Spectrum problems for singular integral operators with Carleman shift

In this paper we are concerned with the complete spectral analysis for the operator T = chi SU in the space Lp(TT) (TT denoting the unit circle), wher e X is the characteristic function of some are of TT, S is the singular int egral operator with Cauchy kernel and U is a Carleman shift operator which satisfies the relations U-2 = I and SU = +/- US, where the sign + or - is t aken in dependence on whether U is a shift operator on Lp(B) preserving or changing the orientation of TT. This includes the identification of the Fre dholm and essential parts of the spectrum of the operator T, the determinat ion of the defect numbers of T - lambdaI, for lambda in the Fredholm part o f the spectrum, as well as a formula for the resolvent operator.