Abstract
Until now factorization has been discussed as a global problem. In this chapter we establish the local character of factorization. This local character is obtained first for factorization relative to a contour and subsequently for generalized factorization.
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Section 1. The main result in Theorem 1.1 was first proved by Subin M.A. Subin: The local principle in the factorization problem. Mat. Issled. 6(1971), 174–180] using a general principle of Rohrl
H. Röhrl: Über das Riemann-Privalovsche Randwertproblem. Math. Ann. 151(1963), 365–423], The proof here is essentially the same as the proof in the scalar case presented in Gohberg and Krupnik
I. Gohberg and N. Ya. Krupnik: Introduction to the theory of one-dimensional singular integral operators. Kishniev: Stiince, 1973(Russian) . German transi: Birkhauser,Verlag, Basel-Boston-Stuggart, 1979.].
Section 2. Proposition 2.1 is due to Simonenko I.B. Simonenko: Some general questions on the theory of the Riemann boundary problem. Izv. Akad. Nauk SSSRSer. Mat. 32 (1968), 1138–1146
I.B. Simonenko: Some general questions on the theory of the Riemann boundary problem. Math USSR Izv. 2(1968), 1091–1099. The other results in this section are from Clancey and Gohberg [14 ,15].We mention that the question of whether every local factorization relative to Lp on an arc γ arises from a restricted factorization relative to Lp remains unanswered. In the case n = 1. p = 2 , these two notions of local factorization relative to Lp are known to be equivalent.
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© 1981 Springer Basel AG
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Clancey, K.F., Gohberg, I. (1981). Local Principles in the Theory of Factorization. In: Factorization of Matrix Functions and Singular Integral Operators. Operator Theory: Advances and Applications, vol 3. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5492-4_10
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DOI: https://doi.org/10.1007/978-3-0348-5492-4_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5494-8
Online ISBN: 978-3-0348-5492-4
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