PIETSCH INTEGRAL OPERATORS DEFINED ON INJECTIVE TENSOR PRODUCTS OF SPACES AND APPLICATIONS

摘要

For X and Y Banach spaces, let X directX_ε Y, be the injective tensor product. If Z is also a Banach space arid U ∈ L(X directX_εY, Z) we consider the operator U~#:X → L(Y,Z), (U~#x)(y) = U(x directX y),x ∈ X, y ∈ Y. We prove that if U ∈ PI(X directX_ε Y, Z), then U~# ∈ I(X, PI(Y, Z)). This result is then applied in the case of operators defined on the space of all X-valued continuous functions on the compact Hausdorff space T. We obtain also an affirmative answer to a problem of J. Diestel and J. J. Uhl about the RNP property for the space of all nuclear operators; namely if X~* and Y have the RNP and Y can be complemented in its bidual, then N(X, Y) has the RNP.