Factorization of singular integral operators with a Carleman backward shift: the case of bounded measurable coefficients
Using the factorization of matrix functions, the authors generalize some of their recently obtained results concerning scalar singular integral operators [J.~Integral Equations Appl.\ 13, No.\,4, 339--383 (2001; Zbl 1009.47034), Oper.\ Theory, Adv.\ Appl.\ 142, 189--211 (2003; Zbl 1057.47055), Math.\ Nachr.\ 280, No.\,9--10, 1157--1175 (2007; Zbl 1130.47028), in order to obtain an operator factorization. In the first two sections, necessary preliminaries are introduced, especially, important properties of the factors of the matrix factorization. The definition of a special matrix factorization is introduced in the third section of the paper, generalizing a previous result, and in the fourth section, a first step towards operator factorization is made. Studying representatives of anti-diagonal matrix functions, the fifth section concludes the operator factorization. Using the results from the previous section, in the sixth section the Fredholm properties of a singular integral operator with a backward shift are characterized, and in the last section, some examples and applications are made which allow a characterization for the set of invertible operators in that class. Also, some applications of the results to the class of generalized Toeplitz plus Hankel operators are presented.
scientific article; zbMATH DE number 5646610
scientific article