Berezin transform on the harmonic Fock space

Abstract

We show that the Berezin transform associated to the harmonic Fock (Segal-Bargmann) space on C n has an asymptotic expansion analogously as in the holomorphic case. The proof involves a computation of the reproducing kernel, which turns out to be given by one of Horn's hypergeometric functions of two variables, and an ad hoc determination of the asymptotic behaviour of the resulting integrals, to which the ordinary stationary phase method is not directly applicable.

Toeplitz Quantization and Asymptotic Expansions: Geometric Construction

“…[5,Theorem 4.1]. (Also, Horn's hypergeometric functions of another kind -namely, Φ 2 in the notation of [23] -appear in the formula for the harmonic Segal-Bargmann kernel on C d , see [21]; it is however unclear if there is any deeper relationship.) |a| }) and v = au, then ρ(u, v) runs from 0 to 1; thus (5.9) holds for some u.…”

Toeplitz Operators with Positive Measures on Harmonic Fock Spaces

“…Let α > 0, p ≥ 1, we use F p h to denote the space of all harmonic functions in L p α (C, dA). Harmonic Fock spaces F p h are defined by Engliš in [3] firstly. Specially, for p = +∞, we write F ∞ h to denote the space of all harmonic functions f which satisfy the following case…”

“…If f belongs to F p h , by the relationship between F p α and F p h , we know f = f 1 + f 2 , where f 1 and f 2 are entire functions and f 2 (0) = 0, more details can be found in [3]. Through straight calculation, we have the orthonormal basis in harmonic Fock space F 2 h are…”

Compact vectorial Toeplitz operators on the Segal-Bargmann space

“…Toeplitz operators on Segal-Bargmann spaces arise in quantization of classical mechanics and are related to pseudodifferential operators (see [1,13,14,16,20,21,22]). They have been studied since the work of Berezin (see [4,5]) in the classical context of complex-valued entire functions (see, e.g., [2,3,11,12,17,18,19]) and also in the more general setting of vector-valued functions (see, e.g., [8,9,10]). The literature concerning these operators is broad and still growing.…”