Singular measures on the unit circle and their reflection coefficients (Q1971646)

Two interesting classes of measures supported on the unit circle are considered. The first is formed by those measures whose sequence of reflection coefficients \(\{a_n\}\) satisfy \(\lim_{n\to\infty}|a_n|= 1\). For this class it is proved that the derived set of the support of the orthogonality measure coincides with the set of limit points of the sequence \(\{-a_{n+ 1}\overline{a_n}\}\). The second class consists of those measures on the unit circle such that the derived set of their support contains a finite number of points \(\{\tau_1,\dots, \tau_N\}\). It is shown that a measure \(\mu\) belongs to this class if and only if for any polynomial \(P\) the operator \(P(U)\) is compact in \(L_2(\mu)\) if and only if \(P(\tau_i)= 0\), \(i= 1,\dots, N\), where \(U\) denotes the multiplication operator.