Projections in the convex hull of isometries on absolutely continuous function spaces
type: Item
instance of: scholarly article
title: Projections in the convex hull of isometries on absolutely continuous function spaces
author: Priyadarshi Dey
author: Zachary Easley
author: Fernanda Botelho
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-12-20T00:00:00Z
review text: A projection \(P\) on a complex Banach space is called \textit{bi-circular} if \(e^{i\alpha} P + e^{i\beta}(I-P)\) is an isometry for all real numbers \(\alpha\) and \(\beta\), and \(P\) is a \textit{generalized bi-circular projection} if \(P + \lambda (1-P)\) is an isometry for some uni-modular scalar \(\lambda \neq 1\). Let \(K\) be a compact and connected Hausdorff space. It is a result of the first-named author [\textit{F.~Botelho}, J. Math. Anal. Appl. 341, No.~2, 1163--1169 (2008; Zbl 1139.47026)] that the average of two isometries on \(C(K)\) is a projection if and only if it is a generalized bi-circular projection. The object of study in this paper is projections in the convex hull of \(n\) surjective linear isometries. It is shown that in general if \(I_1\) and \(I_2\) are surjective isometries on a Banach space \(X\) and a non-trivial convex combination \(\lambda I_1 + (1-\lambda) I_2\) is a non-trivial projection, then \(\lambda = 1/2\). It is shown that there exists a single projection in the convex hull of all isometries in a cyclic group of finite order, and that this projection is the average of all the isometries in the group. These results are then applied in the setting of vector-valued absolutely continuous function on a compact subset of \(\mathbb{R}\) with at least two points, with values in a strictly convex Banach space.
reviewed by: Vegard Lima
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 47B01
Mathematics Subject Classification ID: 47A99
zbMATH DE Number: 7635000
zbMATH Keywords: projections in the convex hull of isometries
zbMATH Keywords: periodic linear isometries
zbMATH Keywords: finite spectrum
zbMATH Keywords: spaces of absolutely continuous functions
MaRDI profile type: Publication
full work available at URL: j.jmaa.2022.126875
OpenAlex ID: W4309825557
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DOI: 10.1016/J.JMAA.2022.126875
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description: scientific article; zbMATH DE number 7635000