## 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.

### MSC:

 46B28 Spaces of operators; tensor products; approximation properties 46B25 Classical Banach spaces in the general theory 47B07 Linear operators defined by compactness properties

### Citations:

Zbl 0597.46021; Zbl 0342.46006
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