##
**Completely continuous operators.**
*(English)*
Zbl 1256.46009

A (bounded, linear) operator \(T: E\to F\) acting between Banach spaces is completely continuous (or Dunford-Pettis; \(T\in CC(E,F)\)) if it maps weakly Cauchy sequences to norm convergent sequences. The class of completely continuous operators forms a closed operator ideal and the identity map on the space \(\ell_1\) is a notable example of a completely continuous operator which is not compact. A bounded subset \(A\) of a Banach space \(X\) is a Dunford-Pettis set (limited set) if each weakly null (respectively, weak* null) sequence in \(X^*\) tends to 0 on \(A\) uniformly. A Banach space has the Dunford-Pettis property (DPP) if every weakly compact operator from \(X\) to an arbitrary Banach space is completely continuous. Banach spaces with DPP include \(L_1(\mu)\)-spaces and \(C(K)\)-spaces.

One of the main results of the paper under review is a list of numerous conditions equivalent to the assertion that for a given Banach space \(X\) each (bounded, linear) operator from \(X\) to \(c_0\) is completely continuous. This characterisation consists of ostensibly technical conditions and it is too long to be quoted here; its proof involves extensively the so-called ‘gliding hump’ technique. The result has many interesting consequences, e.g. it gives a characterisation of limited subsets of a Grothendieck space \(X\) with DPP (\(X\) is a Grothendieck space if weak* null sequences in \(X^*\) converge weakly) which satisfies additionally \(L(X,c_0)=CC(X,c_0)\) as precisely those sets which are weakly precompact. Also, for each Banach space \(X\) which fails the Schur property and each not-purely atomic measure \(\mu\), the authors construct an operator \(T: L_p(\mu, X)\to c_0\) which is not completely continuous. Making use of a result of R. H. Lohman [Can. Math. Bull. 19, 365–367 (1976; Zbl 0342.46006)], it is shown that the property \(L(X,c_0)=CC(X,c_0)\) is inherited by quotients \(X/Y\), where \(Y\) is a closed subspace of \(X\) without an isomorphic copy of \(\ell_1\).

The third section of the paper is devoted to completely continuous operators on tensor products. Denote by \(\otimes_\pi\) the projective tensor product of Banach spaces. One of the results asserts that if \(X\) lacks the Schur property and \(Y\) is a Banach space such that \(L(X,Y^*)=CC(X, Y^*)\), then \(L(X\otimes_\pi Y^*, c_0)\neq CC(X \otimes_\pi Y^*, c_0)\). This theorem is then used to derive a long list of consequences.

An interplay between Pełczyński’s properties (V) and (V\({}^*\)), the Grothendieck property and completely continuous operators is studied in the final section. The authors rediscover a result of F. Räbiger [“Beiträge zur Strukturtheorie der Grothendieck-Räume”, Sitzungsber., Heidelberger Akad. Wiss., Math.-Naturwiss. Kl. 4, 78 S. (1985; Zbl 0597.46021)] which asserts that a Banach space with the property (V) is a Grothendieck space if and only if it does not contains a complemented copy of \(c_0\). The authors also prove that the property \(L(C(K), c_0)=CC(C(K), c_0)\) characterises Grothendieck \(C(K)\)-spaces. Furthermore, relatively weak*-sequentially compact sets and \(V\)-sets in Banach spaces are studied. The proofs of results in this section have often a strong measure-theoretic flavour.

The paper contains an impressive list of sixty references.

One of the main results of the paper under review is a list of numerous conditions equivalent to the assertion that for a given Banach space \(X\) each (bounded, linear) operator from \(X\) to \(c_0\) is completely continuous. This characterisation consists of ostensibly technical conditions and it is too long to be quoted here; its proof involves extensively the so-called ‘gliding hump’ technique. The result has many interesting consequences, e.g. it gives a characterisation of limited subsets of a Grothendieck space \(X\) with DPP (\(X\) is a Grothendieck space if weak* null sequences in \(X^*\) converge weakly) which satisfies additionally \(L(X,c_0)=CC(X,c_0)\) as precisely those sets which are weakly precompact. Also, for each Banach space \(X\) which fails the Schur property and each not-purely atomic measure \(\mu\), the authors construct an operator \(T: L_p(\mu, X)\to c_0\) which is not completely continuous. Making use of a result of R. H. Lohman [Can. Math. Bull. 19, 365–367 (1976; Zbl 0342.46006)], it is shown that the property \(L(X,c_0)=CC(X,c_0)\) is inherited by quotients \(X/Y\), where \(Y\) is a closed subspace of \(X\) without an isomorphic copy of \(\ell_1\).

The third section of the paper is devoted to completely continuous operators on tensor products. Denote by \(\otimes_\pi\) the projective tensor product of Banach spaces. One of the results asserts that if \(X\) lacks the Schur property and \(Y\) is a Banach space such that \(L(X,Y^*)=CC(X, Y^*)\), then \(L(X\otimes_\pi Y^*, c_0)\neq CC(X \otimes_\pi Y^*, c_0)\). This theorem is then used to derive a long list of consequences.

An interplay between Pełczyński’s properties (V) and (V\({}^*\)), the Grothendieck property and completely continuous operators is studied in the final section. The authors rediscover a result of F. Räbiger [“Beiträge zur Strukturtheorie der Grothendieck-Räume”, Sitzungsber., Heidelberger Akad. Wiss., Math.-Naturwiss. Kl. 4, 78 S. (1985; Zbl 0597.46021)] which asserts that a Banach space with the property (V) is a Grothendieck space if and only if it does not contains a complemented copy of \(c_0\). The authors also prove that the property \(L(C(K), c_0)=CC(C(K), c_0)\) characterises Grothendieck \(C(K)\)-spaces. Furthermore, relatively weak*-sequentially compact sets and \(V\)-sets in Banach spaces are studied. The proofs of results in this section have often a strong measure-theoretic flavour.

The paper contains an impressive list of sixty references.

Reviewer: Tomasz Kania (Lancaster)

### MSC:

46B28 | Spaces of operators; tensor products; approximation properties |

46B25 | Classical Banach spaces in the general theory |

47B07 | Linear operators defined by compactness properties |