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

Let with l,k∈N0 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 TΨ and commute on the space of all holomorphic polynomials. By using polar coordinates we represent Ψ as an infinite sum . Then we are able to reduce the above problem to the case of quasi-homogeneous symbols Ψ=Ψjeijθ. We obtain the radial part Ψj(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 TΨjeijθ exists commuting with T.