The Bartle bilinear integration and Carleman operators

Some characterizations of integrable functions in the bilinear sense of Bar tle with respect to the injective tensor product are obtained. As a consequ ence it is shown that the kernels of Carleman compact operators coincide wi th these Bartle integrable functions. This result is applied to prove that every Carleman L-weak-compact operator is compact. An example showing the d ifferent behavior of the integrability with respect to the projective tenso r product is given. A general Fubini theorem in this setting is shown, (C) 1999 Academic Press.