Commuting Toeplitz operators on the Segal–Bargmann space

Abstract

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.