Banach spaces with the (CRP) and dominated operators on \(C(K)\). (English) Zbl 0707.47021

Emmanuele, Giovanni

Ann. Acad. Sci. Fenn., Ser. A I, Math. 16, No. 2, 243-248 (1991).

The main result of the paper is the following: a Banach space E has the compact range property (CRP) if and only if any dominated operator T: C(K)\(\to E\), K an arbitrary Hausdorff spaces, is compact. From this result the following corollaries are derived:
i) A Banach space \(E\) has the (CRP) if and only if any Pietsch integral operator from an arbitrary Banach space \(F\) into \(E\) is compact;
ii) A Banach space \(E\) has the (CRP) if and only if any operator from an arbitrary \(L^ 1\) space into \(E\) is a Dunford-Pettis operator,
iii) If \(E\) is a Banach space with the (CRP) property, but without the (RNP), then there exists a compact Pietsch integral operator that is not nuclear.

Reviewer: G.Emmanuele (Catania)

MSC:

47B38	Linear operators on function spaces (general)
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46B22	Radon-Nikodým, Kreĭn-Milman and related properties
47B07	Linear operators defined by compactness properties
46G10	Vector-valued measures and integration

Keywords:

compact range property; dominated operator; compact Pietsch integral operator that is not nuclear