27/10/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Andreas Hartmann, Université Bordeaux I, France
Extremal functions of kernels of Toeplitz operators
We will essentially discuss two points in the connection with
extremal functions of kernels of Toeplitz operators on Hardy
spaces. The first one concerns divisor properties of such extremal
functions. It turns out that in many situations such a division has
nice properties like being a contraction (case of Hedenmalm's
canonical divisors in the Bergman space), or even an isometry
(inner functions in the Hardy space). Concerning extremal functions
of kernels of a Toeplitz operator, the question has been considered
in the larger class of nearly invariant subspaces by Hitt. He
proved that in the Hilbert space situation , the division by
the extremal function of a nearly invariant subspace is isometric.
The situation changes drastically even for Toeplitz kernels when
one switches to the non Hilbert case ( ), where,
depending on the parameter , one can in general only
expect a control on the division or on the multiplication by the
extremal function. Examples show that two-sided estimates cannot be
expected in general. The second part of the talk will be devoted to
the investigation of invertibility properties of Toeplitz operators
by means of the extremal function. This understands that the
Toeplitz operator is supposed non injective in order that such an
extremal function exists. In this part we have to assume the
Hilbert situation . It turns out that certain parameters
associated with the extremal function, and that have previously
been used by Hayashi to distinguish kernels of Toeplitz operators
from general nearly invariant subspaces, enable us to characterize
the surjectivity of a (non injective) Toeplitz operator.