# SCIENTIFIC PROGRAMS AND ACTIVITIES

January 16, 2017

## August 10-14, 2009 Multivariate Operator Theory Workshop Fields Institute

Organizers:
Kenneth Davidson, University of Waterloo
Ronald Douglas, Texas A&M University
Joerg Eschmeier, Universitaet des Saarlandes
Mihai Putinar, University of California at Santa Barbara

### 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.