Linear systems theory, operator model theory, Beurling-Lax representations for shift-invariant subspaces: extensions to multivariable weighted Bergman space functional models

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