Commuting Toeplitz operators on the Segal-Bargmann space

Consider two Toeplitz operators T-g, T-f 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 [T-g, T-f] = 0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal-Bargmann space over C-n and n > 1, where the commuting property of Toeplitz operators can be realized more easily. (C) 2010 Elsevier Inc. All rights reserved.