Abstract
We prove criteria for the invertibility of the binomial functional operator
in the Lebesgue spaces L p(0,1), 1 < p < ∞, where a and b are continuous functions on (0, 1), I is the identity operator, W,, is the shift operator, W α f=f ○ α, generated by a non-Carleman shift α : [0,1] → [0,1] which has only two fixed points 0 and 1. We suppose that log a’ is bounded and continuous on (0, 1) and that a, b, a’ slowly oscillate at 0 and 1. The main difficulty connected with slow oscillation is overcome by using the method of limit operators.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Antonevich, A.B.Linear Functional Equations. Operator ApproachBirkhäuser Verlag, Basel, Berlin 1995.
Antonevich, A., Belousov, M., Lebedev, A.Functional Differential Equations: II. C*-applications. Parts 1–2Longman, Harlow 1998.
Antonevich, A., Lebedev, A.Functional-Differential Equations: I. C*-theoryLongman Scientific & Technical, Harlow 1994.
Aslanov, V., Karlovich, Yu.I., One-sided invertibility of functional operators in reflexive Orlicz spacesDokl. Akad. Nauk AzSSR45 (1989), no. 11–12, 3–7 [in Russian].
Böttcher, A., Karlovich, Yu.I., Rabinovich, V.S., The method of limit operators for one-dimensional singular integrals with slowly oscillating dataJ. Operator Theory43 (2000), 171–198.
Chicone, C., Latushkin, Yu.Evolution Semigroups in Dynamical Systems and Differential EquationsAMS, Providence, RI 1999.
Drakhlin, M.E., Stepanov, E., Weak convergence of inner superposition operatorsProc. Amer. Math. Soc. 126(1998), 173–179.
Gohberg, I., Feldman, I.A.Convolution Equations and Projection Methods for Their SolutionAMS, Providence, RI 1974.
Karlovich, A.Yu., Criteria for one-sided invertibility of a functional operator in rearrangement-invariant spaces of fundamental typeMathematische Nachrichten 229(2001), 91–118.
Karlovich, A.Yu., Karlovich, Yu.I., One-sided invertibility of binomial functional operators with a shift on rearrangement-invariant spacesIntegr. Equat. Oper. Theory42 (2002), 201–228.
Karlovich, Yu.I., On the invertibility of functional operators with non-Carleman shift in Hölder spacesDiff. Uravn.20 (1984), 2165–2169 [in Russian].
The continuous invertibility of functional operators in Banach spacesDis- sertationes Mathematicae340 (1995), 115–136.
Karlovich, Yu.I., Kravchenko, V.G., A Noether theory for a singular integral operator with a shift having periodic pointsSoviet Math. Dokl. 17(1976), 1547–1551.
Karlovich, Yu.I., Kravchenko, V.G., Litvinchuk, G.S., Invertibility of functional operators in Banach spacesFunctional-Differential Equations Perm. Politekh. Inst. Perm(1990), 18—58 [in Russian].
Karlovich, Yu.I., Lebre, A.B. Algebra of singular integral operators with a Carleman backward slowly oscillating shiftIntegr. Equat. Oper. Theory41 (2001), 288–323.
Karlovich, Yu.I., Mardiev, R., One-sided Invertibility of Functional Operators and the n(d)-normality of Singular Integral Operators with Translation in Hölder SpacesDifferential Equations24 (1988), 350–359.
Kitover, A.K., Spectrum of automorphisms with weight, and the KamowitzScheinberg theoremFunct. Anal. Appl.13 (1979), 57–58.
Kravchenko, V.G., On a Singular Integral Operator with a ShiftSoviet Math. Dokl.15 (1974), 690–694.
Kravchenko, V.G., Litvinchuk, G.S.Introduction to the Theory of Singular Integral Operators with ShiftSeries: Mathematics and its applications 289, Kluwer Academic Publishers, Dordrecht, Boston, London 1994.
Kurbatov, V.G.Functional Differential Operators and EquationsSeries: Mathematics and its applications 473, Kluwer Academic Publishers, Dordrecht, Boston, London 1999.
Litvinchuk, G.S.Boundary Value Problems and Singular Integral Equations with ShiftNauka, Moscow 1977 [in Russian].
Solvability Theory of Boundary Value Problems and Singular Integral Equa- tions with ShiftSeries: Mathematics and its applications 523, Kluwer Academic Publishers, Dordrecht, Boston, London 2000.
Mardiev, R., A criterion for the semi-Noetherian property of one class of singular integral operators with a non-Carleman shiftDokl. Akad. Nauk UzSSR2 (1985), 5–7 [in Russian].
Rabinovich, V.S., Roch, S., Silbermann, B., Fredholm theory and finite section method for band-dominated operatorsIntegr. Equat. Oper. Theory30 (1998), 452–495.
Sarason, D., Toeplitz operators with piecewise quasicontinuous symbolsIndiana Univ. Math. J.26 (1977), 817–838.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this paper
Cite this paper
Karlovich, A.Y., Lebre, A.B., Karlovich, Y.I. (2003). Invertibility of Functional Operators with Slowly Oscillating Non-Carleman Shifts. In: Böttcher, A., Kaashoek, M.A., Lebre, A.B., dos Santos, A.F., Speck, FO. (eds) Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, vol 142. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8007-7_9
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8007-7_9
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9401-2
Online ISBN: 978-3-0348-8007-7
eBook Packages: Springer Book Archive