The theory of Toeplitz operators has come to resemble more and more
in recent years the classical theory of pseudodifferential
operators. For instance, Toeplitz operators possess a symbolic
calculus analogous to the usual symbolic calculus, and by symbolic
means one can construct parametrices for Toeplitz operators and
create new Toeplitz operators out of old ones by functional
operations. If P is a self-adjoint pseudodifferential operator on a
compact manifold with an elliptic symbol that is of order greater
than zero, then it has a discrete spectrum. Also, it is well known
that the asymptotic behavior of its eigenvalues is closely related
to the behavior of the bicharacteristic flow generated by its
symbol. It is natural to ask if similar results are true for
Toeplitz operators. In the course of answering this question, the
authors explore in depth the analogies between Toeplitz operators
and pseudodifferential operators and show that both can be viewed
as the "quantized" objects associated with functions on compact
contact manifolds.
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