Zero product preserving bilinear operators acting in sequence spaces

Abstract

Consider a couple of sequence spaces and a product function $-$ a canonical bilinear map associated to the pointwise product $-$ acting in it. We analyze the class of "zero product preserving" bilinear operators associated with this product, that are defined as the ones that are zero valued in the couples in which the product equals zero. The bilinear operators belonging to this class have been studied already in the context of Banach algebras, and allow a characterization in terms of factorizations through $\ell^r(\mathbb{N})$ spaces. Using this, we show the main properties of these maps such as compactness and summability.