Compact sets of compact operators in absence of $l^{1}$

Abstract

We characterize the compactness of a subset of compact operators between Banach spaces when the domain space does not have a copy of 11. In a recent paper, F. Galaz-Fontes [6] characterizes the (relative) compactness of a set of compact operators from a reflexive and separable Banach space X into another Banach space Y. Here we prove that such characterization is also valid when X does not contain a copy of 11. In order to clarify our terminology we say that a set of bounded linear operators H C L(X, Y) is sequentially weaknorm equicontinuous (called uniformly w-continuous in [6]) if for each weakly-null sequence (xn) in X the sequences (h(Xn)) converge in norm to 0 uniformly in h c H, i.e. sup {fIh(xn)I: h e H} converges to 0. Theorem 1. Let X be a Banach space without a copy of 11 and let H be a subset of bounded operators from X to a Banach space Y. Then H is a relatively compact subset in the space of compact operators K(X, Y) in the uniform topology of operators if and only if it verifies the following two conditions: (1) H is pointwisely relatively compact, i.e. for each x c X the set H(x) = {h(x): h c H} is relatively compact in Y. (2) H is sequentially weak-norm equicontinuous. Note that if X does not contain a copy of 11, then by applying the RosenthalDor Theorem (see [11] and [4] or [3, Ch. IX]) every bounded sequence in X has a weakly-Cauchy subsequence and, therefore, a bounded linear operator h: X Y is compact if and only if it is completely continuous, that is, if and only if it takes weakly convergent sequences in X to convergent ones in Y. This tells us that condition (2) applied to a singleton characterizes compact operators when the space X does not have a copy of 11 [7, 17.7]. Our notation is standard: L(X, Y) is the Banach space of bounded linear operators from the Banach space X to the Banach space Y, endowed with the topology of the uniform convergence on the unit ball Bx of X, K(X, Y) is its closed subspace Received by the editors April 20, 1998. 2000 Mathematics Subject Classification. Primary 47B07, 46B25.