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Some characterizations of integrable functions in the bilinear sense of Bartle with respect to the injective tensor product are obtained. As a consequence it is shown that the kernels of Carleman compact operators coincide with these Bartle integrable functions. This result is applied to prove that every Carleman L-weak-compact operator is compact. An example showing the different behavior of the integrability with respect to the projective tensor product is given. A general Fubini theorem in this setting is shown.

We present some recent results on operators with values in Hardy spaces and vector valued functions. Some compact operators are constructed which are not representable by functions and the non coincidence of the vector valued Hardy spaces with either the projective or the injective tensor product is proved. We improve some previous results on the failure of Fatou’s theorem on radial almost everywhere convergence for analytic Pettis integrable functions.