Colloquium November 28
Date: Tuesday November 28
Time: 16:00 to 17:00
Place: 455 McBryde (Commons Room)
Speaker: Sanne ter Horst of Vrije Universiteit Amsterdam
Title: The Nehari extension problem and relaxed commutant lifting
Abstract
The Nehari extension problem is one of the many metric
constrained problems that fits in the commutant lifting setting. The
commutant lifting theory developed since the 60s provides various
explicit state space descriptions of all solutions to the Nehari
extension problem. However, for these descriptions the state space
will be infinite dimensional. In the special case that the Hankel
operator associated with the data has finite rank this can be overcome
by constructing a finite dimensional realization for the data.
Recently a relaxation of the commutant lifting theorem was introduced,
which provides a setting in which a family, indexed by natural
numbers, of relaxed versions of the classical interpolation and
extension problems can be formulated. In this talk, we consider the
classical Nehari extension problem as well as its relaxed versions. We
will see how these problems fit in the (relaxed) commutant lifting
setting, and obtain an explicit description of all their solutions.
For the relaxed versions these descriptions involve state space
formulas where the state is finite dimensional with the dimension
depending on the index. Moreover, it is observed how the classical
Nehari extension problem appears as a limit case of the relaxed
versions as the index goes to infinity.
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