OPERATORS TAKING VALUES IN LEBESGUE-BOCHNER SPACES

It is a subtle fact of the theory of regular operators on Banach lattices that every linear operator T : L-1(mu) -> L-1(nu) is norm-bounded if and only if it is regular. We generalize this result to the setting of operators taking values in a Lebesgue-Bochner space. Our main result asserts that every linear operator T : L-1(mu) -> L-1(nu, X) is norm-bounded if and only if it is dominated. We show that this result is no longer true for Lebesgue-Bochner domain spaces. As a consequence of the main theorem, we obtain a generalized version of the Grothendieck inequality for linear norm-bounded operators from L-1(mu) to a Lebesgue-Bochner space L-1(nu, X).