[Paper Review] Closed ideals in $\mathcal{L}(X)$ and $\mathcal{L}(X^*)$ when $X$ contains certain copies of $\ell_p$ and $c_0$

Ben Wallis|arXiv (Cornell University)|Jul 12, 2015

Advanced Banach Space Theory2 references3 citations

TL;DR

This paper establishes that if a Banach space $X$ contains a complemented copy of $ \'ell_p $ for $ p \in (1,2) $ and a copy of either $ \ell_q $ or $ c_0 $, then the operator algebras $ \mathcal{L}(X) $ and $ \mathcal{L}(X^*) $ each contain a continuum of closed ideals. Furthermore, under additional conditions on $ q $, these ideals are not linearly ordered, demonstrating a rich and nontrivial ideal structure in these operator algebras.

ABSTRACT

Suppose $X$ is a real or complexified Banach space containing a complemented copy of $\ell_p$, $p\in(1,2)$, and a copy (not necessarily complemented) of either $\ell_q$, $q\in(p,\infty)$, or $c_0$. Then $\mathcal{L}(X)$ and $\mathcal{L}(X^*)$ each admit continuum many closed ideals. If in addition $q\geq p'$, $\frac{1}{p}+\frac{1}{p'}=1$, then the closed ideals of $\mathcal{L}(X)$ and $\mathcal{L}(X^*)$ each fail to be linearly ordered. We obtain additional results in the special cases of $\mathcal{L}(\ell_1\oplus\ell_q)$ and $\mathcal{L}(\ell_p\oplus c_0)$, $1

Motivation & Objective

• To determine the structure of closed ideals in $ \mathcal{L}(X) $ and $ \mathcal{L}(X^*) $ when $ X $ contains specific classical sequence spaces.
• To extend known results on closed ideal lattices in $ \mathcal{L}(\ell_p \oplus \ell_q) $ to broader classes of Banach spaces.
• To show that the closed ideal lattice in $ \mathcal{L}(X) $ and $ \mathcal{L}(X^*) $ is not linearly ordered under certain conditions.
• To analyze the role of finitely strictly singular and superstrictly cosingular operators in generating such ideals.
• To provide a unified framework using operator ideals and factorization techniques to prove the existence of continuum-many closed ideals.

Proposed method

• Use of finitely strictly singular (FSS) and superstrictly cosingular (SSCS) operator ideals as key tools to construct closed ideals.
• Application of the Principle of Small Perturbations to establish equivalence of basic sequences and control operator norms.
• Construction of bounded linear operators via factorization through $ \ell_\infty^{k_n} $ and $ c_0 $-sums of $ \ell_2^n $, leveraging separable injectivity.
• Definition of operators $ J_n $, $ Q_n $, and $ U $ to build a global operator $ \widetilde{J}RU $ that factors through $ \theta I_{2,p,0} $, proving the desired diagrammatic factorization.
• Use of $ \ell_2 $-factorability of certain operators to establish ideal inclusions and incomparability.
• Adaptation of proofs from [SZ14] and [SSTT07] to generalized Banach spaces containing $ \ell_p $, $ \ell_q $, or $ c_0 $.

Experimental results

Research questions

• RQ1 Under what conditions on a Banach space $ X $ does $ \mathcal{L}(X) $ admit a continuum of closed ideals?
• RQ2 When do the closed ideals in $ \mathcal{L}(X) $ and $ \mathcal{L}(X^*) $ fail to be linearly ordered?
• RQ3 How does the presence of complemented $ \ell_p $ and non-complemented $ \ell_q $ or $ c_0 $ subspaces affect the ideal structure of $ \mathcal{L}(X) $?
• RQ4 What role do FSS and SSSC operator ideals play in generating new closed ideals in $ \mathcal{L}(X) $?
• RQ5 Can the factorization techniques used for $ \mathcal{L}(\ell_p \oplus c_0) $ and $ \mathcal{L}(\ell_1 \oplus \ell_q) $ be generalized to other Banach spaces?

Key findings

• If $ X $ contains a complemented copy of $ \ell_p $ for $ p \in (1,2) $ and a copy of $ \ell_q $ or $ c_0 $, then $ \mathcal{L}(X) $ and $ \mathcal{L}(X_{\mathbb{C}}) $ each contain a chain of closed ideals of cardinality continuum, all contained in the FSS ideal.
• The dual space $ \mathcal{L}(X^*) $ and $ \mathcal{L}(X_{\mathbb{C}}^*) $ each contain a continuum-sized chain of closed ideals within the SSSC ideal.
• When $ 1 < p < 2 < p' \leq q < \infty $, the closed ideals in $ \mathcal{L}(X) $ and $ \mathcal{L}(X^*) $ are not linearly ordered, as they contain two incomparable ideals.
• For $ \mathcal{L}(\ell_p \oplus c_0) $ and $ \mathcal{L}(\ell_1 \oplus \ell_q) $ with $ 1 < p < 2 < q < \infty $, the operator algebras contain continuum-many closed ideals.
• The ideals $ \mathcal{FSS} \cap [\mathcal{G}_{\ell_2}] $ and $ [\mathcal{G}_T] $ in $ \mathcal{L}(\ell_p, c_0) $ are incomparable, and $ \mathcal{FSS} $ is properly contained in $ \mathcal{FSS} + [\mathcal{G}_T] $, showing non-trivial ideal structure.
• The factorization of operators through $ \theta I_{2,p,0} $ via $ \widetilde{J}RU $ confirms the existence of new closed ideals in $ \mathcal{L}(X) $, extending results from $ \mathcal{L}(\ell_p \oplus \ell_q) $ to broader classes of spaces.