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.
