Spectral Properties of Toeplitz Operators Acting on Gabor Type Reproducing Kernel Hilbert Spaces

This is a survey presenting an overview of the results describing the behavior of the eigenvalues of compact Gabor–Toeplitz operators and Gabor multipliers. We introduce Gabor–Toeplitz operators and Gabor multipliers as Toeplitz operators defined in the context of general reproducing kernel Hilbert spaces. In the first case the reproducing kernel Hilbert space is derived from the continuous Gabor reproducing formula, and in the second case, out of the discrete Gabor reproducing formula, based on tight Gabor frames. The extended metaplectic representation provides all affine transformations of the phase-space. Both classes of operators satisfy natural transformation properties with respect to this group, and both have natural interpretations from the point of view of phase space geometry. Toeplitz operators defined on the Fock space of several complex variables are at the background of the topic. The Berezin transform of general reproducing kernel Hilbert spaces applied to both kinds of Toeplitz operators shares in both cases the same natural phase-space interpretation of the Fock space model. In the first part of the survey we discuss the dependence of the eigenvalues on symbols and generating functions. Then we concentrate on Szegö type asymptotic formulae in order to analyze the dependence on the symbol and on Schatten class cutoff phenomena in dependence on the generating function. In the second part we restrict attention to symbols which are characteristic functions of phase space domains, called localization domains in the current context. The corresponding Toeplitz operators are called localization operators. We present results expressing mutual interactions between localization domains and generating functions from the point of view of the eigenvalues of the localization operators. In particular, we discuss asymptotic boundary forms quantifying these interactions locally at the boundary points of localization domains. Our approach to localization operators is motivated by the principles of the semiclassical limit. We finish the survey with a list of open problems and possible future research directions.