Toeplitz operators on the Fock space: Radial component effects

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 Ta = 0 implies a(r)=0 a.e. For each compact set M there exists a Toeplitz operator Ta such that sp Ta =ess-sp Ta = 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.