Weighted approximation of entire functions and Toeplitz operators in Segal-Bargmann spaces.
by
Dariusz Cichon
Instytut Matematyki UJ, Krakow

Toeplitz operators defined in Segal-Bargmann spaces (i.e. the space of entire functions, which are square-integrable with respect to the Gaussian measure d\mu) are related to weighted approximation via graph norms induced by them. The important question of describing the adjoint of a Toeplitz operator can be solved by proving the density of polynomials or exponentials in its graph norm. Positive results are proved for Toeplitz operators induced by so called ëxponential polynomials", in particular for ordinary polynomials. An example disproving the possibility of polynomial approximation may be indicated.

The situation becomes even more interesting in case of vector-valued Segal-Bargmann spaces. The results obtained in the scalar case cannot be easily carried over to the vector one, e.g. it is not known if (operator-valued) polynomials induce Toeplitz operators with polynomials dense with respect to the graph norm.

Part of of the presented results come from the paper written with Harold S. Shapiro.

Date received: June 10, 2002