Finite rank commutators and semicommutators of quasihomogeneous Toeplitz operators (Q841661)

The authors study the finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols acting on the Bergman space over the unit disk. Recall that a function \(\phi\) is called quasihomogeneous of degree \(p \in \mathbb{Z}\) if it has the form \(e^{ip\theta}f\), where \(f\) is a radial function. The main results of the paper are as follows. Let \(p\) and \(s\) be two positive integers and let \(f\) and \(g\) be two integrable radial functions such that \(T_{e^{ip\theta}f}\), \(T_{e^{is\theta}g}\) and \(T_{e^{i(p+s)\theta}fg}\) (resp., \(T_{e^{ip\theta}f}\) and \(T_{e^{is\theta}g}\)) are bounded Toeplitz operators. If the semicommutator \((T_{e^{ip\theta}f}, T_{e^{is\theta}g}]= T_{e^{i(p+s)\theta}fg} - T_{e^{ip\theta}f}T_{e^{is\theta}g}\) (resp., the commutator \([T_{e^{ip\theta}f}, T_{e^{is\theta}g}]= T_{e^{ip\theta}f}T_{e^{is\theta}g}-T_{e^{is\theta}g}T_{e^{ip\theta}f}\)) has finite rank, then it is equal to zero. Let \(p\geq s\) be two positive integers and let \(f\) and \(g\) be two integrable radial functions such that \(T_{e^{ip\theta}f}\), \(T_{e^{-is\theta}g}\) and \(T_{e^{i(p-s)\theta}fg}\) (resp., \(T_{e^{ip\theta}f}\) and \(T_{e^{-is\theta}g}\)) are bounded Toeplitz operators. If the semicommutator \((T_{e^{ip\theta}f}, T_{e^{-is\theta}g}]\) (resp., the commutator \([T_{e^{ip\theta}f}, T_{e^{-is\theta}g}]\)) has finite rank \(N\), then \(N\) is at most equal to the quasihomogeneous degree~\(s\). Examples are given of rank one semicommutators and commutators of quasihomogeneous Toeplitz operators whose quasihomogeneous degrees have opposite signs.