Abstract
We prove a factorization theorem for multilinear operators acting in topological products of spaces of (scalar) p-summable sequences through a product. It is shown that this class of multilinear operators called product factorable maps coincides with the well-known class of the zero product preserving operators. Due to the factorization, we obtain compactness and summability properties by using classical functional analysis tools. Besides, we give some isomorphisms between spaces of linear and multilinear operators, and representations of some classes of multilinear maps as n-homogeneous orthogonally additive polynomials.
Highlights
The objective of the paper is to present a factorization theorem for multilinear operators de...ned on the topological product of spaces of p-summable sequences through the product of scalar sequences
We prove a factorization theorem for multilinear operators acting in topological products of spaces of p-summable sequences through a product
Such a factorization has been studied for multilinear operators de...ned on Banach algebras and vector lattices, and in the last years it has been studied for Banach spaces
Summary
The objective of the paper is to present a factorization theorem for multilinear operators de...ned on the topological product of spaces of p-summable sequences through the product of (multiple) scalar sequences. The author together with other mathematicians have investigated the class of multilinear operators acting in the topological product of Banach function spaces and integrable functions factoring through the pointwise product and the convolution operation, respectively (see [12,13,14]) Motivated by these ideas, in this paper we introduce the notion of product factorability for multilinear operators de...ned on topological products of spaces of (scalar) p-summable sequences, and we prove that this class coincides with the class of zero product preserving multilinear maps.
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