Zero-product preserving operators and product-factorability of bilinear maps

Abstract

The present dissertation deals with bilinear operators acting in pairs of Banach spaces that factor through a canonical product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices −orthosymmetric maps−,∗- algebras −zero product preserving operators−, and classical and harmonic analysis −integral bilinear operators. We purpose the use of a generic product as a linearizing tool for bilinear maps. Concretely, in this dissertation we introduce a certain bilinear map, called product, by some inclusion and norm equality requirements and present a factorization through the product given in terms of a summability condition for bilinear continuous operators acting in topological product of Banach spaces. If we specialize the product and the domain space of the bilinear map, this factorization also concerns about zero product preserving bilinear maps. In a second step, we center our attention to the pointwise product and convolution product particularly. In the case of pointwise product, we consider the bilinear maps acting in couples of Banach function spaces and sequence spaces. We obtain that a bilinear map can be pointwise product factorable if and only if it is zero product preserving. In the sequel, we notice that the same result works if we take into account convolution product and the bilinear maps acting in a product of Hilbert spaces of integrable functions, respectively, a product of Banach algebras of integrable functions. In this case, we get that all bilinear maps that are 0-valued for couples of functions whose convolution equals zero have a factorization through convolution. The other objective of the dissertation is to apply these factorizations to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps by some concavity properties of operators. In addition to them, we give also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces,we adapt and apply our results to the case of some particular classes of integral bilinear operators and kernel operators and explain some consequences in a more applied context.