Calculation of the Defect Numbers of the Generalized Hilbert and Carleman Boundary Value Problems with Linear Fractional Carleman Shift

Abstract

The defect numbers of the generalized Hilbert and Carleman boundary value problems with a direct or an inverse linear fractional Carleman shift of order 2 (α (α (t)) ≡ t) on the unit circle are computed. The approach followed consists of the reduction of the mentioned problems to singular integral equations with linear fractional Carleman shift and of the factorization of Hermitian matrix functions with negative determinant. Mathematics Subject Classification (2000). Primary 47G10; Secondary 47A68.

On the Solvability of Singular Integral Equations with Reflection on the Unit Circle

“…[1,2,7,11,15,17,18,20,21] for different concrete examples of weighted shift operators of this form). A (weighted) shift operator is called a Carleman shift operator if W 2 = I T .…”

Solvability of singular integral equations with rotations and degenerate kernels in the vanishing coefficient case

“…Particularly, the theory of SIES and boundary value problems for analytic functions founded by Hilbert and Poincaré are particularly applied in many theories such as the theory of the limit problems for differential equations with second order partial derivatives of mixed type, the theory of the cavity currents in an ideal liquid, the theory of infinitesimal bonds of surfaces with positive curvature, the contact theory of elasticity, and that of physics of plasma. Moreover, the theory of SIOS contributes theoretically in a significant way not only to the theory of Fredholm operators (Noetherian operators according to the terminology of Russian-language literature) and to that of one-sided invertible operators but also to the theory of general and abstract operators within C * -algebras (see [1,6,9,14,15,25,26,28,29] and references therein). In the previous decades, the theory of SIES has been considered an attractive object of study due to a great variety of reasons.…”