The oscillator semigroup. (English) Zbl 0687.47034

Howe, Roger

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)

Keywords:

symplectic group; Cayley transform; oscillator semigroup; semigroup of continuous linear operators; Gaussian kernels; algebra of functions under twisted convolution

Citations:

Zbl 0644.00001