- November 24, 2006, Friday - 15h15 (Room P 3.10):
Toeplitz operators with special symbols in weighted
Bergman spaces
Alexey N. Karapetyants
(Rostov State University, Russia )
Abstract. 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.
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