Maximal non-compactness of embeddings between sequence spaces
Anna Kneselová
TL;DR
This work investigates the ball measure of non-compactness $α(T)$ for embeddings between sequence spaces, with a focus on identifying when the identity embedding is maximally non-compact. It develops general sufficient criteria for maximal non-compactness of embeddings into $c_0$, and introduces a span-based approach to preclude MN behavior for embeddings into $\ell^{\infty}$, subsequently applying these results to the Lorentz sequence spaces $\ell^{p,q}$ to obtain complete inclusion relations and exact norms for identity maps. The findings show that many embeddings $\ell^{p_{1},q_{1}} \to \ell^{p_{2},q_{2}}$ are maximally non-compact when they sit inside $c_0$, while several embeddings into $\ell^{\infty}$ are not MN, with the canonical inclusion $c_{0}\to \ell^{\infty}$ being MN as an isometry. Overall, the paper provides precise, quantitative criteria for MN behavior in Lorentz-space embeddings and clarifies when maximal non-compactness occurs in this two-parameter scale of sequence spaces.
Abstract
We will focus on studying the ball measure of non-compactness $α(T)$ 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.