Tensor products and operators in spaces of analytic functions

Let X be an infinite dimensional Banach space. The paper proves the non-coi ncidence of the vector-valued Hardy space H-p(T. X) with neither the projec tive nor the injective tensor product of H-p(T) and X, for 1 < p < infinity . The same result is proved for some other subspaces of L-p. A characteriza tion is given of when every approximable operator from X into a Banach spac e 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 oper ators from X into H-p(T)(1 less than or equal to p less than or equal to in finity) which are not representable. An analytic Pettis integrable function F:T --> X is constructed whose Poisson integral does not converge pointwis e.