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

• Autorzy: Joachim Toft
• Języki publikacji: EN

Abstrakty

EN

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 𝒟'. 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.

Kategorie tematyczne

• 35A18: Wave front sets
• 47B65: Positive operators and order-bounded operators
• 35A21: Propagation of singularities
• 43A35: Positive definite functions on groups, semigroups, etc.
• 35S05: Pseudodifferential operators

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 179
• Numer: 1
• Strony: 63-80

Daty

wydano

2007

Twórcy

autor

Joachim Toft

• Department of Mathematics and Systems Engineering, Växjö University, S-351 95 Växjö, Sweden

Identyfikatory

DOI

10.4064/sm179-1-6