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**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)].

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 |