Hyponormal Toeplitz operators with polynomial symbols
For a Toeplitz operator \(T_f\) on the Hardy space \(H^2\) of the unit circle \(S\) defined by \(T_f g= P(fg)\), where \(f\in L^\infty (S)\), and \(P: L^2 (S)\to H^2\) is the orthogonal projection, a well-known result of \textit{C. C. Cowen} [Proc. Am. Math. Soc. 103, No. 3, 809-812 (1988; Zbl 0668.47021)]\ gives necessary and sufficient conditions for \(T_f\) to be hyponormal \((T^*_f T_f\geq T_f T^*_f)\), viz., that for \(f= f_1+ \overline {f}_2\) with \(f_1, f_2\in H^2\) there exists a constant \(C\) and a function \(g\) in the unit disk \((H^\infty)_1\) such that \(f_2= C+ T_{\overline {g}} f_1\). The present author applies this result to the case where \(f_1\) and \(f_2\) are trigonometric polynomials with \(f_1\) of degree \(n\) and (as the author shows, necessarily) \(f_2\) is of degree \(\leq n\). The Cowen criterion is equivalent to being able to solve the functional equation \(\overline {f}_2- g\overline {f}_1 =h\) for \(g\in (H)_1\) and \(h\in H_2\). But a method for determining whether this can be done in the present context is given as a classical algorithm of I. Schur for characterizing the first \(n\) Taylor coefficients of functions in \((H^\infty)_1\), using a sequence of functions \(\Phi_0, \Phi_1, \dots\;\). The author shows that \(T_f\) is hyponormal if and only if \(|\Phi_k (c_0, \dots, c_k) |\leq 1\) for \(0\leq k\leq n-1\) and where the \(c_0, \dots, c_{n-1}\) are specific complex numbers derived from the coefficients of \(f_1\) and \(f_2\).
scientific article; zbMATH DE number 765291