## Compact operators whose adjoints factor through subspaces of $$l_p$$.(English)Zbl 1008.46008

Starting with the well-known Grothendieck’s characterization: a subset $$K$$ of a Banach space $$X$$ is relatively compact if and only if there exists $$x=(x_n)_n\in c_0(X)$$ such that $$K\subseteq \{\sum_n a_n x_n$$; $$(a_n)_n\in B_{\ell_1}\}$$ (where $$B_Y$$ denotes the unit ball of the Banach space $$Y$$), the authors introduce a stronger property: $$K\subseteq X$$ is relatively $$p$$-compact, $$1\leq p\leq \infty$$, if there exists $$x=(x_n)_n\in \ell_p(X)$$ such that $$K\subseteq \{\sum_n a_n x_n$$; $$(a_n)_n\in B_{\ell_{p^\ast}}\}$$ ($$p^\ast$$ being the conjugate exponent of $$p$$); for $$p=\infty$$, $$\ell_p(X)$$ should be replaced by $$c_0(X)$$, and $$\infty$$-compactness is the usual compactness. For $$1\leq p_1\leq p_2\leq \infty$$, $$p_1$$-compactness implies $$p_2$$-compactness. Replacing the space $$\ell_p(X)$$ of strongly $$p$$-summable series by the space $$\ell_p^w(X)$$ of weakly ones, one gets the notion of weak $$p$$-compactness. The notions of $$p$$-compact operator, weakly $$p$$-compact operator, $$p$$-approximation property are then defined in an obvious way. $$p$$-compact and weakly $$p$$-compact operators are characterized by a specific factorization of the adjoint through some subspace of $$\ell_p$$.
The spaces $${\mathcal K}_p(X,Y)$$ and $${\mathcal W}_p(X,Y)$$ of all the $$p$$-compact (resp. weakly $$p$$-compact) operators, equipped with their factorization norm $$\kappa_p$$ and $$\omega_p$$, respectively, are Banach operator ideals. The authors compare these operators with $$p$$-summing and $$p$$-nuclear ones. In particular, they show that if $$T: X\to Y$$ is $$p$$-compact, then $$T^\ast: Y^\ast \to X^\ast$$ is $$p$$-summing. The $$p$$-approximation property for $$X$$ is shown to be equivalent to the density, for every Banach space $$Y$$, of the ideal $${\mathcal F}(X,Y)$$ of the finite-rank operators in $${\mathcal K}_p(X,Y)$$ for the norm $$\omega_p$$. It is not known whether $$\omega_p$$ can be replaced by $$\kappa_p$$. The $$p_2$$-approximation property implies the $$p_1$$-AP if $$1\leq p_1\leq p_2\leq \infty$$. The authors show that every Banach space has the $$2$$-approximation property (this result comes, as used in the beginning of the proof of their Lemma 6.5, from the fact that every quotient of $$\ell_2$$ has the usual approximation property), but for each $$p>2$$, there are Banach spaces which fail the $$p$$-AP. In the last section, they introduce the $$(q,p)$$-approximation property, $$1\leq p\leq q\leq \infty$$. $$p$$-AP implies $$(\infty,p)$$-AP, and they compare it with the different notions of $$p$$-approximation properties, using tensor products, introduced previously by P. Saphar [Isr. J. Math. 13, 379-399 (1972; Zbl 0257.46108)] and O. I. Reinov [Teor. Oper. Teor. Funkts. 1, 145-165 (1983; Zbl 0561.46033) and Math. Nachr. 109, 125–134 (1982; Zbl 1231.47018)].
Reviewer: Daniel Li (Lens)

### MSC:

 46B28 Spaces of operators; tensor products; approximation properties 46B50 Compactness in Banach (or normed) spaces 47B07 Linear operators defined by compactness properties

### Citations:

Zbl 0257.46108; Zbl 0561.46033; Zbl 1231.47018
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