Commuting Toeplitz operators on the Segal–Bargmann space

Consider two Toeplitz operators Tg, Tf on the Segal–Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg,Tf]=0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal–Bargmann space over Cn and n>1, where the commuting property of Toeplitz operators can be realized more easily.