Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Joachim Toft

Studia Mathematica, Tome 178 (2007), p. 63-80 / Harvested from The Polish Digital Mathematics Library

Résumé

Let WF⁎ be the wave front set with respect to $C^{∞}$, quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C₀^{∞}$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u∗_{B}φ(x) = ∫ u(x-y)φ(y)B(x,y)dy$, where $B ∈ C^{∞}$ is appropriate, and prove that if $(u∗_{B}φ,φ) ≥ 0$ for every $φ ∈ C₀^{∞}$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

Détail

Publié le : 2007-01-01

Zbl 1110.35301

EUDML-ID : urn:eudml:doc:286203