On a relation between singular integral operators with a Carleman linear-fractional shift and matrix characteristic operators without shifts (Q1598349)

In the \(L_2({\mathbb T})\)-setting it is studied the singular operator with shift \[ A = \sum_{k=0}^{n-1} \left[ a_k(t) {\mathbf I} + b_k(t) {\mathbf S}_{\mathbb T} \right] {\mathbf W}^k, \] where \({\mathbf S}_{\mathbb T}\) is the Cauchy singular integral operator and \({\mathbf W}\) is the shift on the unit circle \({\mathbb T}\) rotating \({\mathbb T}\) on \(\frac{2\pi}{m}, m\in {\mathbb N}\). This operator is reduced to a matrix characteristic singular integral operator without shift. A similarity transformation is constructed. An analogous problem in the \(L_2({\mathbb T})\)-setting is also considered for the operator \[ {\mathbf B} = a {\mathbf I}_{\mathbb R} + b {\mathbf Q}_{\mathbb R} + c {\mathbf S}_{\mathbb R} + d {\mathbf Q}_{\mathbb R} {\mathbf S}_{\mathbb R}, \] with linear fractional shift \({\mathbf Q}_{\mathbb R}\) on the real line \({\mathbb R}\). The results are applied to the study of integral operators with endpoint singularities.