OPERATOR THEORY SEMINAR

SPRING 2007

Bill Ross, November 14, 2006

The gang in Tom Kriete's office, November 14, 2006. Left to right: Katie Quertermous, Katherine Heller, Rebecca Schmitz, Bill Duren, Matthew Pons, and Tom Kriete

## RECENT PAST SCHEDULES:

The seminar meets Tuesdays 3:30-4:30 PM in KER 326. Special lectures outside this hour may be announced from time to time.

Spring 2006 , Fall 2006

Spring 2005 , Fall 2005

Spring 2004 , Fall 2004

Spring 2003 , Fall 2003

Fall 2002

Feb. 19-23

Tuesday, Feb. 20

James Rovnyak

Schwarz-Pick inequalities for the Schur-Agler class on the polydisk and unit ball

Feb. 26 - Mar. 2

Tuesday, Feb. 27

James Rovnyak

Schwarz-Pick inequalities for the Schur-Agler class on the polydisk and unit ball, continued

Mar. 5-9

Tuesday, Mar. 6

NO MEETING: SPRING BREAK

Title

Mar. 12-16

SPECIAL LECTURES

Monday, March 12, 2 PM

Anna-Maria Persson, University of Lund

On the spectrum of the Cesaro operator on spaces of analytic functions

Monday, March 12, 3:30 PM

Marcus Carlsson, University of Lund

On the index of invariant subspaces in spaces of vector-valued analytic functions

Mar. 19-23

Tuesday, Mar. 20

James Rovnyak

Schwarz-Pick inequalities for the Schur-Agler class on the polydisk and unit ball, continued

Mar. 26-30

SPECIAL LECTURE

Friday, March 30, 3:30 PM, KER 205

Joe Ball, Virginia Tech

Models for commuting pairs of Hilbert space contraction operators

The Sz.-Nagy dilation theorem asserts that any contraction operator T on a Hilbert space H can be dilated to a unitary operator U on a larger Hilbert space K. Later versions of this dilation theorem give a detailed picture of the geometry of how the original space H sits inside the dilation space K. This geometry in turn leads simultaneously to the Sz.-Nagy-Foias model for the contraction operator T and to a discrete-time Lax-Phillips scattering system with unitary evolution given by U. Moreover, the dilation space K can be identified with the space of finite-energy trajectories of an embedded discrete-time, conservative, input/state/output linear system. The system matrix for the input/state/output linear system, with appropriate minimality assumptions, is just a Halmos dilation (Julia operator) for the contraction operator T. Each of these three paradigms has a functional model determined by the same Schur-class function on the unit disk (the Sz.-Nagy-Foias characteristic function for T, the scattering matrix for the Lax-Phillips scattering system, and the transfer function for the input/state/output linear system). In this talk I will discuss recent progress on finding an analogue of all these ideas for the case where the single contraction operator T is replaced by a commuting pair of contractions (T1,T2). The Ando dilation theorem giving a commuting unitary-pair dilation (U1,U2) for a commuting contractive pair (T1,T2) is just the first step. This reports on joint work with Victor Vinnikov of Ben Gurion University.

Apr. 2-6

Tuesday, Apr. 3

James Rovnyak

Operator identities in the study of canonical differential systems

Beamer lecture

Apr. 9-13

Tuesday, Apr. 10

Katherine Heller

Adjoints of composition operators on Hilbert spaces of analytic functions

Apr. 16-20

Tuesday, Apr. 17

No lecture

Apr. 23-27

Tuesday, Apr. 3

No lecture

Apr. 30 - May 4

Tuesday, May 1

Vladimir Bolotnikov, The College of William and Mary

Carathéodory-Julia type theorems for operator-valued Schur functions

Abstract