679 research outputs found
On the Life and Work of S. Helgason
This article is a contribution to a Festschrift for S. Helgason. After a
biographical sketch, we survey some of his research on several topics in
geometric and harmonic analysis during his long and influential career. While
not an exhaustive presentation of all facets of his research, for those topics
covered we include reference to the current status of these areas.Comment: Final versio
Fock spaces corresponding to positive definite linear transformations
Suppose is a positive real linear transformation on a finite dimensional
complex inner product space . The reproducing kernel for the Fock space of
square integrable holomorphic functions on relative to the Gaussian measure
is described
in terms of the holomorphic--antiholomorphic decomposition of the linear
operator . Moreover, if commutes with a conjugation on , then a
restriction mapping to the real vectors in is polarized to obtain a
Segal--Bargmann transform, which we also study in the Gaussian-measure setting
The Restriction Principle and Commuting Families of Toeplitz Operators on the Unit Ball
On the unit ball B^n we consider the weighted Bergman spaces H_\lambda and
their Toeplitz operators with bounded symbols. It is known from our previous
work that if a closed subgroup H of \widetilde{\SU(n,1)} has a
multiplicity-free restriction for the holomorphic discrete series of
\widetilde{\SU(n,1)}, then the family of Toeplitz operators with H-invariant
symbols pairwise commute. In this work we consider the case of maximal abelian
subgroups of \widetilde{\SU(n,1)} and provide a detailed proof of the pairwise
commutativity of the corresponding Toeplitz operators. To achieve this we
explicitly develop the restriction principle for each (conjugacy class of)
maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform.
In particular, we obtain a multiplicity one result for the restriction of the
holomorphic discrete series to all maximal abelian subgroups. We also observe
that the Segal-Bargman transform is (up to a unitary transformation) a
convolution operator against a function that we write down explicitly for each
case. This can be used to obtain the explicit simultaneous diagonalization of
Toeplitz operators whose symbols are invariant by one of these maximal abelian
subgroups
Reflection positive affine actions and stochastic processes
In this note we continue our investigations of the representation theoretic
aspects of reflection positivity, also called Osterwalder--Schrader positivity.
We explain how this concept relates to affine isometric actions on real Hilbert
spaces and how this is connected with Gaussian processes with stationary
increments
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