The symbols of a Frechet algebra of pseudodifferential operators on a polycylinder

Abstract

H. O. Cordes investigated a C*-algebra C of singular integral operators on a polycylinder Ω=ℝn×B and defined symbol homomorphisms {Mathematical expression} where {Mathematical expression} is a certain compact space, {Mathematical expression} and Cx is the Laplace comparison algebra of the compact space B (cf. [2]). We give a characterization of the Frechet algebra C∞⊂C obtained as the closure of a certain algebra of ψ dos in the topology generated by all HS norms. We define Bρ:L(HS)→L(HS)(ρe{open}S) by analogy with [4, 7] and prove that Bρ maps C into C∞ continuously. As a corollary we get {Mathematical expression}, generalizing surjectivity results from [4] and [7]. It seems that no characterization of γ(C∞) is known, but it is clear that {Mathematical expression} where {Mathematical expression} is the Frechet algebra studied in [1] and [7]. © 1988 Birkhäuser Verlag.