Joseph Ball (Virginia Tech)

Tuesday, November 17, 2015, 14:00 – 15:00, Math -101

Please Note the Unusual Day!

Abstract:

It is well known that subspaces of the Hardy space over the unit disk which are
invariant under the backward shift operator occur as the image of an observability
operator associated with a discrete-time linear system with stable state-dynamics,
as well as the functional-model space for a Hilbert space C·0-contraction operator,
while forward shift-invariant subspaces have a Beurling-Lax representation in
terms of an inner function. We discuss several variants of these statements in the
context of (1) weighted Bergman spaces on the unit disk (the single-variable context)
as well as (2) a weighted Fock space of formal power series in a collection
of d freely noncommuting indeterminates. The first case gives a model theory for
n-hypercontractions and Beurling-Lax representations for forward shift-invariant
subspaces of a weighted Bergman space on the unit disk; the second case gives a
model theory for n-hypercontractive operator d-tuples and Beurling-Lax representations
for the weighted-Bergman multishift invariant subspaces of a weighted Fock
space. The talk reports on joint work with Vladimir Bolotnikov of the College of
William and Mary