Seminar in operator theory and operator algebras (MATH 9310)
Fall 2009


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


September 1

James Rovnyak, UVa

The operator Fejer-Riesz theorem

September 8

David Sherman, UVa

The generator problem for von Neumann algebras

Can every von Neumann algebra acting on a separable Hilbert space be generated by a single operator? After explaining what this question means, I will survey some of its long history and sketch a few results. This should be a good vehicle for illustrating some of the most basic ideas in von Neumann algebras.

Ward Henson, a logician specializing in applications to functional analysis, will give the department colloquium on September 10 at 4 PM in Kerchof 317.

September 15

Stefanos Orfanos, University of Cincinnati

Quasidiagonality and crossed products of C*-algebras

We describe quasidiagonality and its connection to other important properties of C*-algebras, and then show a couple of results concerning group C*-algebras and crossed products.

September 22

Bill Ross, University of Richmond

Self adjoint extensions of symmetric operators I

It is an old problem in mathematical physics to determine the self adjoint extensions of symmetric operators (if they even exist at all). In this work, I'll discuss an approach which involves developing a model theory for these operators in terms of de Branges spaces. Since this is not the operator theory usually covered in our seminar, I'll be sure to include an exposition of the basics of unbounded operators (namely Schrodinger operators).

September 29

Bill Ross, University of Richmond

Self adjoint extensions of symmetric operators II

See previous week's abstract.

October 6

No meeting - reading day

October 13

Sivaram Narayan, Central Michigan University

Normal weighted composition operators on the Hardy space

Let φ be an analytic selfmap of the open unit disc U and let ψ be an analytic function on U such that the weighted composition operator Wψ,φ defined by Wψ,φf = ψ f ° φ is bounded on the Hardy space H2(U).

In this talk we present a characterization of weighted composition operators that are unitary, showing that in contrast to the unweighted case, every automorphism of U induces a unitary weighted composition operator. A conjugation argument, using these unitary operators, allows us to describe all normal weighted composition operators for which the inducing map φ fixes a point in U. In general, we show that if Wψ,φ is normal, then φ must be univalent and ψ must be nonzero at each point of U. Descriptions of spectra are provided for the operator Wψ,φ when it is unitary or when it is normal and φ fixes a point in U.

This is a joint work with Paul S. Bourdon.

October 20

Sivaram Narayan, Central Michigan University

Compact weighted composition operators on the Hardy space

Suppose ψ is an analytic function on the open unit disk D and φ is an analytic self-map of D. The weighted composition operator Wφ, ψ is defined on the Hardy space H2(D) as follows:

(Wφ, ψf)(z) = ψ(z)f(φ(z)),

where z is in D and f is in H2(D). In this talk we provide necessary and sufficient conditions for certain classes of φ and ψ such that Wφ, ψ is compact or Hilbert-Schmidt.

October 27

Craig Kleski, UVa

Diagonalization

In 1909, Weyl proved that every bounded self-adjoint operator A on a Hilbert space is D+K, where D is diagonal and K is compact. Von Neumann improved this result by showing that K can be made arbitrarily small in norm. A theorem of Berg in 1971 expanded the result to normal operators. This implies that the Weyl-von Neumann-Berg theorem can be reinterpreted as a result about abelian C* algebras. In 1976, Voiculescu proved a noncommutative version of the Weyl-von Neumann-Berg theorem. Voiculescu's theorem furnishes answers to important operator-theoretic and operator-algebraic questions, and includes applications as varied as reducible operators, representations of separable C* algebras, and BDF theory.

November 3

Mrinal Raghupathi, Vanderbilt University

Nevanlinna-Pick interpolation and Fuchsian groups

Given a Riemann surface R, a set of n points z1,...,zn in R, and a set of scalars w1,...,wn in C, the Nevanlinna-Pick problem is concerned with finding necessary and sufficient conditions for the existence of a holomorphic map f from R to C such that f(zj) = wj, j=1,...,n.

In this talk we will describe an approach based on viewing the algebra of bounded holomorphic functions on R as a fixed-point subalgebra for the action of a Fuchsian group on H(D). We will present two approaches to the problem. The first is based on Sarason's pioneering duality approach; the second is based on McCullough's ideas for the extremal problem on the annulus.

The Virginia Operator Theory and Complex Analysis Meeting (VOTCAM) will be held at Virginia Commonwealth University in Richmond on Saturday, November 7.

November 10

Ugur Gul, Sabanci University (visiting UVa)

The Cauchy integral formula and composition operators

The Cauchy integral is the reproducing kernel for the Hardy space of the upper half-plane. Hence inserting an inducing map into this reproducing kernel gives a nice integral representation of a composition operator. This talk is about how to push this slick but fundemental idea forward in order to get a picture of a spectral theory of some classes of composition operators. Several examples will be illustrated as well.

November 17

Marian Robbins, Cal Poly - San Luis Obispo (visiting UVa)

The numerical range of an operator on a Hilbert space

We will look at some of the general theory of the classical numerical range of an operator, and then look at the numerical ranges of some composition operators on the Hardy space of the disk.

November 24

No meeting - day before Thanksgiving break

December 1

Katie Quertermous, UVa

Composition operators and crossed products

In this talk, we consider unital C*-algebras generated by composition operators acting on the Hardy space. Let F be the collection of all composition operators induced by linear-fractional, non-automorphism self-maps of the unit disk that fix the point 1. We will investigate the structure of C*(F) as well as the structures of the C*-algebras generated by the compact operators and a single composition operator from F. These C*-algebras, modulo the compact operators, are isomorphic to unitized crossed products, and, in the single operator case, we can calculate the K-theory of the C*-algebra.

December 8

Katie Quertermous, UVa

Composition operators and crossed products II

See previous week's abstract.




You can reminisce about previous semesters at the links below:
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