Strongly exposed points, Helson-Szegö weights and Toeplitz operators (Q1265578)

For a contractive function \(b\) in \(H^\infty\), \(M(b)\) denotes the range of \(T_b\) and \(H(b)\) denotes the range of \((1- T_b T_{\overline b})^{1/2}\), where \(T_b\) is the Toeplitz operator on \(H^2\). Both \(M(b)\) and \(H(b)\) are called de Branges space. Put \(f= a^2/(1- b)^2\), where \(| a|^2+| b|^2= 1\) and \(a\) is a nonzero function in \(H^\infty\). Note that \(H^p\) \((p= 1,2,\infty)\) is the Hardy space. \textit{D. E. Sarason} [Oper. Theory, Adv. Appl. 41, 485-496 (1989; Zbl 0678.46019)] showed that \(f\) is an exposed point in \(H^1\) if and only if \(M(a)\) is dense in \(H(b)\). When \(f\) is an extreme point in \(H^1\), the author shows that \(f\) is an strongly exposed point in \(H^1\) if and only if \(M(a)= M(b)\).