A theorem of Nehari type (Q1182727)

Let \(A^ 2\) denote the Bergman space of square integrable analytic functions on the open unit disc \(D\); let \(J\), \(T_ z\) and \(T_{\bar z}\) be the operators in \(A^ 2\) given by \(J(f)(z)=f(\bar z)\), \(T_ z(f)=P(zf)\), and \(T_{\bar z}(f)=P(\bar zf)\), and let \(P\) be the orthogonal projection from \(L^ 2(D)\) onto \(A^ 2\). In this paper it is proved that an operator \(S\) in \(A^ 2\) verifies \(ST_ z=T_{\bar z}S\) if and only if it is a Hankel operator; i.e., there exists \(\phi\in L^ \infty(D)\) such that \(S(f)=P(J(\phi(f))\).