Positivity of a system of differential operators
type: Item
instance of: scholarly article
title: Positivity of a system of differential operators
zbMATH Open document ID: 0607.34008
DOI: 10.1016/0022-0396(87)90041-6
author: B. George
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/26871
review text: A necessary and sufficient condition for the positivity of the system of Weyl differential operators \(p^ w(x,D)\) with symbol \[ p(x,\xi)=\left[ \begin{matrix} ax^ 2+b\xi^ 2\\ \alpha x\xi \end{matrix} \begin{matrix} \alpha x\xi \\ cx^ 2+d\xi^ 2\end{matrix} \right],\quad a,b,c,d\geq 0 \] and \(ad+bc\neq 0\) is obtained. This condition depends only on ad, bc and \(\alpha\). In particular, it is shown that \((abcd)^{1/2}\geq \alpha^ 2/4\) is a sufficient condition. On the other hand, if \(abcd=0\), then \(p^ w(x,D)\geq 0\) if and only if \(\alpha =0\). This last result generalizes an example of Hörmander.
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 3981568
zbMATH Keywords: positivity
zbMATH Keywords: Weyl differential operators
MaRDI profile type: Publication
OpenAlex ID: W2060061440
cites work: The weyl calculus of pseudo-differential operators
cites work: Q3215519
cites work: Lower bounds for pseudo-differential operators
description: scientific article; zbMATH DE number 3981568