Seminar in operator theory and operator algebras (MATH 9310)
Spring 2013

The seminar is organized by David Sherman. We meet Tuesdays 3:30-4:30 in Kerchof 326.

January 15

Sergey Belyi, Troy University

On system realizations of Herglotz-Nevanlinna functions

In this overview talk we discuss realization problems for linear systems of Livsic type (L-systems) with an unbounded state-space operator. The main object of our study is a class of operator-valued Herglotz-Nevanlinna (H-N) functions that can be realized as a linear-fractional transformation of the transfer function of an L-system. We will show how direct and inverse problems for such type of systems are solved and provide a complete description of realizable HN operator-functions in a finite-dimensional Hilbert space. Various subclasses of the class of realizable H-N functions will be described in connection with specific properties of realizing L-systems. We will also talk about the general realization problem with non-canonical systems. As an important application, we will discuss the realization of certain scalar H-N functions by L-systems that are based on a non-self-adjoint Schrodinger operator in L2[a, +∞).

This talk is based on joint work with Yury Arlinskii and Eduard Tsekanovskii and will survey all the above mentioned developments and connections.

Also there is a colloquium on Jan 17 by Kate Juschenko of Vanderbilt.

January 22 - February 12

No meetings

February 19

starts at 3:15

Vrej Zarikian, US Naval Academy

Bimodules over Cartan subalgebras, and Mercer's extension theorem

The abstract is here.

Also there is a colloquium at 4:30 by Stephen Curran of MIT.

February 26

starts at 3:15

Craig Kleski, UVa

Peaking for operator systems

The Bishop-de Leeuw theorem shows how various notions of peaking for function spaces are related. In this talk, I will discuss progress toward a noncommutative version of this theorem.

March 5

Martino Lupini, York University

C*-algebras and omitting types

I will give an introduction to model theory for operator algebras. I will then explain how many important classes of C*-algebras can be characterized by the model-theoretic notion of omitting types. I will conclude by presenting some applications to the theory of UHF and AF algebras.

March 12


March 19

David Blecher, University of Houston

Generalized invertibility and outers in noncommutative Hardy spaces

We review noncommutative Hp spaces (Arveson's von Neumann algebraic generalization of the Hardy spaces of the disk), and discuss generalized notions of invertibility and outers, focusing on some work in progress with Labuschagne. Hopefully the seminar will be just a little interactive, with some voluntary input from the audience on the purely function theoretic case.

March 26

Chris Ramsey, University of Waterloo

Triangular operator algebras

A subalgebra of a C*-algebra is triangular if its intersection with its adjoint is a maximal abelian self-adjoint algebra. In a uniformly hyperfinite algebra the closed union of a chain of upper triangular matrix algebras is triangular, this will be the main object for discussion. I will talk about the isometric automorphism groups and dilation theory of these algebras. In particular, we can describe the C*-envelope of the semidirect product of one of these algebras by an automorphism.

April 2

Rob Martin, University of Cape Town

Nearly invariant subspaces and symmetric operators

Here is the abstract.

April 9

Stephen Hardy, UVa

"Ultra"-techniques in Banach space theory I

An introduction to the theory of ultrafilters and ultraproducts along with applications to Banach space theory, based on Stefan Heinrich's 1980 paper "Ultraproducts in Banach space theory".

April 16

Stephen Hardy, UVa

"Ultra"-techniques in Banach space theory II

Additional applications of ultraproducts to the study of local properties of Banach spaces. Based on Stefan Heinrich's 1980 paper "Ultraproducts in Banach space theory".

April 23

Mor Katz, UVa

The boundary Caratheodory–Fejer problem

The Caratheodory–Fejer problem is to determine whether a finite sequence of complex numbers comprises the initial Taylor coefficients of an analytic function in the Pick class about some point x of the domain. In "The boundary Caratheodory–Fejer interpolation problem", Jim Agler, Zinaida A. Lykova and N.J. Young give a new solvability criterion to the situation where x lies on the boundary, and derive a parametrization of all solutions for the real case. We will discuss their work and follow similar steps to derive a parametrization for the "almost real" case.

April 30

Mor Katz, UVa

Essential normality for a class of composition operators

A Hilbert space operator is essentially normal if its commutator with its adjoint is compact. Recent work of Cowen-Gallardo and Hammond-Moorhouse-Robbins has produced a pointwise formula for the adjoint of a rationally-induced composition operator whose constituent parts contain multiple-valued analytic functions which do not individually represent well-defined operators. We show how to work with this formula to produce legitimate operator expressions involving the adjoint and apply this to find a criterion for essential normality of composition operators induced by functions which extend analytically across the boundary of the unit disk.

You can reminisce about previous semesters at the links below:
Fall 2012 Spring 2012 Fall 2011 Spring 2011 Fall 2010 Spring 2010 Fall 2009 Spring 2009 Fall 2008 Spring 2008 Fall 2007 Spring 2007 Fall 2006 Spring 2006 Fall 2005 Spring 2005 Fall 2004 Spring 2004 Fall 2003 Spring 2003 Fall 2002