University of Georgia
Mathematics Department Colloquium
Cora Sadosky
Howard University
April 8, 1999
How a general dilation construction pays off in analysis
Abstract:
Positive definite functions admit integral representation as
Fourier transforms of positive measures. This famous result, since
its discovery in the early part of the century, has shown to have
major applications in probability, harmonic analysis, and even
operator theory. Parallel applications require a more general
integral representation theorem, that includes as a bonus an
extension property. The procedure involves extending a form to a
new containing space while preserving its norm and invariance
properties. This dilation cum representation extends to positive
definite vector-valued functions, and to completely positive maps,
leading to non-commutative results. More surprisingly, with
appropriate definitions one can carry such a dilation construction,
not only in ordinary (Lax-Phillips) scattering systems, but in
scattering structures with several (not necessarily commuting)
evolution groups. The corresponding integral representations
provide applications to function spaces in several variables (e.g.,
the Hardy space in the polydisk and symplectic spaces). (Joint work
with Mischa Cotlar.)