Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means

Abstract

For a sequence x = (x(k)), we denote the difference sequence by Delta x = (xk x(k-1)). Let u = (u(k))(k=0)(infinity) and v = (v(k))(k=0)(infinity) be the sequences of real numbers such that u(k) not equal 0, vk not equal 0 for all k is an element of N. The difference sequence spaces of weighted means lambda(u, v, Delta) are defined as lambda(u, v, Delta) = {x = (x(k)) : W(x) is an element of lambda}, where lambda = c, c(0) and l(infinity) and the matrix W = (w(nk)) is defined by w(nk) = {u(n) (v(k) - v(k+1)); (k n) In this paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on lambda(u, v, Delta). Further, we characterize some classes of compact operators on these spaces by using the Hausdorff measure of noncompactness. (C) 2011 Elsevier Ltd. All rights reserved.