## Banach spaces in which Dunford-Pettis sets are relatively compact.(English)Zbl 0761.46010

This paper is concerned with the study of the class of Banach spaces in which Dunford-Pettis sets are relatively compact (in symbols (DPrcP)). The interest in this class of spaces is due to the following result: $$E^*$$ has the (DPrcP) iff $$E$$ does not contain $$\ell^ 1$$. This fact is then used to show that if $$E$$, $$F$$ do not contain $$\ell^ 1$$ and $$L(E,F^*)=K(E,F^*)$$, then $$E\otimes F$$ does not contain $$\ell^ 1$$, so answering a question by Ruess. The above family of Banach spaces is also utilized to prove that if $$E$$ has the Dunford-Pettis property and $$F$$ the (DPrcP) then every dominated operator from $$C(K,E)$$ into $$F$$ is a Dunford-Pettis operator (we note that if $$E$$ is finite dimensional this is true for all $$F$$, but if $$E$$ is infinite dimensional this can be false). At the end, several results showing that special Banach lattices as well as spaces of compact operators have the (DPrcP) are presented.

### MSC:

 46B28 Spaces of operators; tensor products; approximation properties 46B22 Radon-Nikodým, Kreĭn-Milman and related properties 47L05 Linear spaces of operators
Full Text:

### References:

 [1] J. Diestel andT. J. Morrison, The Radon-Nikodym property for the space of operators, I. Math. Nachr.92, 7-12 (1979). · Zbl 0444.46021 [2] J.Diestel and J. J.Uhl, jr., Vector Measures. Math. Surveys 15, Amer. Math. Soc. 1977. [3] N.Dinculeanu, Vector Measures. Berlin 1967. [4] G. Emmanuele, A dual characterization of Banach spaces not containingl 1. Bull. Polish. Acad. Sci. Math.34, 155-160 (1986). · Zbl 0625.46026 [5] G. Emmanuele, On the containment ofc 0 by spaces of compact operators. Bull. Sci. Math.115, 177-184 (1991). · Zbl 0749.46013 [6] M. Feder, On subspaces of spaces with an unconditional basis and spaces of operators. Illinois J. Math.24, 196-205 (1980). · Zbl 0411.46009 [7] P. Meyer Nieberg, Zur schwachen Kompaktheit in Banachverbanden. Math. Z.134, 303-315 (1973). · Zbl 0268.46010 [8] K.Musial, Martingales of Pettis integrable functions. In Measure Theory, Proceedings, Oberwolfach 1979, LNM794, Berlin-Heidelberg-New York 1980. · Zbl 0433.28010 [9] A. Pelczynski, On Banach spaces containingL 1. Studia Math.30, 231-246 (1968). [10] G. Pisier, Une propriété de stabilité de la class des espaces ne contenant pasl 1. C. R. Acad. Sci. Paris Ser. A-B286, 747-749 (1978). · Zbl 0373.46033 [11] W. Ruess, Compactness and Collective Compactness in Spaces of Compact Operators. J. Math. Anal. Appl.84, 400-417 (1981). · Zbl 0477.47027 [12] W.Ruess, Duality and Geometry of Spaces of Compact Operators. In Functional Analysis: Surveys and recent Results III. 59-78, Amsterdam-New York 1984. [13] A. Ryan, The Dunford-Pettis property and the projective tensor product. Bull. Polish. Acad. Sci. Math.35, 785-792 (1987). · Zbl 0656.46057 [14] M.Talagrand, Pettis integral and Measure Theory. Mem. Amer. Math. Soc.307 (1984). · Zbl 0582.46049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.