Convolution factorability of bilinear maps and integral representations
The paper under review deals with the analysis of a particular class of bilinear maps acting in products of Banach algebras of functions with the convolution product, that are often called zero product preserving operators. These maps are primarily defined as those that are equal to zero when the convolution of the involved elements is zero. This class is in a sense classical, and can be related with other classes of operators, for example, bilinear maps on Banach algebras that preserve disjointness, which can be understood as a generalization of the so-called Lamperti operators. Note that, for the case of Banach function spaces, to be disjoint or to have a product that is equal to zero a.e. are equivalent properties for couples of functions. So this is related to the contents of this paper, in which the convolution product is considered instead. A factorization theorem is proved for zero product preserving operators. It is proved the relevant fact that the above mentioned operators are the ones that factor through the convolution product bilinear map, composed with a linear operator. This result is given in Lemma 3.3, and Theorem 3.4, which deals with the case of zero product preserving bilinear operators with values in a Banach space \(Y\) as \(B:L^1(\mathbb T) \times U(\mathbb T) \to Y,\) where \(U(\mathbb T) \in \{ L^p(\mathbb T)\, (1 \le p< \infty), \, C(\mathbb T),\,R(\mathbb T) \},\) \(\mathbb T\) is a compact group and \(R(\mathbb T)\) is the associated Wiener algebra. Once this result is proved, the door for analyzing compactness and summability of this class of maps is open. This is done in the rest of the paper, in which some nice results are shown. For example, Corollary 4.4 establishes that, for \(U(\mathbb T) = L^1(\mathbb T)\) or \( C(\mathbb T),\) every Hilbert space-valued zero product preserving bilinear map \(B:L^1(\mathbb T) \times U(\mathbb T) \to H,\) is \(2\)-summing, and satisfies a summing inequality and an integral domination involving the convolution product.
scientific article; zbMATH DE number 6939367
scientific article