CONVOLUTION-CONTINUOUS BILINEAR OPERATORS ACTING ON HILBERT SPACES OF INTEGRABLE FUNCTIONS

Abstract

We study bilinear operators acting on a product of Hilbert spaces of integrable functions zero-valued for couples of functions whose convolution equals zero that we call convolution-continuous bilinear maps. We prove a factorization theorem for them, showing that they factor through l(1). We also present some applications for the case when the range space has some relevant properties, such as the Orlicz or Schur properties. We prove that l(1) is the only Banach space for which there is a norming bilinear map which equals zero exactly in those couples of functions whose convolution is zero. We also show some examples and applications to generalized convolutions.