Functional Analysis and Applications Seminar 

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 $H^2$, 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 ( $p \neq 2$), where, depending on the parameter $p \gt 1$, 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 $p = 2$. 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.