On positive operators on some ordered Banach spaces
type: Item
instance of: scholarly article
title: On positive operators on some ordered Banach spaces
zbMATH Open document ID: 1113.46011
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: A well-known theorem due to B.\,Maurey [cf.\ \textit{B.\,Maurey} and \textit{G.\,Pisier}, Stud.\ Math.\ 58, 45--90 (1976; Zbl 0344.47014)] asserts that if \(B\) is a Banach space having cotype \(q\geq2\) and \(K\) is a compact Hausdorff space, then each continuous linear map \(T:C(K)\rightarrow B\) is \(r\)-summing, for every \(r>q\). In the paper under review, among other results, the author obtains some variations (and a sketch of an alternative proof) of this theorem. The present author defines \(i(X)=\inf \left\{p\mid(\sum_{i=1}^{n}\left\| x_{i}\right\| ^{p})^{\frac{1}{p}}\leq C_{p}\left\| \sum_{i=1}^{n}x_{i}\right\| \right\}\), where \(x_{1},\dots,x_{n}\in X\), where \(X\) is a normal cone contained in a Banach space \(B\) and \(C_{p}\) is some constant. Then he presents an interesting counterpart of the result of Maurey, where \(i(X)\), in some sense, can be seen as a kind of cotype of the cone \(X\). More precisely, if \(K\) is a compact Hausdorff space and \(B\) is a Banach space, ordered by a normal cone \(X\) such that \(i(X)<\infty\), then, for \(r>i(X)\), each positive continuous operator \(T:C(K)\rightarrow B\) is \(r\)-summing and \(\Pi_{r}(T)\leq C\left\| T\right\|\), where \(C\) is some constant. Reciprocally, if \(r>1\) enjoys the preceding properties, then \(r\geq i(X)\). Some applications to conical measures are also presented. The last section of the paper is devoted to the study of positive operators with values in an \(L^{1}\)-space. These results can be regarded, in some sense, as a dual counterpart of the main result of the previous section.
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 5070756
zbMATH Keywords: summing operators
zbMATH Keywords: ordered Banach spaces
author: Richard Becker
reviewed by: Daniel Pellegrino
MaRDI profile type: Publication
full work available at URL: s11117-003-2785-6
OpenAlex ID: W1983326582
cites work: Cones contained in a Banach space and factorization of positive operators defined on its dual with values in a space \(L^ 1\)
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cites work: Factorization of operators through \(L_{p\infty}\) or \(L_{p1}\) and non- commutative generalizations
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DOI: 10.1007/S11117-003-2785-6
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description: scientific article; zbMATH DE number 5070756