Commuting Toeplitz operators with quasi-homogeneous symbols on the Segal–Bargmann space

Let T = T z l z ¯ k with l , k ∈ N 0 be a Toeplitz operator with monomial symbol acting on the Segal–Bargmann space over the complex plane. We determine the symbols Ψ of polynomial growth at infinity such that T Ψ and T z l z ¯ k commute on the space of all holomorphic polynomials. By using polar coordinates we represent Ψ as an infinite sum Ψ ( r e i θ ) = ∑ j = − ∞ ∞ Ψ j ( r ) e i j θ . Then we are able to reduce the above problem to the case of quasi-homogeneous symbols Ψ = Ψ j e i j θ . We obtain the radial part Ψ j ( r ) in terms of the inverse Mellin transform of an expression which is a product of Gamma functions and a trigonometric polynomial. If we allow operator symbols of higher growth at infinity, we point out that in some of the cases more than one Toeplitz operator T Ψ j e i j θ exists commuting with T.