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Remarks on the classical Banach operator ideals

J. Diestel, B. Faires

1976
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Proceedings of the American Mathematical Society
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Sufficient conditions are given that the X-tensor product of two operators be weakly compact. Suppose W, X, Y, Z denote Banach spaces and W ®XX denotes the topological tensor product of W and X under the least reasonable tensor crossnorm X.UT:W-*Y and S: X -* Z ave continuous linear operators then a continuous linear operator W ®x X -> Y ®^ Z is induced which may or may not share certain special properties enjoyed by F and S. The present note is concerned with the classical Banach operator
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... anach operator ideal of weakly compact operators. While the A-tensor product of weakly compact linear operators need not be weakly compact, if either of the operators in question is compact (Theorem 2) or if one of the operator's domains is a C(K)-or an Lx(p:)-space then the A-tensor product of weakly compact operators is again a weakly compact operator (Theorem 4). In proving that the A-tensor product of a weakly compact operator and a compact operator is again weakly compact no real use is made of weak compactness; indeed, Theorem 2 shows that for any classical injective Banach operator ideal an analogous statement holds. A basic tool used in the proof of Theorem 4 is a recent, as yet unpublished result of W. J. Davis, T. Figiel, W. B. Johnson, and A. Pelczynski which states that every weakly compact linear operator between Banach spaces factors through a reflexive Banach space. We wish to thank Professors Davis, Figiel, Johnson, and Pelczynski for communicating their result. We also wish to thank Professor D. R. Lewis for conversations which led to the present proof of Theorem 4; this proof avoids the use of the representation theory of weakly compact operators on C(K)-and L, (/¿/spaces which we originally employed in proving Corollary 5. Let / denote a classical Banach operator ideal; i.e., for each pair of Banach spaces X, Y, I(X; Y) is a closed subspace of L(X; Y) in the uniform norm topology containing the finite rank operators F(X; Y) from X to Y and possessing the ideal property that if W, Z are Banach spaces and F: W -> X, R: Y -> Z are bounded linear operators and 5 G I(X; Y), then RST: W ^> Z is a member of I(W; Z). The reader is referred to [12] for a rather complete discussion of Banach operator ideals-classical and otherwise. Among the classical examples of such structures one finds the classes of

doi:10.1090/s0002-9939-1976-0454701-6
fatcat:2x5mp6ma2vdb5hyhtnhjw4z4au