Singular integral operators with coefficients of a special structure related to operator equalities (Q1042636)

The Cauchy singular integral operator along a contour \(\Gamma\) \[ (S_{\Gamma}\varphi)(t)=\frac{1}{\pi i}\int\limits_{\Gamma}\frac{\varphi(\tau)}{\tau-t} \,d\tau \] is considered and operator equalities are used for the study of matrix characteristic singular integral operators. The Gohberg-Krupnik matrix equality is used for the study of questions connected with the invertibility problem. The author also considers a singular integral operator with orientation preserving shift on the unit circle and coefficients generated by piecewise constant functions, functions \(t\) and \(t^{-1}\), and possessing special properties. Using the operator equality, conditions for invertibility of this operator are obtained. Conditions for the invertibility of matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure are likewise obtained.