Some connections between Pettis integration and opertor theory
type: Item
instance of: scholarly article
title: Some connections between Pettis integration and opertor theory
zbMATH Open document ID: 0706.46032
DOI: 10.1216/RMJ-1987-17-4-683
author: Elizabeth Bator
author: David Race
author: Paul Lewis
published in: Rocky Mountain Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Suppose X is a Banach space, (\(\Omega\),\(\Sigma\),\(\mu\)) is a probability space, f: \(\Omega\to X\) is a scalarly measurable function such that \(x^*f\in L^ 1(\mu)\) for each \(x^*\in X^*\) and \(T_ f\) is the operator from \(X^*\) to \(L^ 1(\mu)\) whose value at \(x^*\) is \(x^*f\). \textit{R. E. Huff} [Proc. Am. Math. Soc. 96, No.3, 402-404 (1986; Zbl 0611.28005)] and \textit{E. M. Bator} [ibid. 95, No.2, 265-270 (1985; Zbl 0589.46033)] have given elegant exposés of a number of previously elusive results regarding the Pettis integral (including the Geitz- Talagrand core theorem) via a careful study of operators like \(T_ f\). Now, taking up where Bator and Huff left off, the authors generalize the ideas implicit in those studies to general operators. In so doing they obtain some very interesting (initial) characterizations of weak\({}^*\)- weak continuous operators along with further simplifications for results of Odell, Riddle, Saab, Uhl and others on when a Banach space finds itself without a subspace isomorphic to \(\ell^ 1\).
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 4159417
zbMATH Keywords: scalarly measurable function
zbMATH Keywords: Pettis integral
zbMATH Keywords: Geitz-Talagrand core theorem
zbMATH Keywords: \(weak^ *\)-weak continuous operators
MaRDI profile type: Publication
full work available at URL: rmj-1987-17-4-683
OpenAlex ID: W1971405823
description: scientific article; zbMATH DE number 4159417