, Volume 44, Issue 1, pp 10-37

Toeplitz operators on the Fock space: Radial component effects

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Abstract

The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbolsa=a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for whichT a =0 impliesa(r)=0 a.e. For each compact setM there exists a Toeplitz operatorT a such that spT a =ess-spT a =M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication.

Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.

This work was partially supported by CONACYT Project 27934-E, México.
The first author acknowledges the RFFI Grant 98-01-01023, Russia.