Characterizations of Compact Operators on $\ell_{p}$ Type Fractional Sets of Sequences

TL;DR

This paper investigates the compactness of operators on fractional difference sequence spaces of $ ext{ell}_p$ type, providing conditions and characterizations using Hausdorff measures and recent results, especially for bounded sequence spaces.

Contribution

It introduces new $ ext{ell}_p$ fractional difference sequence spaces via Euler gamma functions and characterizes compact operators on these spaces, advancing understanding of their structure.

Findings

Sufficient conditions for compactness when the target space is $ ext{ell}_0$.

Exact characterization of compact matrix operators on bounded sequence spaces.

Application of Hausdorff measure of noncompactness to fractional difference spaces.

Abstract

Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some $\ell_{p}$ type fractional difference sequence spaces via Euler gamma function. Although we characterize compactness conditions on those spaces using the main tools of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is $\ell _{\infty }$. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.