Özet:
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.
