Factorization of Singular Integral Operators with a Carleman Shift and Spectral Problems

Abstract

In this paper we study singular integral operators with a linear fractional Carleman shift preserving the orientation on the unit circle. The main goal is to characterize the spectrum of some of these operators. To this end a special factorization of the operator is derived with the help of a factorization of a matrix function in a suitable algebra. After developing methods which permit us to obtain a factorization of a matrix function for some classes of interest, the spectral analysis of some types of singular integral operators with a Carleman shift is done.