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

Let T = Tz(l)z(-k), with l, k is an element of N-0 be a Toeplitz operator with monomial symbol acting on the Segal-Bargmann space over the complex plane. We determine the symbols of polynomial growth at infinity such that T-psi, and Tz(l)z(-k) commute on the space of all holomorphic polynomials.. By using polar coordinates we represent psi as an infinite sum psi(re(i0)) = Sigma(infinity)(j) = psi(j)(r)e(ij0). Then we are able to reduce the above problem to the case of quasi-homogeneous symbols psi = psi(j)e(ij0). We obtain the radial part psi(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-psi jeij theta exists commuting with T. (C) 2011 Elsevier Inc. All rights reserved.