Functional Analysis and Applications Seminar 

24/11/2006, 15:15 — 16:15 — Room P3.10, Mathematics Building
Alexey N. Karapetyants, CINVESTAV, Mexico

Toeplitz operators with special symbols in weighted Bergman spaces

We study Toeplitz operators in a weighted Bergman space on the unit disc with a power type weight related to the boundary of the disc. We deal with special symbols connected to the three types of hyperbolic geometry in the unit disc (elliptic, parabolic and hyperbolic pencils). That is, in each of the mentioned three cases the symbols are constant on geodesics orthogonal to the trajectories forming a pencil. The spectrum of each of the Toeplitz operator seems to be quite accidental, the definite tendency starts appearing only as the exponent of the weight tends to infinity. The correspondence principle (F. Berezin) suggests that the limit set of those spectra has to be strictly connected with the range of the initial symbol. This is definitely true for continuous symbols. Given a continuous symbol a, the limit set of spectra does coincide with the range of a. The new effects appear when we consider more complicated symbols. In particular, in the case of piecewise continuous symbols the limit set coincides with the range of a together with the line segments connecting the one-sided limit points of piecewise continuous symbol. Note that these additional line segments may essentially enlarge the limit set comparing to the range of a symbol.