Factorization of Functional Operators with Involutive Rotation on the Unit Circle

Aleksandr Karelin,Anna Tarasenko

doi:10.4236/am.2020.1111076

Abstract

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.

Highlights

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
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]

Summary

Introduction

A large number of works have been dedicated to Riemann boundary value problems and to the related singular integral equations. For homogeneous equations with such operators, the number of linearly independent solutions was calculated [10], the case when the coefficients of singular integral operators with four values has been considered and conditions for the non-triviality of the kernel of such operators were found. Following this method, we studied a Riemann boundary value problem with a shift inward of the domain with piecewise constant coefficients taking two values [8] [9]. The conditions of existence and uniqueness of the homogeneous problem were found, as well as the formula for calculating the number of linearly independent solutions

Factorization of the Operators with Carlemann Rotation on the Unit Circle

Uniqueness of Factorization and Partial Indices

Conclusion