Michael Hartz, Department of Pure Mathematics, University of Waterloo
"Nevanlinna-Pick spaces and dilations"
I
will
present
some
of
the
results
in
my
thesis.
A
large
part
of
the
talk
will
be
devoted
to
Nevanlinna-Pick
spaces
and
their
multiplier
algebras.
These
algebras
occupy
an
important
place
at
the
interface
between
operator
algebras,
operator
theory
and
complex
analysis.
I
will
talk
about
the
classification
problem
for
these
algebras
from
three
different
points
of
view:
Multiplier
algebras
associated
to
embedded
discs
in
the
Drury-Arveson
space,
multiplier
algebras
of
homogeneous
varieties,
and
the
Borel
complexity
of
the
classification
problem.
Moreover,
I
will
indicate
that
the
Hardy
space
on
the
unit
disc
is
essentially
the
only
Nevanlinna-Pick
space
whose
multiplication
operators
are
all
hyponormal.
Finally,
I
will
talk
about
dilations
and
von
Neumann's
inequality.
While
it
is
known
that
there
are
three
commuting
contractions
which
do
not
satisfy
von
Neumann's
inequality,
I
will
present
a
result
which
shows
that
von
Neumann's
inequality
holds
for
all
multivariable
weighted
shifts.
This
provides
a
positive
answer
to
a
question
of
Lubin
and
Shields.