Maximal non-compactness of embeddings between sequence spaces

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.

💡 Research Summary

The paper investigates the ball measure of non‑compactness α(T) for embedding operators between sequence spaces, with a particular focus on when the identity embedding is maximally non‑compact, i.e., when α(T) equals the operator norm ‖T‖. After recalling the definition of α(T) as the infimum of radii r such that the image of the unit ball can be covered by finitely many balls of radius r, the authors establish the basic inequality 0 ≤ α(T) ≤ ‖T‖ and note that α(T)=0 characterises compact operators while α(T)=‖T‖ defines maximal non‑compactness.

The first substantive result (Proposition 2.1) gives a sufficient condition for maximal non‑compactness of the identity embedding I:X→Y when both X and Y are subspaces of c₀, every coordinate of any element of Y is bounded by its norm, and the embedding norm satisfies 0 < ‖I‖ ≤ 1. The proof proceeds by contradiction: assuming α(I)<‖I‖, one constructs a standard basis vector e_j that cannot be covered by the finitely many balls guaranteed by the definition of α(I), contradicting the assumption.

An example (Example 2.2) shows that dropping the coordinate‑bound condition can lead to a non‑maximally non‑compact embedding, even when the other hypotheses hold.

The authors then introduce the concept of a rearrangement‑invariant lattice: a sequence space closed under taking absolute values and under decreasing rearrangements, with the norm invariant under such rearrangements. Theorem 2.3 proves that if both X and Y are rearrangement‑invariant lattices contained in c₀, then the identity embedding is automatically maximally non‑compact. The argument uses a scaling factor λ∈(0,1) to construct a sequence ε=λx* and shows that the distance between x* and ε in the target space exceeds any prescribed radius smaller than ‖I‖, again contradicting the definition of α(I).

Conversely, Theorem 2.4 provides a sufficient condition for an embedding into ℓ^∞ to fail maximal non‑compactness. Defining the “span” σ_ℓ = sup_{y∈B_X}(sup_n y_n – inf_n y_n), the theorem states that if ‖I‖ ≤ σ_ℓ < 2‖I‖ then α(I) ≤ σ_ℓ/2 < ‖I‖, so the embedding is not maximally non‑compact. The proof builds a finite family of constant sequences y_k whose radii cover the image of the unit ball.

The main application concerns Lorentz sequence spaces ℓ^{p,q}. For 0<p,q≤∞, the norm is defined via the decreasing rearrangement a* by  ‖a‖{p,q}= (∑{n=1}^∞ (n^{1/p−1/q} a*n)^q)^{1/q} if q<∞,  ‖a‖{p,∞}= sup_n n^{1/p} a*_n if q=∞. The paper first shows that for any finite p or q, ℓ^{p,q}⊂c₀, guaranteeing that the previous theorems apply. Then it characterises the inclusion ℓ^{p₁,q₁}⊂ℓ^{p₂,q₂}: the inclusion holds precisely when p₁≤p₂ and either p₁<p₂ or q₁≤q₂. Counter‑examples are supplied for the remaining parameter regimes.

Exact norms of the identity embeddings I:ℓ^{p₁,q₁}→ℓ^{p₂,q₂} are computed. In most cases the norm equals 1, but when p or q equals ∞, a constant depending on the parameters appears. Using these explicit norms together with Theorem 2.3, the authors conclude that whenever the source and target Lorentz spaces are rearrangement‑invariant lattices (which they always are) and both lie in c₀, the embedding is maximally non‑compact. In particular, for finite p,q with p₁<p₂ the embedding is maximally non‑compact. For embeddings into ℓ^∞, Theorem 2.4 applies, showing that α(I) is strictly smaller than ‖I‖, and the exact bound α(I)≤σ_ℓ/2 is provided.

Overall, the paper establishes a clear dichotomy: embeddings between Lorentz sequence spaces are either maximally non‑compact (when the target is a proper subspace of c₀) or, when the target is ℓ^∞, they fail maximal non‑compactness with a quantifiable gap. The work unifies the study of non‑compactness measures with the structural properties of rearrangement‑invariant lattices, offering a template that could be extended to function‑space embeddings such as Sobolev or Besov spaces, and suggesting future investigations into non‑linear operators.