Maximal non-compactness of embeddings between sequence spaces

TL;DR

This paper investigates the non-compactness measure of embedding operators between sequence spaces, especially Lorentz spaces, providing conditions for maximal non-compactness and exact norms of identity operators.

Contribution

It offers necessary and sufficient conditions for maximal non-compactness of identity operators and characterizes inclusions and norms between Lorentz sequence spaces.

Findings

Characterized when identity operators are maximally non-compact.

Determined exact norms of identity operators between Lorentz spaces.

Provided criteria for inclusions between Lorentz sequence spaces.

Abstract

We will focus on studying the ball measure of non-compactness $\alpha(T)$ for various particular instances of embedding operators in sequence spaces. Our first main goal is to find necessary and sufficient conditions for an identity operator to be maximally non-compact. Next, we will focus on studying Lorentz sequence spaces $\ell^{p,q}$ and their basic properties. We will characterize the inclusions between Lorentz sequence spaces depending on the values of $p$ and $q$. Then we will try to determine exact values of the norms of the identity operators between these embedded spaces. Lastly, we will determine whether these identity operators are maximally non-compact by using our general theorems.