Derivatives of orbital function and an extension of Berezin-Gel'fand's theorem
type: Item
instance of: scholarly article
title: Derivatives of orbital function and an extension of Berezin-Gel'fand's theorem
zbMATH Open document ID: 1372.15015
author: Tin-Yau Tam
author: William C. Hill
published in: Special Matrices
publication date: 2017-01-09T00:00:00Z
review text: The authors use the nonsmooth analytic approach employed by \textit{A. S. Lewis} [SIAM J. Matrix Anal. Appl. 21, No. 2, 379--381 (1999; Zbl 1047.90511)] in his proof of Lidskii's theorem on eigenvalue perturbation to obtain a generalization of the Lie theoretic abstract version of Lidskii's theorem due to \textit{F. A. Berezin} and \textit{I. M. Gel'fand} [Transl., Ser. 2, Am. Math. Soc. 21, 193--238 (1962; Zbl 0195.42302); translation from Tr. Mosk. Mat. Obshch. 5, 311--351 (1956)]. This generalization is in the context of Eaton triples,\ an Eaton triple \((V,G,F)\) consisting of a finite-dimensional real inner product space \(V\), a closed subgroup \(G\) of the orthogonal group on \(V\), and a nonempty closed convex cone \(F\subset V\). To obtain this generalization, the authors first extend some results due to \textit{A. S. Lewis} [Math. Oper. Res. 21, No. 3, 576--588 (1996; Zbl 0860.49017)] on derivatives of spectral functions to the context of orbital functions (a function \(f\) defined on a \(G\)-invariant subset \(U\) of \(V\) is said to be an orbital function if it is constant on each orbit \(Gz\) of \(z\in U\)).
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 6672753
zbMATH Keywords: orbital function
zbMATH Keywords: Berezin-Gel'fand theorem
zbMATH Keywords: Eaton triple
zbMATH Keywords: nonsmooth analysis
zbMATH Keywords: subdifferential
zbMATH Keywords: Clarke generalized gradient
zbMATH Keywords: Lebourg mean value theorem
zbMATH Keywords: finite reflection group
zbMATH Keywords: eigenvalue perturbation
zbMATH Keywords: real inner product space
zbMATH Keywords: orthogonal group
reviewed by: Juan-Enrique Martínez-Legaz
MaRDI profile type: Publication
full work available at URL: spma-2016-0033
OpenAlex ID: W2563442935
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DOI: 10.1515/SPMA-2016-0033
description: scientific article; zbMATH DE number 6672753