Tensor Products and Operators in Spaces of Analytic Functions

Let X be an infinite dimensional Banach space. The paper proves the non‐coincidence of the vector‐valued Hardy space H p (T, X ) with neither the projective nor the injective tensor product of H p (T) and X , for 1 < p < ∞. The same result is proved for some other subspaces of L p . A characterization is given of when every approximable operator from X into a Banach space of measurable functions F ( S ) is representable by a function F : S → X * as x ↦ 〈 F (·), x 〉. As a consequence the existence is proved of compact operators from X into H p (T) (1 ⩽ p < ∞) which are not representable. An analytic Pettis integrable function F :T → X is constructed whose Poisson integral does not converge pointwise.