Factorization of Functional Operators with Involutive Rotation on the Unit Circle

Following the classical definition of factorization of matrix-functions, we introduce a definition of factorization for functional operators with involutive rotation on the unit circle. Partial indices are defined and their uniqueness is proven. In previous works, the main research method for the study scalar singular integral operators and Riemann boundary value problems with Carlemann shift were operator identities, which allowed to eliminate shift and to reduce scalar problems to matrix problems without shift. In this study, the operator identities were used for the opposite purpose: to transform operators of multiplication by matrix-functions into scalar operators with Carlemann linear-fractional shift.


Introduction
A large number of works have been dedicated to Riemann boundary value problems and to the related singular integral equations. We point out some monographs that have already become classic on this subject [1] [2] [3] [4]. A special place is occupied by problems with shift in boundary conditions and equations with shift [5]. Listed monographs and their authors played a significant role in the development of this topic.
The problem of factorization of matrix functions is closely connected with the solution of matrix Riemann boundary value problems, for which effective solution methods have not yet been found [5] p. 24, Theorem 6. This explains the interest in and motivation for the study of this topic. In [6], we constructed operator identities with invertible operators, which transform a singular integral operator A with involutive fractional linear shift into a vector singular integral operator D without shift. Applications have been identified in which the main method of investigation was operator identities [7] [8] [9] [10].
Simplicity of the shift under consideration permits us, when studying the operator A, to avoid associated operators, and to avoid the appearance of compact operators and to obtain the operator identity, which directly connects the class of singular integral operators with shift and the class of matrix characteristic singular integral operators without shift. For an orientation-preserving shift, this corresponds to a similarity transform In [7], based on the known results on factorization [11], invertibility condi-

Factorization of the Operators with Carlemann Rotation on the Unit Circle
Let T denote the unit circle. We review definitions that we are going to use [5] [12].
Factorization of non-degenerate matrix function were matrix functions It is known [5] [11] that the partial indices are invariants of the factorization and do not depend on a particular type of representation, and that the numbers 1 2 , κ κ are uniquely defined.
We use the following notations for projectors acting in space where I is the identity operator, operator W is the rotation operator: We introduce similar notation for identity operator I in the space ( )  , g z g z g z g z z D D + − ≠ ∈  . We also provide other forms of representation (3), of operator A and of (4) through the projectors

A t B t A t B t A t B t A t B t
We call integers 1 κ and 2 κ partial indices of A.
In works [6]  L T : We also note that the operator of multiplication by a function transforms into To describe the similarity transformation structure we need some definitions and operators.
Let Γ and γ be contours, and let γ ⊂ Γ . The extension of a function ( ) f t , t γ ∈ , to \ γ Γ by the value zero, will be denoted by ( )( ) The restriction of a function ( ) t ϕ , t ∈ Γ to γ will be denoted by ( )( ) is determined by the composition of the operators M GN Π Z .
In our case, these operators have the following form where T + and T − are the upper and the lower parts of the unit circle, respectively, 2  2  2  2  2  2  2  2  2  2 , , ,