Function Spaces and Their Operators
A Conference in Honor of Richard Rochberg on the Occasion of his 65th Birthday

Krzysztof Nowak from Drexel University; Spectral properties of Calderon-Toeplitz and Gabor-Toeplitz operators; May 31, 2008.

Abstract: This is an overview talk presenting a series of developments on Toeplitz operators defined in the context of Calderon and Gabor reproducing formulae. We start by discussing the relationship with classical time-frequency localization and Toeplitz operators defined on Fock and Bergman spaces. We move next to the dependence on function parameters, the symbol function and the function generating the reproducing formula. At the central stage we restrict attention to operators with symbols being characteristic functions of bounded domains. Operators of that class are called localization operators. We present several results explaining the nature of the relationship between the eigenvalues and the domain of localization.