## The oscillator semigroup.(English)Zbl 0687.47034

The mathematical heritage of Hermann Weyl, Proc. Symp., Durham/NC 1987, Proc. Symp. Pure Math. 48, 61-132 (1988).
[For the entire collection see Zbl 0644.00001.]
A semigroup, termed by the author, the oscillator semigroup is defined. Several realizations of this semigroup are presented. One realization is a semigroup of continuous linear operators (those determined by a class of Gaussian kernels) on $$L^ 2({\mathbb{R}}^ n)$$. Another is an algebra of functions under twisted convolution. The structure of the semigroup is described and numerous applications to analysis are given.
Reviewer: J.P.Holmes

### MSC:

 47D03 Groups and semigroups of linear operators 22A20 Analysis on topological semigroups 47B38 Linear operators on function spaces (general)

Zbl 0644.00001