Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means

Abstract

a b s t r a c tIn the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators on some sequence spaces of weighted means. Furthermore, by using the Hausdorff measure of noncompactness, we apply our results to characterize some classes of compact operators on those spaces. Background, notation and preliminariesIt seems to be quite natural, in view of the fact that matrix operators between BK -spaces are continuous, to find necessary and sufficient conditions for the entries of an infinite matrix to define a compact operator between such spaces. This can be achieved in many cases by applying the Hausdorff measure of noncompactness. In this section, we give some related definitions, notation and preliminary results. Compact operators and matrix transformationsLet X be a normed space. Then, we write S X for the unit sphere in X , that is, S X = {x ∈ X : x = 1}. If X and Y are Banach spaces, then B(X, Y ) denotes the set of all bounded (continuous) linear operators L : X → Y , which is a Banach space with the operator norm given by L = sup x∈S X L(x) Y for all L ∈ B(X, Y ). A linear operator L : X → Y is said to be compact if the domain of L is all of X and for every bounded sequence (x n ) in X , the sequence (L(x n )) has a subsequence which converges in Y . We write C(X, Y ) for the class of all compact operators in B(X, Y ). An operator L ∈ B(X, Y ) is said to be of finite rank if dim R(L) < ∞, where R(L) is the range space of L. An operator of finite rank is clearly compact.By w, we shall denote the space of all complex sequences. If x ∈ w, then we writewe write φ for the set of all finite sequences that terminate in zeros. Further, we use the conventions that e = (1, 1, . . .) and e (k) is the sequence whose only non-zero term is 1 in the kth place for each k ∈ N, where N = {0, 1, 2, . . .}. Any vector subspace of w is called a sequence space. We shall write ∞ , c and c 0 for the sequence spaces of all bounded, convergent and null sequences, respectively. Further, by 1 and p (1 < p < ∞), we denote the sequence spaces of all absolutely and p-absolutely convergent series, respectively. Moreover, we write bs, cs and cs 0 for the sequence spaces of all bounded, convergent and null series, respectively.

On B(m)${B^{(m)}}$-difference sequence spaces using generalized means and compact operators

“…provided the expression on the right side exists and is nite, which is the case whenever ∈ (see [21]). Let The function : M → [0, ∞) is called the Hausdor measure of noncompactness.…”

“…This can be achieved with the help of Hausdor measure of noncompactness. Recently several authors, among them Malkowsky and Rakočević [17], Djolović [11], Mursaleen and Noman [21,22,25], Başarir and Kara [5][6][7], Mohiuddine et al [20], have established some identities or estimates for the operator norms and the Hausdor measure of noncompactness of matrix operators from an arbitrary space to an arbitrary space. Let us recall some de nitions and well-known results.…”

Some $m$th-order Difference Sequence Spaces of Generalized Means and Compact Operators

“…This can be achieved with the help of Hausdorff measure of noncompactness. Recently several authors, namely, Malkowsky and Rakočević [13], Dojolović et al [9], Dojolović [7], Mursaleen and Noman ([19], [18]), Başarir and Kara [5] etc. have established some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of matrix operators from an arbitrary BK space to arbitrary BK space.…”

“…Lemma 5.1. [18] Let X denote any of the sequence spaces c 0 or l ∞ . If A ∈ (X, c), then we have (i) α k = lim n→∞ a nk exists for all k ∈ N 0 ,…”

Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces

“…A A refer to: [36,38]. In other words, we emphasize on the spaces` 1 , c , c 0 and` p of bounded, convergent, null and absolutely p summable sequences, respectively, that is to say that…”