About radial Toeplitz operators on Segal-Bargmann and $l^2$ spaces

We discuss Toeplitz operators on the Segal-Bargmann space as functional realizations of anti-Wick operators on the Fock space. In the special case of radial symbols we exploit the isometric mapping between the Segal-Bargmann space and $l^2$ complex sequences in order to establish conditions such that an equivalence between Toeplitz operators and diagonal operators on $l^2$ holds. We also analyze the inverse problem of mapping diagonal operators on $l^2$ into Toeplitz form. The composition problem of Toeplitz operators with radial symbols is reviewed as an application. Our notation and basic examples make contact with Quantum Mechanics literature.