The commutant of analytic Toeplitz operators on Bergman space (Q953237)

Let \(L^2_a(\mathbf D)\) be the Bergman space of all square-integrable holomorphic functions on the unit disc~\(\mathbf D\), and \(M_f:g\mapsto fg\) the operator on \(L^2_a(\mathbf D)\) of multiplication by a bounded holomorphic function~\(f\) on~\(\mathbf D\). Suppose that the Taylor expansion of \(f\) is \(f(z)=\sum_j b_j z^{p_j}\), with \(b_j\neq0\) and integers \(0<p_0<p_1<p_2<\dots\). The author proves that the commutant of \(M_f\) coincides with the commutant of~\(M_{z^s}\), where \(s\) is the greatest common divisor of \(p_0,p_1,p_2,\dots\). A~second result -- seemingly independent of the first and not directly related to~it -- concerns winding numbers of the curves \(f(e^{it})\) for strongly irreducible operators~\(M_f\).