The Bartle Bilinear Integration and Carleman Operators1

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-weakcompact 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