On some applications of the essential norms of singular integral operators

In this paper, we calculate the norms and essential norms of singular operators with the Cauchy kernel. We also determine the conditions under which these quantities coincide. Using the essential norms, conditions are obtained under which singular integral operators with bounded coefficients are Noetherian in the weighted space L₂. In the absence of a weights, the presented conditions coincide with the corresponding conditions of I. Simonenko.