We generalise the classical Weyl pseudo-differential calculus on Rd to the setting of two d-tuples of operators A = (A1,..., Ad) and B = (B1,..., Bd) acting on a Banach space generating bounded C0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl calculus extends to symbols in the standard symbol class S0 provided appropriate bounds can be established. Using transference techniques we obtain boundedness of the H∞-functional calculus (and even the Hormander calculus), for the abstract harmonic oscillator.
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- Canonical commutation relations
- H¥-functional calculus
- Pseudo-differential calculus
- Spectral multipliers
- Transference of C-groups
- Twisted convolution
- UMD spaces
- Weyl pairs