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Integral operators in Banach spaces

Giving a new proof of a result of Grothendieck, it is shown that a weak operator limit of a bounded net of integral maps is again integral and that every integral map is a weak operator limit of a net of finite dimensional maps which are uniformly bounded in nuclear norm. It is also shown that the subnuclear maps defined by Ruckle are precisely the integral maps. Using this result some questions of Ruckle are answered. 1. In a recent paper [6] Ruckle has introduced a class of operators called subnuclear operators and asked whether every such operator is nuclear. In this paper we show that the subnuclear maps are precisely the integral maps of Grothendieck [3] and give a characterization of integral maps as weak operator limits of bounded nets of nuclear maps. 2. Definitions and preliminary results. Let E and F be Banach spaces and S(E, F) the space of all bounded linear operators from E to F. Throughout the paper the term "operator" or "map" will mean a bounded linear transformation. We will denote by E 0 6F the completion of the algebraic tensor product E 0) F in the e-norm defined by EXi X yi|= sup E (J, xi(g yi) i=1 f Ilf 11;g1,fe E*; lS51 R<1 EF* i=1 Similarly, we denote by E0(T,F the completion of E0 F in the 7rnorm defined by x]E i 8) yi ::=inf E: || r4 || IIY! III i=l X j=1 where the infimum is taken over all