[Paper Review] Compactness and an approximation property related to an operator ideal

Anil Kumar Karn, D. P. Sinha|arXiv (Cornell University)|Jul 9, 2012

Advanced Banach Space Theory3 references3 citations

TL;DR

This paper introduces an $σ$-approximation property ($\mathcal{A}$-a.p.) for Banach spaces relative to an operator ideal $\mathcal{A}$, characterizing it via the density of finite rank operators in $\mathcal{A} \circ \mathcal{K}$ and $\mathcal{K} \circ \mathcal{A}$. Key results show that for injective $\mathcal{A}$ with the $\ell_\infty$-extension property, $X$ has the $\mathcal{A}$-a.p. iff $(\mathcal{A}^{\min})^{\text{inj}} = \mathcal{A}^{\min}$, and dually for $X^*$, extending classical approximation properties to operator ideal settings.

ABSTRACT

For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact operators. We introduce a notion of an $\mathcal A$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in ${\mathcal A}\circ{\mathcal K}$ and ${\mathcal K}\circ{\mathcal A}$. We propose the notions of $\ell_{\infty}$-extension and $\ell_{1}$-lifting properties for an operator ideal $\mathcal A$ and study ${\mathcal A}\circ{\mathcal K}$, ${\mathcal}\circ{\mathcal A}$ and the $\mathcal A$-approximation property where $\mathcal A$ is injective or surjective and/or with the $\ell_{\infty}$-extension or $\ell_{1}$-lifting property. In particular, we show that if $\mathcal A$ is an injective operator ideal with the $\ell_\infty$-extension property, then we have: (a) $X$ has the $\mathcal A$-approximation property if and only if $({\mathcal A}^{min})^{inj}(Y,X)={\mathcal A}^{min}(Y,X)$, for all Banach spaces $Y$. (b) The dual space $X^*$ has the $\mathcal A$-approximation property if and only if $(({\mathcal A}^{dual})^{min})^{sur}(X,Y)=({\mathcal A}^{dual})^{min}(X,Y)$, for all Banach spaces $Y$.}For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$,

Motivation & Objective

• To generalize the classical approximation property by introducing an $\mathcal{A}$-approximation property (a.p.) for a Banach space $X$ relative to an operator ideal $\mathcal{A}$.
• To characterize the $\mathcal{A}$-a.p. in terms of the density of finite rank operators in the composition ideals $\mathcal{A} \circ \mathcal{K}$ and $\mathcal{K} \circ \mathcal{A}$.
• To investigate the role of the $\ell_\infty$-extension and $\ell_1$-lifting properties in determining the behavior of composition ideals and the $\mathcal{A}$-a.p.
• To establish duality results linking the $\mathcal{A}$-a.p. of $X$ and the $\mathcal{A}^\text{dual}$-a.p. of $X^*$, particularly for injective and surjective ideals.
• To extend known results on $p$-summing operators and their approximation properties to a general operator ideal framework.

Proposed method

• Introduce the kernel procedure $\text{com}: \mathcal{A} \mapsto \mathcal{K} \circ \mathcal{A} \circ \mathcal{K}$, defining the compact-level object associated with $\mathcal{A}$.
• Define the $\mathcal{A}$-approximation property as the condition that $\overline{\mathcal{F}}(Y,X)$ is dense in $\mathcal{A} \circ \mathcal{K}(Y,X)$ and $\mathcal{K} \circ \mathcal{A}(Y,X)$ for all Banach spaces $Y$.
• Introduce the $\ell_\infty$-extension and $\ell_1$-lifting properties for operator ideals, generalizing projective properties in operator ideal theory.
• Establish that the composition of injective (surjective) ideals is injective (surjective), and that ideals with $\ell_\infty$-extension (or $\ell_1$-lifting) preserve these properties under composition.
• Use duality and accessibility properties to relate $({\mathcal{A}}^{\min})^{\text{inj}}$ and $(({\mathcal{A}}^{\text{dual}})^{\min})^{\text{sur}}$ to the $\mathcal{A}$-a.p. via norm density conditions.
• Apply the kernel procedure to relate $({\mathcal{A}}^{\text{com}})^{\text{inj}}$ and $({\mathcal{A}}^{\text{com}})^{\text{sur}}$ to $({\mathcal{A}}^{\min})^{\text{inj}}$ and $({\mathcal{A}}^{\text{dual}})^{\min}$, respectively, under accessibility assumptions.

Experimental results

Research questions

• RQ1 When is the $\mathcal{A}$-approximation property of a Banach space $X$ equivalent to the equality $({\mathcal{A}}^{\min})^{\text{inj}}(Y,X) = {\mathcal{A}}^{\min}(Y,X)$ for all Banach spaces $Y$?
• RQ2 How does the $\mathcal{A}$-a.p. of $X^*$ relate to the surjective hull of $({\mathcal{A}}^{\text{dual}})^{\min}$?
• RQ3 What role do the $\ell_\infty$-extension and $\ell_1$-lifting properties play in preserving injectivity/surjectivity and duality in composition ideals?
• RQ4 Under what conditions does $({\mathcal{A}}^{\min})^{\text{inj}} = {\mathcal{A}}^{\min}$ imply the $\mathcal{A}$-a.p. for $X$?
• RQ5 How do the results generalize known characterizations of the $p$-summing approximation property for $\Pi_p$?

Key findings

• For an injective operator ideal $\mathcal{A}$ with the $\ell_\infty$-extension property, $X$ has the $\mathcal{A}$-approximation property if and only if $({\mathcal{A}}^{\min})^{\text{inj}}(Y,X) = {\mathcal{A}}^{\min}(Y,X)$ for all Banach spaces $Y$.
• For the dual space $X^*$, the $\mathcal{A}$-a.p. holds if and only if $(({\mathcal{A}}^{\text{dual}})^{\min})^{\text{sur}}(X,Y) = ({\mathcal{A}}^{\text{dual}})^{\min}(X,Y)$ for all Banach spaces $Y$.
• If $\mathcal{A}$ is right accessible and surjective, then $X$ has the $\mathcal{A}$-a.p. if and only if $({\mathcal{A}}^{\min})^{\text{sur}}(Y,X) = {\mathcal{A}}^{\min}(Y,X)$ for all $Y$.
• If $\mathcal{A}$ is right accessible and surjective, then $X^*$ has the $\mathcal{A}$-a.p. if and only if $(({\mathcal{A}}^{\text{dual}})^{\min})^{\text{inj}}(X,Y) = ({\mathcal{A}}^{\text{dual}})^{\min}(X,Y)$ for all $Y$.
• The results generalize Theorems 4.5 and 4.6 from [8] on $p$-summing operators, showing that the $\Pi_p$-approximation property is a special case of the $\mathcal{A}$-a.p. for $\mathcal{A} = \Pi_p$.
• The $\ell_\infty$-extension and $\ell_1$-lifting properties are shown to be preserved under composition of ideals and are instrumental in characterizing the injective and surjective hulls of $\mathcal{A}^{\min}$.