Reduction of singular integral operators with flip and their Fredholm property
- 32 Downloads
This paper deals with singular integral operators with a reverting orientation Carleman shift defined on the classic Lebesgue space and having essentially bounded functions as coefficients. We will use similarity relations to show that the mentioned operators are equivalent tomatrix Toeplitz plus Hankel operators acting on the corresponding Hardy space. The main purpose is to extract Fredholm characteristics of the initial operators (in the form of necessary and sufficient conditions). Namely, Fredholm criteria are obtained for some of the operators under study when they have coefficients in the classes of continuous, piecewise continuous, and semi-almost-periodic functions. In addition, Fredholm index formulas are also provided in some of these cases.
Key words and phrasesSingular integral operator Carleman shift Toeplitz operator Hankel operator Fredholm property
2000 Mathematics Subject Classification47G10 47B35 47A53 45E05 45F15 47A20
Unable to display preview. Download preview PDF.
- 1.H. Bart and V. E. Tsekanovshii, Matricial Coupling and Equivalence after Extension, Oper. Theory Adv. Appl. 59, 143 (1992).Google Scholar
- 3.A. Böttcher, Yu. I. Karlovich, and I. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Funtions (Birkhauser Verlag, Bassel, 2002).Google Scholar
- 6.L. P. Castro and E. M. Rojas, Similarity Transformation Methods for Singular Integral Operators with Reflection on Weighted Lebesgue Spaces, Int. J.Mod.Math. 3, 19 (2008).Google Scholar
- 8.T. Ehrhardt, Factorization Theory for Toeplitz plus Hankel Operators and Singular Integral Operators with Flip (Habilitation Thesis, Technishe Universtität Chemnitz, Chemnitz, 2004).Google Scholar
- 10.A. A. Karelin, Relation between Singular Integral Operators with an Orientation-reversing and Orientation-preserving Shifts, Proceedings of Math. Inst. of National Academy of Sciences of Belarus, Vol. C, 121 (2004).Google Scholar
- 13.V. G. Kravchenko, A. B. Lebre, and J. S. Rodríguez, Factorization of Singular Integral Operators with a Carleman Shift via Factorization of Matrix Functions, Oper. Theory Adv. Appl. 142, 189 (2003).Google Scholar
- 16.S. G. Mikhlin and S. Prössdorf, Singular Integral Operators (Springer-Verlag, Berlin, 1980).Google Scholar
- 18.S. Roch and B. Silbermann, Algebras of Convolution Operators and Their Image in the Calkin Algebra, Rep. Akad.Wiss. DDR, Karl-Weierstrass-Inst.Math. 05/90 (Berlin, 1990).Google Scholar