THE WEYL CALCULUS FOR GROUP GENERATORS SATISFYING THE CANONICAL COMMUTATION RELATIONS

摘要

We generalise the classical Weyl pseudo-differential calculus on R-d to the setting of two d-tuples of operators A = (A(1), ..., A(d)) and B = (B-1, ..., B-d) acting on a Banach space generating bounded C-0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl calculus extends to symbols in the standard symbol class S-0 provided appropriate bounds can be established. Using transference techniques we ob- tain boundedness of the H'-functional calculus (and even the Hormander calculus), for the abstract harmonic oscillator L = 1/2 Sigma(d)(j=1) (A(j)(2) + B-j(2)) - 1/2 d.