Toeplitz operators on the Fock space: Radial component effects Article Received: 15 January 2001 Revised: 26 August 2001 DOI:
Cite this article as: Grudsky, S.M. & Vasilevski, N.L. Integr equ oper theory (2002) 44: 10. doi:10.1007/BF01197858 Abstract
The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbols
a=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 which T =0 implies a a(r)=0 a.e. For each compact set M there exists a Toeplitz operator T such that sp a T =ess-sp a T = 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.
AMS Classification 47B35 47E20
This work was partially supported by CONACYT Project 27934-E, México.
The first author acknowledges the RFFI Grant 98-01-01023, Russia.
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