A Note on the Spectrum of Composition Operators on Spaces of Real Analytic Functions

[EN] In this paper the spectrum of composition operators on the space of real analytic functions is investigated. In some cases it is completely determined while in some other cases it is only estimated.The research of the authors was partially supported by MEC and FEDER Project MTM2013-43540-P and the work of of Bonet by the Grant GV Project Prometeo II/2013/013. The research of Domanski was supported by National Center of Science, Poland, Grant No. DEC-2013/10/A/ST1/00091.Bonet Solves, JA.; Domanski, P. (2017). A Note on the Spectrum of Composition Operators on Spaces of Real Analytic Functions. Complex Analysis and Operator Theory. 11(1):161-174. https://doi.org/10.1007/s11785-016-0589-5S161174111Belitskii, G., Lyubich, Y.: The Abel equation and total solvability of linear functional equations. Studia Math. 127, 81–97 (1998)Belitskii, G., Lyubich, Y.: The real analytic solutions of the Abel functional equation. Studia Math. 134, 135–141 (1999)Belitskii, G., Tkachenko, V.: One-Dimensional Functional Equations. Springer, Basel (2003)Belitskii, G., Tkachenko, V.: Functional equations in real analytic functions. Studia Math. 143, 153–174 (2000)Bonet, J., Domański, P.: Power bounded composition operators on spaces of analytic functions. Collect. Math. 62, 69–83 (2011)Bonet, J., Domański, P.: Hypercyclic composition operators on spaces of real analytic fucntions. Math. Proc. Cambridge Phil. Soc. 153, 489–503 (2012)Bonet, J., Domański, P.: Abel’s functional equation and eigenvalues of composition operators on spaces of real analytic functions. Integr. Equ. Oper. Theor. 81, 455–482 (2015). doi: 10.1007/s00020-014-2175-4Cartan, H.: Variétés analytiques réelles et variétés analytiques complexes. Bull. Soc. Math. France 85, 77–99 (1957)Domański, P.: Notes on real analytic functions and classical operators, Topics in Complex Analysis and Operator Theory (Winter School in Complex Analysis and Operator Theory, Valencia, February 2010), Contemporary Math. 561 (2012) 3–47. Amer. Math. Soc, Providence (2012)Domański, P., Goliński, M., Langenbruch, M.: A note on composition operators on spaces of real analytic functions. Ann. Polon. Mat. 103, 209–216 (2012)Domański, P., Langenbruch, M.: Composition operators on spaces of real analytic functions. Math. Nachr. 254–255, 68–86 (2003)Domański, P., Langenbruch, M.: Coherent analytic sets and composition of real analytic functions. J. reine angew. Math. 582, 41–59 (2005)Domański, P., Langenbruch, M.: Composition operators with closed image on spaces of real analytic functions. Bull. Lond. Math. Soc. 38, 636–646 (2006)Domański, P., Vogt, D.: The space of real analytic functions has no basis. Studia Math. 142, 187–200 (2000)Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North Holland, Amsterdam (1986)Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon, Oxford (1997)Smajdor, W.: On the existence and uniqueness of analytic solutions of the functional equation $$\varphi (z)=h(z,\varphi [f(z)])$$ φ ( z ) = h ( z , φ [ f ( z ) ] ) . Ann. Polon. Math. 19, 37–45 (1967

Absorbers: Definitions, properties and applications

Roughly speaking, the absorber is a set, which includes, after finite number of initial states, each trajectory of a transformation of space into itself. This paper deals with the exact definition of absorbers for linear operators, the study of the properties, the applications to “classical” dynamics and to solvability of operator equations. It is expected that the description of the structure of absorbers will add new insights to the recent discussion of nature and content of notion of attractiveness for nonlinear dynamics

Poincare' normal forms and simple compact Lie groups

We classify the possible behaviour of Poincar\'e-Dulac normal forms for dynamical systems in $R^n$ with nonvanishing linear part and which are equivariant under (the fundamental representation of) all the simple compact Lie algebras and thus the corresponding simple compact Lie groups. The ``renormalized forms'' (in the sense of previous work by the author) of these systems is also discussed; in this way we are able to simplify the classification and moreover to analyze systems with zero linear part. We also briefly discuss the convergence of the normalizing transformations.Comment: 17 pages; minor corrections in revised versio

Generalized Induced Norms

Let ||.|| be a norm on the algebra M_n of all n-by-n matrices over the complex field C. An interesting problem in matrix theory is that "are there two norms ||.||_1 and ||.||_2 on C^n such that ||A||=max{||Ax||_2: ||x||_1=1} for all A in M_n. We will investigate this problem and its various aspects and will discuss under which conditions ||.||_1=||.||_2.Comment: 8 page

Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

We obtain full description of eigenvalues and eigenvectors of composition operators Cϕ : A (R) → A (R) for a real analytic self map ϕ : R → R as well as an isomorphic description of corresponding eigenspaces. We completely characterize those ϕ for which Abel’s equation f ◦ ϕ = f + 1 has a real analytic solution on the real line. We find cases when the operator Cϕ has roots using a constructed embedding of ϕ into the so-called real analytic iteration semigroups.(1) The research of the authors was partially supported by MEC and FEDER Project MTM2010-15200 and MTM2013-43540-P and the work of Bonet also by GV Project Prometeo II/2013/013. The research of Domanski was supported by National Center of Science, Poland, Grant No. NN201 605340. (2) The authors are very indebted to K. Pawalowski (Poznan) for providing us with references [26,27,47] and also explaining some topological arguments of [10]. The authors are also thankful to M. Langenbruch (Oldenburg) for providing a copy of [29].Bonet Solves, JA.; Domanski, P. (2015). Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integral Equations and Operator Theory. 81(4):455-482. https://doi.org/10.1007/s00020-014-2175-4S455482814Abel, N.H.: Determination d’une function au moyen d’une equation qui ne contient qu’une seule variable. In: Oeuvres Complètes, vol. II, pp. 246-248. Christiania (1881)Baker I.N.: Zusammensetzung ganzer Funktionen. Math. Z. 69, 121–163 (1958)Baker I.N.: Permutable power series and regular iteration. J. Aust. Math. Soc. 2, 265–294 (1961)Baker I.N.: Permutable entire functions. Math. Z. 79, 243–249 (1962)Baker I.N.: Fractional iteration near a fixpoint of multiplier 1. J. Aust. Math. Soc. 4, 143–148 (1964)Baker I.N.: Non-embeddable functions with a fixpoint of multiplier 1. Math. Z. 99, 337–384 (1967)Baker I.N.: On a class of nonembeddable entire functions. J. Ramanujan Math. Soc. 3, 131–159 (1988)Baron K., Jarczyk W.: Recent results on functional equations in a single variable, perspectives and open problems. Aequ. Math. 61, 1–48 (2001)Belitskii G., Lyubich Y.: The Abel equation and total solvability of linear functional equations. Studia Math. 127, 81–97 (1998)Belitskii G., Lyubich Yu.: The real analytic solutions of the Abel functional equation. Studia Math. 134, 135–141 (1999)Belitskii G., Tkachenko V.: One-Dimensional Functional Equations. Springer, Basel (2003)Belitskii G., Tkachenko V.: Functional equations in real analytic functions. Studia Math. 143, 153–174 (2000)Bonet J., Domański P.: Power bounded composition operators on spaces of analytic functions. Collect. Math. 62, 69–83 (2011)Bonet J., Domański P.: Hypercyclic composition operators on spaces of real analytic functions. Math. Proc. Camb. Philos. Soc. 153, 489–503 (2012)Bracci, F., Poggi-Corradini, P.: On Valiron’s theorem. In: Proceedings of Future Trends in Geometric Function Theory. RNC Workshop Jyväskylä 2003, Rep. Univ. Jyväskylä Dept. Math. Stat., vol. 92, pp. 39–55 (2003)Contreras, M.D.: Iteración de funciones analíticas en el disco unidad. Universidad de Sevilla (2009). (Preprint)Contreras M.D., Díaz-Madrigal S., Pommerenke Ch.: Some remarks on the Abel equation in the unit disk. J. Lond. Math. Soc. 75(2), 623–634 (2007)Cowen C.: Iteration and the solution of functional equations for functions analytic in the unit disc. Trans. Am. Math. Soc. 265, 69–95 (1981)Cowen C.C., MacCluer B.D.: Composition operators on spaces of analytic functions. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)Domański, P.: Notes on real analytic functions and classical operators. In: Topics in Complex Analysis and Operator Theory (Winter School in Complex Analysis and Operator Theory, Valencia, February 2010). Contemporary Math., vol. 561, pp. 3–47. Am. Math. Soc., Providence (2012)Domański P., Goliński M., Langenbruch M.: A note on composition operators on spaces of real analytic functions. Ann. Polon. Math. 103, 209–216 (2012)P. Domański M. Langenbruch 2003 Language="En"Composition operators on spaces of real analytic functions Math. Nachr. 254–255, 68–86 (2003)Domański P., Langenbruch M.: Coherent analytic sets and composition of real analytic functions. J. Reine Angew. Math. 582, 41–59 (2005)Domański P., Langenbruch M.: Composition operators with closed image on spaces of real analytic functions. Bull. Lond. Math. Soc. 38, 636–646 (2006)Domański P., Vogt D.: The space of real analytic functions has no basis. Studia Math. 142, 187–200 (2000)Fuks D.B., Rokhlin V.A.: Beginner’s Course in Topology. Springer, Berlin (1984)Greenberg M.J.: Lectures on Algebraic Topology. W. A. Benjamin Inc., Reading (1967)Hammond, C.: On the norm of a composition operator, PhD. dissertation, Graduate Faculty of the University of Virginia (2003). http://oak.conncoll.edu/cnham/Thesis.pdfHandt T., Kneser H.: Beispiele zur Iteration analytischer Funktionen. Mitt. Naturwiss. Ver. für Neuvorpommernund Rügen, Greifswald 57, 18–25 (1930)Heinrich T., Meise R.: A support theorem for quasianalytic functionals. Math. Nachr. 280(4), 364–387 (2007)Karlin S., McGregor J.: Embedding iterates of analytic functions with two fixed points into continuous group. Trans.Am. Math. Soc. 132, 137–145 (1968)Kneser H.: Reelle analytische Lösungen der Gleichung $${\varphi(\varphi(x))=e^x}$$ φ ( φ ( x ) ) = e x und verwandter Funktionalgleichungen. J. Reine Angew. Math. 187, 56–67 (1949)Königs, G.: Recherches sur les intégrales de certaines équations fonctionnelles. Ann. Sci. Ecole Norm. Sup. (3) 1, Supplément, 3–41 (1884)Kuczma M.: Functional Equations in a Single Variable. PWN-Polish Scientific Publishers, Warszawa (1968)Kuczma M., Choczewski B., Ger R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Milnor, J.: Dynamics in One Complex Variable. Vieweg, Braunschweig (2006)Schröder E.: über iterierte Funktionen. Math. Ann. 3, 296–322 (1871)Shapiro J.H.: Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics. Springer, New York (1993)Shapiro, J.H.: Notes on the dynamics of linear operators. Lecture Notes. http://www.mth.msu.edu/~hapiro/Pubvit/Downloads/LinDynamics/LynDynamics.htmlShapiro, J.H.: Composition operators and Schröder functional equation. In: Studies on Composition Operators (Laramie, WY, 1996), Contemp. Math., vol. 213, pp. 213–228. Am. Math. Soc., Providence (1998)Szekeres G.: Regular iteration of real and complex functions. Acta Math. 100, 203–258 (1958)Szekeres G.: Fractional iteration of exponentially growing functions. J. Aust. Math. Soc. 2, 301–320 (1961)Szekeres G.: Fractional iteration of entire and rational functions. J. Aust. Math. Soc. 4, 129–142 (1964)Szekeres G.: Abel’s equations and regular growth: variations on a theme by Abel. Exp. Math. 7, 85–100 (1998)Trappmann H., Kouznetsov D.: Uniqueness of holomorphic Abel function at a complex fixed point pair. Aequ. Math. 81, 65–76 (2011)Viro, O.: 1-manifolds. Bull. Manifold Atlas. http://www.boma.mpim-bonn.mpg.de/articles/48 (a prolonged version also http://www.map.mpim-bonn.mpg.de/1-manifolds#Differential_structures )Walker P.L.: A class of functional equations which have entire solutions. Bull. Aust. Math. Soc. 39, 351–356 (1988)Walker P.L.: The exponential of iteration of e x −1. Proc. Am. Math. Soc. 110, 611–620 (1990)Walker P.L.: On the solution of an Abelian functional equation. J. Math. Anal. Appl. 155, 93–110 (1991)Walker P.L.: Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57, 723–733 (1991

Gluon Polarization from QCD Sum Rules

The gluon polarization $\Delta G$ in a nucleon can be defined in a gauge invariant way as the integral over the Ioffe-time distribution of polarized gluons. We argue that for sufficiently regular polarized gluon distributions $\Delta G$ is dominated by contributions from small and moderate values of the Ioffe-time z < 10. As a consequence $\Delta G$ can be estimated with 20% accuracy from the first two even moments of the polarized gluon distribution, and its behavior at small values of Bjorken x or, equivalently, at large Ioffe-times z. We employ this idea and compute the first two moments of the polarized gluon distribution within the framework of QCD sum rules. Combined with the color coherence hypothesis we obtain an upper limit for $\Delta G \sim 2 \pm 0.5$ at a typical scale $\mu^2 \sim 1 GeV^2$.Comment: 12 pages, Latex, 2 figures include

Результати експериментальних досліджень ефективності застосування гідравлічного грейфера з центральним гвинтовим якорем

В статье рассматриваются результаты экспериментальных исследований грейферного ковша с центральным винтовым якорем. Данные экспериментальные исследования изменения массы разрабатываемого грунта и энергоемкости процесса копания в зависимости от параметров винтового якоря позволяют говорить о высокй эффективности применения грейферов с центральным винтовым якорем на прочных грунтах.В статье рассматриваются результаты экспериментальных исследований грейферного ковша с центральным винтовым якорем. Данные экспериментальные исследования изменения массы разрабатываемого грунта и энергоемкости процесса копания в зависимости от параметров винтового якоря позволяют говорить о высокй эффективности применения грейферов с центральным винтовым якорем на прочных грунтах.У статті розглядаються результати експериментальних досліджень грейферного ковша з центральним гвинтовим якорем. Дані експериментальні дослідження зміни маси грунту, що розробляється і енергоємності процесу копання в залежності від параметрів гвинтового якоря дозволяють говорити про високу ефективність застосування грейферів з центральним гвинтовим якорем на міцних грунтах

The Abel equation and total solvability of linear functional equations

We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx) - φ(x) = γ(x). The smooth situation can also be considered in this way