An application of averaging operators to multilinearity
type: Item
instance of: scholarly article
title: An application of averaging operators to multilinearity
zbMATH Open document ID: 0804.46032
DOI: 10.1007/BF01459517
author: J. Taskinen
published in: Mathematische Annalen
publication date: 1994-04-12T00:00:00Z
full work available at URL: https://eudml.org/doc/165147
review text: For compact metric spaces \(K\), \(S\) and a continuous surjection \(\varphi: K\to S\), let the linear isometry \(\varphi^ 0: C(S)\to C(K)\) be given by \(\varphi^ 0 f= f\circ\varphi\). \(\varphi\) is said to admit a (regular) averaging operator if there exists a (contractive) projection from \(C(K)\) onto \(\varphi^ 0(C(S))\). As an application of the existence (due to A. Pełczyński) of averaging operators, the author proves that certain multilinear operators between spaces \(C(K)\) of continuous functions can be represented as products of bounded linear operators and the (pointwise) multiplication operator on \(C(K)\). More precisely (and more generally), let \(K\) and \(S\) be compact spaces, \(K\) metric uncountable, let \(P\) be a continuous symmetric \(M\)-linear map \(C(K)\times\cdots\times C(K)\to C(S)\), \(M>1\). Assume that, for each \(t\in S\), the map \[ f_ 1(x_ 1)\cdots f_ M(x_ M)\to P(f_ 1,\dots,f_ M)(t),\quad\text{where } f_ j\in C(K), \] can be extended to a continuous linear form \(m^{(t)}\) on \(C(K^ M)\) which depends continuously on \(t\) when \(C(K^ M)^*\) is endowed with the \(\text{weak}^*\)-topology. Then there exist continuous linear operators \(A\in L(C(K),C(K))\), \(B\in L(C(K),C(S))\) such that \[ P(f_ 1,\dots,f_ M)= B\left(\prod^ M_{j=1} Af_ j\right)\qquad\text{for all }f_ 1,\dots,f_ M\in C(K). \] Conversely, if \(P\) can be represented in this form, then the assumptions on \(m^{(t)}\) hold. -- There is a similar result for continuous symmetric \(M\)-linear maps from \(C(K)\times\cdots \times C(K)\) into a Banach space \(X\). The author also discusses the case when \(K\) is non-metrizable.
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 47H99
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46B25
zbMATH DE Number: 537331
zbMATH Keywords: multilinear operator
zbMATH Keywords: integral multilinear form
zbMATH Keywords: polynomial operator
zbMATH Keywords: almost Milutin space
zbMATH Keywords: Cantor set
zbMATH Keywords: products of bounded linear operators and the pointwise multiplication operator
zbMATH Keywords: compact metric spaces
zbMATH Keywords: continuous surjection
zbMATH Keywords: averaging operators
zbMATH Keywords: multilinear operators
reviewed by: Klaus Dieter Bierstedt
MaRDI profile type: Publication
cites work: Q3943468
cites work: On a Lemma of Milutin Concerning Averaging Operators in Continuous Function Spaces
cites work: On a problem of Pełczyński: Milutin spaces, Dugundji spaces and AE(0-dim)
cites work: An injective characterization of Peano spaces
cites work: Classical Banach spaces
OpenAlex ID: W2076200730
description: scientific article; zbMATH DE number 537331