Weighted strongly elliptic operators on Lie groups
type: Item
instance of: scholarly article
title: Weighted strongly elliptic operators on Lie groups
zbMATH Open document ID: 0815.58023
author: A. F. M. ter Elst
author: Derek W. Robinson
published in: Journal of Functional Analysis
publication date: 1994-12-20T00:00:00Z
full work available at URL: weighted-strongly-elliptic-operators-on-lie-groups(c4bd377d-5f71-4f4d-bb05-3d27ba406b9b).html
review text: Let \(({\mathcal X},G,U)\) be a continuous representation of a Lie group \(G\) by bounded operators \(g \to U(g)\) on the Banach space \(\mathcal X\) and let \(({\mathcal X},{\mathfrak g},dU)\) denote the representation of the Lie algebra \(\mathfrak g\) obtained by differentiation. Let \(a_ 1, \dots, a_{d'}\) be a Lie algebra basis, \(A_ i = dU(a_ i)\), \(A^ \alpha = A_{i_ 1} \cdots A_{i_ k}\) whenever \(\alpha = (i_ 1, \dots, i_ k)\), and \(H = \sum_ \alpha c_ \alpha A^ \alpha\) where the \(c_ \alpha\) are complex coefficients satisfying a weighted strongly elliptic condition in which different directions may have different weights. The authors prove that the closure \(\overline{H}\) generates a holomorphic semigroup \(S\) with holomorphy sector which contains a non-empty subsector determined by the coefficients and independent of the representation. Also elliptic regularity properties for the operators and their powers are established, and as a corollary optimal growth bounds for the eigenfunctions of the anharmonic oscillators \(P^{2m} + Q^{2n}\).
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35J30
zbMATH DE Number: 704513
zbMATH Keywords: strongly elliptic operator
zbMATH Keywords: holomorphic semigroup
zbMATH Keywords: Lie groups
reviewed by: Jürgen Marschall
MaRDI profile type: Publication
OpenAlex ID: W2026547111
DOI: 10.1006/JFAN.1994.1136
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description: scientific article; zbMATH DE number 704513