectrum of the generalized difference operator Δνr over the sequence spaces bvp (1

The operator Δνr denotes a triangular sequential band matrix (ank) defined by Δνr(x)=(Δνrx)k=∑i=0r(-1)i(ri)νk-ixk-i, with xk=νk=0 for k<0, where x∈ℓp or bvp, r,k∈ℕ0={0,1,2,3,...}; the set nonnegative integers and ν=(νk) is either a constant or strictly decreasing sequence of positive real numbers satisfying certain conditions.

Finally, we obtain the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of the operator Δνr over the sequence spaces ℓp and bvp.

These results are more general and comprehensive than the spectrum of the difference operators Δ, Δν, Δ2, Δuv2, and Δr and include some other special cases such as the spectrum of the operators B(r,s), B(r,s,t), and Δab over the sequence spaces ℓp or bvp (1