Hausdorff measure of noncompactness of matrix operators on some sequence spaces of a double sequential band matrix

Abstract

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.

A Generalization of the Averaged Hausdorff Distance

“…In order to use the (p, q)-averaged distance, we discretize P , by taking 11 uniformly distributed points over P . We assume two archives: X 1 is obtained from P by changing (0, 1) for (0, 10), including an outlier, and adding 1 10 to the remaining ordinates. X 2 is obtained from P by adding 5 to each ordinate.…”

“…x ∈ X is close to some y ∈ Y and vice versa. The metric d H is used in Brownian motion [15], matrix theory [1], dynamical systems [2], or fractal geometry [7], among other research areas. In the theory of evolutionary multiobjective optimization (EMO), the closeness of a set A to certain PF determines the approximation (called convergence in the EMO literature) of the outcome, and the closeness of PF to A determines the spread (maximal gap).…”