Toeplitz Operators with Special Symbols on Segal–Bargmann Spaces

Abstract

We study the boundedness of Toeplitz operators on SegalBargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.Mathematics Subject Classification (2010). 47B35, 43A85, 22E30.

Orthogonal polynomials, Laguerre Fock space, and quasi-classical asymptotics

“…Surprisingly, this "Laguerre Fock space" turns out to coincide with the space of entire functions discovered by Barut and Girardello [7] in the construction of coherent states that nowadays bear their name. (Similar spaces were also obtained in [1] while working with ensembles of non-Hermitian matrices and in [17,19,23].) The associated Toeplitz operators and their asymptotics just mentioned, however, up to the authors' knowledge seem not to have previously appeared in the literature: it turns out that they again satisfy the correspondence principle (5), but with the Poisson bracket coming from the flat metric on the punctured complex plane C\{0} (which is somewhat surprising).…”