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 ${\ell }^1$. 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 ${\ell }^1$ 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.