Factorization of singular integral operators with a Carleman shift via factorization of matrix functions: the anticommutative case

Abstract

Key words Singular integral operators with shift, operator factorization, factorization of matrix functions MSC (2000) Primary: 47G10; Secondary: 47A68 Dedicated to Professor Frank-Olme Speck on the occasion of his 60th birthdayThis paper deals with what we call modified singular integral operators. When dealing with (pure) singular integral operators on the unit circle with coefficients belonging to a decomposing algebra of continuous functions it is known that a factorization of the symbol induces a factorization of the original operator, which is a representation of the operator as a product of three singular integral operators where the outer operators in that representation are invertible.The main purpose of this paper is to obtain a similar operator factorization for the case of singular integral operators with a backward shift and to extract from there some consequences for their Fredholm characteristics. At the end of the paper it is shown that the operator factorization is also possible for other classes of singular integral operators, namely those including either a conjugation operator or a composition of a conjugation with a forward shift operator.

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 .…”

“…As a consequence, it is obtained a pure vector singular integral operator which has the same Fredholm properties as the initial one but with a "double" symbol matrix. In much of the cases, the so-called Gohberg-Krupnik-Litvinchuk identity (see e.g., [16,17,20]) and other explicit operator equivalence relations (c.f., e.g., [5,17]) are main ingredients for such analysis. In this way, the solvability of a (scalar) SIES associated with the SIOS is equivalently reformulated as a matrix factorization problem for corresponding matrices (which are built based on the new matrix coefficients); for these and other methods see, for instance, [6,10,19,20,23].…”

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.…”

Factorization of singular integral operators with a Carleman backward shift: The case of bounded measurable coefficients

“…In our previous works ( [8], [9] and [10]), it was shown that such a representation can also be constructed in the case of singular integral operators with linear fractional shift and continuous coefficients from a decomposing Banach algebra which is invariant under the action of the shift. Recall that a decomposing Banach algebra of continuous functions on T is one on which the projections P + and P − act continuously.…”

“…However, at that time it was not recognized that there is a connection between this factorization of the given operator and the factorization of a certain matrix function in a certain algebra. In [10], the case where the coefficients of the functional operators belong to a decomposing algebra of continuous functions which is invariant under the action of the shift was considered. Although the general method used in that paper is similar to the one we use here, the situation in [10] was much simpler, since the characteristic parameters of the factorization of a matrix function related to the given operator T A,B are of a very special type.…”