[Paper Review] A note on lineability

Geraldo Botelho, Diogo Diniz+2 more|ArXiv.org|May 16, 2009

Advanced Banach Space Theory5 references3 citations

TL;DR

This paper establishes that the set of compact operators from a superreflexive Banach space $E$ to a Banach space $F$, excluding absolutely $p$-summing operators, is $\aleph_1$-lineable under certain conditions—specifically, when $E$ has a complemented infinite-dimensional unconditional basis or $F$ contains an infinite unconditional basic sequence. The proof constructs an injective linear map from $\ell_1$ into the difference set, demonstrating an uncountable-dimensional subspace within the set, thus extending prior results on $\aleph_0$-lineability.

ABSTRACT

In this note we answer a question concerning lineability of the set of non-absolutely summing operators.

Motivation & Objective

• To resolve an open question from [1] on whether the set of non-absolutely summing operators can be $\mu$-lineable for $\mu > \aleph_0$.
• To extend known $\aleph_0$-lineability results for non-absolutely summing operators to uncountable lineability.
• To establish $\aleph_1$-lineability for the set of norm-attaining operators at a fixed point in $\mathcal{L}(E;F)$ under suitable geometric conditions on $E$.
• To demonstrate that the lineability dimension is optimal in certain cases, such as when $E = \ell_p$, $F = \ell_q$ with $p > 1$.
• To unify and generalize techniques from [1] and [4] to prove lineability in both operator ideals and norm-attaining operator sets.

Proposed method

• Decompose $\mathbb{N}$ into infinitely many pairwise disjoint infinite subsets $(A_j)_{j=1}^\infty$ to construct subspaces with unconditional bases.
• Use the unconditional basis property to define complemented subspaces $E_j$ of $E_0 \subset E$ or $F_j \subset F$, ensuring bounded projections with norm at most $\varrho$, the unconditional basis constant.
• Construct operators $u_j$ on each $E_j$ or $F_j$ that are compact but not absolutely $p$-summing, using known results from [2].
• Extend each $u_j$ to $\widetilde{u}_j$ on $E$ via composition with projections and inclusions, ensuring $\widetilde{u}_j \in \mathcal{K}(E;F)\setminus\Pi_p(E;F)$.
• Define a linear map $T: \ell_1 \to \mathcal{K}(E;F)$ by $T((a_n)) = \sum_{j=1}^\infty a_j \widetilde{u}_j$, leveraging disjoint supports to ensure injectivity.
• Prove $T(\ell_1) \subset (\mathcal{K}(E;F)\setminus\Pi_p(E;F)) \cup \{0\}$, showing the image is an $\aleph_1$-dimensional subspace contained in the difference set.

Experimental results

Research questions

• RQ1 Can the set $\mathcal{L}(E;F)\setminus\Pi_p(E;F)$ be $\mu$-lineable for $\mu > \aleph_0$ under suitable Banach space conditions?
• RQ2 Is $\mathcal{K}(E;F)\setminus\Pi_p(E;F)$ $\aleph_1$-lineable when $E$ is superreflexive and contains a complemented infinite-dimensional subspace with an unconditional basis?
• RQ3 Does the same lineability technique apply to the set of norm-attaining operators $\mathcal{NA}^{x_0}(E;F)$?
• RQ4 What is the maximal possible lineability dimension for $\mathcal{K}(E;F)\setminus\Pi_p(E;F)$ in natural examples like $E = \ell_p$, $F = \ell_q$?
• RQ5 Can the construction be generalized to other operator ideals beyond absolutely summing and compact operators?

Key findings

• The set $\mathcal{K}(E;F)\setminus\Pi_p(E;F)$ is $\aleph_1$-lineable if $E$ is superreflexive and contains a complemented infinite-dimensional subspace with an unconditional basis.
• The same conclusion holds if $F$ contains an infinite unconditional basic sequence, even without assuming $E$ has such a subspace.
• The construction yields an injective linear map $T: \ell_1 \to \mathcal{K}(E;F)$ whose image lies in $\left(\mathcal{K}(E;F)\setminus\Pi_p(E;F)\right) \cup \{0\}$, proving uncountable lineability.
• The dimension of the subspace constructed is exactly $\aleph_1$, which is optimal in cases such as $E = \ell_p$, $F = \ell_q$ with $p > 1$, since $\dim \mathcal{L}(\ell_p;\ell_q) = \aleph_1$.
• The method extends to norm-attaining operators: $\mathcal{NA}^{x_0}(E;F)$ is $\aleph_1$-lineable if $E$ contains an isometric copy of $\ell_q$ for some $1 \leq q < \infty$.
• The proof relies on disjoint supports and bounded projections, ensuring injectivity of the map $T$ and preserving the desired operator properties in the image.