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Statistical convergence on composite vector valued sequence space

100%

Basu A. , Srivastava P. D.

tom Vol. 29

75-90

In this paper, we have defined the idea of statistical convergence and statistically Cauchy sequence over the generalized class of composite vector valued sequence space F(Ek, f). The class F(Ek, f) is in-troduced and discussed by Ghosh and Srivastava [7], where F is a normal paranormed sequence space, Ek's are Banach spaces and f is a modulus function. We have established some results of Fridy, Connor and Rath and Tripathy, such as, decomposition of statistically convergent sequences, equivalence of statistical convergence and statistical Cauchy convergence and sequentially completeness of the space of bounded statistically con-vergent sequences of F [E, f].

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

100%

Panigrahi B. L. , Srivastava P. D.

2012

tom 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.