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

Abstract

In this paper, we generalize our recent results concerning scalar singular integral operators with a Carleman backward shift, allowing more general coefficients, bounded measurable functions on the unit circle. Our aim is to obtain an operator factorization for singular integral operators with a backward shift and bounded measurable coefficients, from which such Fredholm characteristics as the kernel and the cokernel can be described. The main tool is the factorization of matrix functions. In the course of the analysis performed, we obtain several useful representations, which allow us to characterize completely the set of invertible operators in that class, thus providing explicit examples of such operators

Generalized Toeplitz Plus Hankel Operators: Kernel Structure and Defect Numbers

“…These operators are similar to the classical Toeplitz plus Hankel operator (2) but the flip operator J of (3) is replaced by another operator J α generated by a linear fractional shift α changing the orientation of the circle T. Areas of particular interest to us are the kernels and cokernels of such operators and we are going to derive an explicit description of these spaces in the case where the generating functions a and b belong to the space L ∞ and satisfy an additional algebraic relation. Note that generalized Toeplitz plus Hankel operators have been previously considered in [10,11] but under more restrictive assumptions. Moreover, in the present work we single out certain classes of generalized Toeplitz plus Hankel operators which are subject to Coburn-Simonenko theorem.…”

“…Note that in the last relations, the mapping α is understood as acting on the unit circle T. A proof of this result can be given by using relations (10). We omit the details here but mention that they can be found in the proof of Proposition 2.2 of [11].…”

“…In the present paper we study the properties of the operators T Recall that the kernel and cokernel dimensions of generalized Toeplitz plus Hankel operators and even more general operators have been studied in [10,11]. The approach of [10,11] is based on a special factorization of the operators in question, which in turn involves factorizations of matrix functions. Thus assuming that the corresponding operators are Fredholm, the authors express the kernel and cokernel dimensions in terms of the partial indices of some matrix valued functions.…”

Matrix Functions Consimilar to the Identity and Singular Integral Operators

“…It should be mentioned that the concept of antisymmetric factorization appeared quite recently in works devoted to Toeplitz plus Hankel operators (see, for instance, [2] and [1]) and separately (and independently), however without that name, in the theory of singular integral operators with a linear fractional shift developed by the authors of the present paper (see [5][6][7][8]). We observe that the factorization of matrix functions fulfilling especial conditions arise quite naturally when we look for the factorization of singular integral operators (or Toeplitz operators) being overweight, so to speak, with operators of the type of the operator U introduced above.…”

“…It suffices to say, for instance, that the formula for the calculation of the dimension of the kernel of this kind of operators depends not only on the partial indices of the factorization of the corresponding matrix function (as is the case of a SIOC), but also on other characteristics of that factorization (see [5][6][7][8]). …”