Certain remarks on the multiplicators of the Cauchy type integrals and Sarason algebras (Q1056886)

Let D be a Douglas algebra, that is \(H^{\infty}\subset D\subset L^{\infty};\) its Sarason algebras are algebras \(QD=D\cap \bar D\) and \(CD-C^*\)-algebra, generated by a set of inner functions, invertible in D. It is established that if a Toeplitz operator \(T_{{\bar \mu}}\), \(\mu \in H^{\infty}\), acts in \(C(T)\cap H^{\infty}\), then it acts in all Sarason algebras. By using this fact a number of Sarason algebras which are Dirichlet algebras are described. Clearly are shown the functions from \(clos_{\infty}(H^{\infty}+\overline{H^{\infty}}+C)\setminus (H^{\infty}+\overline{H^{\infty}}+C).\)