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Factorization of singular integral operators with a Carleman shift via factorization of matrix functions: the anticommutative case
Author(s) -
Kravchenko V. G.,
Lebre A. B.,
Rodríguez J. S.
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510543
Subject(s) - mathematics , factorization , singular integral , fourier integral operator , operator (biology) , invertible matrix , singular integral operators , incomplete lu factorization , singular integral operators of convolution type , shift operator , operator theory , spectral theorem , algebra over a field , strictly singular operator , operator norm , quasinormal operator , unit circle , pure mathematics , finite rank operator , compact operator , matrix decomposition , microlocal analysis , mathematical analysis , integral equation , algorithm , computer science , eigenvalues and eigenvectors , repressor , banach space , physics , quantum mechanics , chemistry , biochemistry , transcription factor , programming language , extension (predicate logic) , gene
This 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. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)