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Bergman-Toeplitz Operators: a Pseudodifferential Approach
by
Nikolai Vasilevski
CINVESTAV del I.P.N., Mexico City
We show that the problem of studying the Bergman-Toeplitz operators with functional symbols on a plane domain is, in a certain sense, the problem of studying the pseudodifferential operators on a half-line.
The principal point is the use of the unitary operator R: L2(D) --> L2(R+), where D is the upper half-plane in C, which reduces Toeplitz operators Ta acting on L2(D) to the unitary equivalent operators RTaR*, acting on L2(R+). The operator R is an exact analog of the Bargmann transform, which maps the Fock space F2(Cn) onto L2(Rn). The passing from Toeplitz operators Ta to the operators RTaR* is nothing but an analog of the Berezin reducing of operators with anti-Wick symbols (i.e., Toeplitz operators) on the Fock space to Weyl pseudodifferential operators on L2(Rn). This is how pseudodifferential operators appear in the context of Bergman-Toeplitz operators.
The class of pseudodifferential operators we obtain is quite interesting in itself, and extends the class of operators studied by H. Cordes. The symbols of these pseudodifferential operators are slowly varying in the additive sense at infinity and in the multiplicative sense at zero in the variable x. With respect to the dual variable the operators are a mixture of additive Wiener-Hopf operators and multiplicative Mellin convolutions.
http://www.math.cinvestav.mx/~nvasilev
Date received: March 13, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-07.