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Spectrum and fine spectrum of generalized second order forward difference operator ∆2uvw on sequence space l1

Panigrahi B. L., Srivastava P. D.

Demonstratio Mathematica

2012

Vol. 45, nr 3

593-609

The purpose of this paper is to determine spectrum and fine spectrum of newly introduced operator ∆²uvw on sequence space l1. The operator ∆²uvw on sequence space l1 is defined by ∆²uvw x= (unxn + vn−1 xn−1 + wn−2 xn−2) ∞ n=0 with x−1, x−2 = 0, where x = (xn) ∈l1 , u= (uk) is either constant or strictly increasing sequence of positive real numbers with U = lim k→∞ u k, v = (vk) is a sequence of real numbers such that vk ≠ 0 for each k∈N0 with V = lim k →∞ vk ≠ 0 and w = (wk) is a non-increasing sequence of positive real numbers such that wk ≠ 0 for each k∈N0 with W = lim k→∞ wk ≠ 0. In this paper we have obtained the results on spectrum and point spectrum for the operator ∆²uvw over sequence space l1. We have also obtained the results on continuous spectrum σc(∆²uvw, l1), residual spectrum σr(∆²uvw, l1) and fine spectrum of the operator ∆²uvw on sequence space l1.