Nonlocal elliptic operators for compact Lie groups
type: Item
instance of: scholarly article
title: Nonlocal elliptic operators for compact Lie groups
zbMATH Open document ID: 1213.47012
DOI: 10.1134/S1064562410020262
published in: Doklady Mathematics
publication date: 2010-11-17T00:00:00Z
review text: The paper considers the class of nonlocal operators corresponding to shifts along orbits under the action of a compact Lie group on a smooth manifold. Suppose that a compact Lie group \(G\) acts on a smooth closed manifold \(M\). Let \(T^*_{G}M:=\{(x, \xi) \in T^*M \mid \langle\xi, X_{M}(x) = 0\;\forall X \in \mathcal(G)\}\) and \(S^*_{G}M\) be the subspace consisting of covectors with unit length in \(T^*_{G}M\). Then \(G\) acts on \(T^*_{G}M\) and \(S^*_{G}M\) by differentials. Denote the crossed product of the algebra \(C(S^*_{G}M)\) and \(G\) corresponding to the action of \(G\) by \(C(S^*_{G}M) \rtimes G\). The paper introduces a definition of ellipticity for nonlocal operators. Namely, a nonlocal operator on \(L^{2}(M, {\mathbb{C}}^{N})\) is called elliptic if and only if its principal symbol is invertible in the algebra Mat\(_{N}( C(S^*_{G}M) \rtimes G )^{+}\). The main results of the paper are Theorems 1,2 which assert that elliptic nonlocal operators are Fredholm and give an index formula for these operators in topological terms.
reviewed by: Le Anh Vu
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 5816916
zbMATH Keywords: nonlocal operator
zbMATH Keywords: nonlocal elliptic operator
zbMATH Keywords: index of operator
zbMATH Keywords: symbol of operator
Wikidata QID: Q115247486
author: Anton Savin
author: Boris Sternin
MaRDI profile type: Publication
cites work: The index of elliptic operators. II
cites work: Elliptic operators and compact groups
cites work: Q4222027
cites work: The index of elliptic operators over V-manifolds
cites work: Pseudodifferential operators on Hilbert bundles
cites work: Equivariant index formulas for orbifolds
full work available at URL: s1064562410020262
OpenAlex ID: W1984175445
description: scientific article; zbMATH DE number 5816916