The sequence spaces l ∞ (B, p), c (B, p), and c 0 (B, p) of non-absolute type derived by the double sequential band matrix B(r,s) have recently been defined. In this work, we establish identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on these spaces that are paranormed spaces. Further, we find the necessary and sufficient condition for compactness of L A in the class (X, l ∞ (q)) (where X is any of the spaces l ∞ (B, p), c (B, p) or c 0 (B, p)) and characterize some classes of compact operators on these spaces by using the Hausdorff measure of the noncompactness technique.Keywords: Hausdorff measure of noncompactness; double sequential matrix; sequence space; paranormed space
Preliminaries and backgroundThe first measure of noncompactness, the function α, was defined and studied by Kuratowsky [] in . Darbo [], using this measure, generalized both the classical Schauder fixed point principle and (a special variant of ) Banach's contraction mapping principle for so called condensing operators. The Hausdorff measure of noncompactness χ was introduced by Goldenstein et al. [] in .Recently, the Hausdorff measure of noncompactness turned out to be very useful in the classification of compact operators between Banach spaces. Many authors characterized the classes of compact operators given by infinite matrices on some sequence spaces by using the Hausdorff measure of noncompactness.
The new sequence spaces X(r, s, t; ) for X ∈ {l ∞ , c, c 0 } have been defined by using generalized means and difference operator. In this work, we establish identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on some new difference sequence spaces X(r, s, t; ) where X ∈ {l ∞ , c, c 0 , l p } (1 ≤ p < ∞), as derived by using generalized means. Further, we find the necessary and sufficient conditions for such operators to be compact by applying the Hausdorff measure of noncompactness. Finally, as applications we characterize some classes of compact operators between these new difference sequence spaces and some other BK-spaces.Keywords: sequence space; difference operators; matrix transformation; generalized means; compact operators; Hausdorff measure of noncompactness
Preliminaries and backgroundThe study of sequence spaces has been very useful in many branches of analysis. Recently, some new sequence spaces have been defined by using matrix domain of a suitable matrix. Beside this, the Hausdorff measure of noncompactness is very useful in the classification of compact operators between Banach spaces.The difference sequence spaces were introduced for the first time by Kizmaz in []. Afterwards, many authors have introduced and studied some new sequence spaces defined by using the difference operator. In this paper we obtain some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on new difference sequence spaces defined by Manna et al. Further, we find the necessary and sufficient condi-
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.