Convolution factorability of bilinear maps and integral representations
type: Item
instance of: scholarly article
title: Convolution factorability of bilinear maps and integral representations
zbMATH Open document ID: 1423.46059
author: Ömer Gök
author: Ezgi Erdogan
published in: Indagationes Mathematicae. New Series
publication date: 2018-09-20T00:00:00Z
review text: 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 …
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 47H60
Mathematics Subject Classification ID: 44A35
zbMATH DE Number: 6939367
zbMATH Keywords: bilinear operators
zbMATH Keywords: factorization
zbMATH Keywords: zero product preserving map
zbMATH Keywords: summability
zbMATH Keywords: Hilbert-Schmidt operators
zbMATH Keywords: integral representation
MaRDI profile type: Publication
full work available at URL: j.indag.2018.06.003
OpenAlex ID: W2810080218
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Wikidata QID: Q129587790
DOI: 10.1016/J.INDAG.2018.06.003
description: scientific article; zbMATH DE number 6939367