Invited Speaker Abstracts
James Agler (UCSD)
Shur models, realizations, and the generalization of theorems
of Caratheodory, Julia and Wolf to several variables
Classical theorems due to Caratheodory, Julia and Wolf form one
of the cornerstones for understanding the complex geometry of an
analytic function of one complex variable near points on the boundary
of the domain of definition where the function attains its maximum.
Using Hilbert space methods, we show how this theory can be generalized
to several variables.
Co-authors: John McCarthy and Nicholas Young
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William Arveson (UC Berkeley)
Standard Hilbert modules and the K-homology of algebraic varieties
Standard Hilbert modules are the appropriate Hilbert space counterparts
of free modules of finite rank over polynomial algebras in several
variables. We describe the role of standard Hilbert modules in allowing
one to generate concrete K-homology classes of algebraic varieties,
focusing on the operator-theoretic conjectures and unsolved problems
that emerge from these ideas.
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Joseph Ball (VPISU)
Hardy algebras associated with $W*$-correspondences: examples
and applications
Given a $W^{*}$-algebra $M$ and a $W^{*}$-correspondence over $M$,
as developed in a series of papers of Paul Muhly and Baruch Solel,
one can associate a Fock space which is itself a $W^{*}$-correspondence
over $M$ (the direct sum of all $n$-fold self-tensor products of
$E$ over $n=0,1,2, \dots$) and a tensor algebra (the weak-$*$ closed
algebra generated by the left diagonal action of $M$ and the left
creation operators generated by elements of $E$). If one also fixes
a representation $\sigma$ mapping $M$ into an algebra of operators
on some Hilbert space $H$, then one arrives at a generalized Hardy
algebra ${\mathcal F}(E, \sigma)$ of operators acting on a Hilbert
space ${\mathcal F}^{2}(E, \sigma)$.One can view this object either
as an analogue of the lower triangular Toeplitz matrices acting
on $\ell^{2}$, or, equivalently, as an analogue of the Hardy algebra
(bounded analytic functions on the unit disk) acting as multiplication
operators on the Hardy space $H^{2}$ over the unit disk. The generality
of this construction masks the full array of detail encoded in particular
examples. This talk reviews recent work of the speaker and collaborators
(Quanlei Fang, Gilbert Groenewald and Sanne ter Horst) on fleshing
out the Muhly-Solel theory for particular apparently disparate special
cases. Themes include: Toeplitz structure, generalized input/state/output
linear systems and associated transfer function realizations, Nevanlinna-Pick
interpolation (full-value as well as tangential), and generalized
de Branges-Rovnayk kernels. We also discuss possible extensions
of the formalism to handle transfer-function realization and Nevanlinna-Pick
interpolation in settings which do not immediately fit into this
formalism, e.g., noncommutative Schur-Agler classes.
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Ronald Douglas (TAMU)
On Resolutions of Hilbert Modules
One approach to the study of general Hilbert modules over natural
function algebras is via resulutions by canonical modules. Although
methods from algebra provide motivation for such an effort, interpretation
of the very successful model theory of Sz.-Nagy-Foias for contractions
provides justification within operator theory itself. In the multivariate
case or the case in which one allows canonical modules other than
the Hardy space over the disk, the subject has more questions than
answers although some interesting results, sometimes indirectly
related to this topic have been obtained in the last decade or so.
In this talk I will review some recent work in which the canonical
modules are either the Drury-Arveson module or more general quasi-free
or "nice" kernel Hilbert space modules mainly over the
unit ball in C^m. One result is a generalization of a result of
Sz.-Nagy and Foias characterizing when certain quotient Hilbert
modules are similar to the Hardy module. Another determines when
certain nice short resolutions, which coincide with dilations, exist.
Most of the work is joint with Ciprian Foias, Gadadhar Misra, and
Jaydeb Sarkar.
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Stefan Richter (University of Tennessee)
Extremals for families of commuting operators.
A d-tuple T=(T_1,..,T_d) of commuting Hilbert space operators is
called a spherical contraction if the column operator x ->(T_1
x, ... T_d x) is contractive, and T is said to be an extremal spherical
contraction if the only way to extend T to another spherical contraction
acting on a larger Hilbert space is by taking direct sums. In this
talk I will discuss a result that identifies the extremal spherical
contractions. As a Corollary one obtains a spectral description
of operators of the form S direct sum U, where S is a direct sum
of d-shifts acting on a Drury-Arveson Hardy space and U is a spherical
unitary operator tuple. This may be considered an extension of the
Wold decomposition Theorem of isometric operators. The results are
from joint work with Carl Sundberg.
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Kristian Seip (Trondheim University)
Bounded analytic functions on the infinite polydisc
The algebra of bounded analytic functions on the infinite polydisc
appears naturally in the study of Dirichlet series. I will review
some recent results from the function theory of this algebra, regarding
boundary values, homogeneous polynomials, Bohr radii, and interpolating
sequences. I will also describe some open problems. A key result
to be presented is the hypercontractivity of the Bohnenblust--Hille
inequality for homogeneous polynomials (joint work with J. Ortega-Cerdà,
A. Defant, L. Frerick, and M. Ounaïes). Results from joint
work with E. Saksman and with J. Marzo will also be discussed.
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Dror Varolin (Stony Brook University)
Weighted-L^2 extension of holomorphic functions from hypersurfaces
in C^n
We discuss the problem of extension of holomorphic functions from
a possibly singular hypersurface in C^n with weighted-L^2 growth
conditions. We begin with a review of the literature, and then discuss
some sufficient conditions for extension. Those conditions are known
not to be necessary, as we will show by example. Finally we state
some open problems.
