Abstract
In this paper, on \({\mathbb {D}}\) we define Cowen–Douglas function introduced by the Cowen–Douglas operator \(M_\phi ^*\) on Hardy space \({\mathcal {H}}^2({\mathbb {D}})\), moreover, we give a necessary and sufficient condition to determine when \(\phi\) is a Cowen–Douglas function, where \(\phi \in {\mathcal {H}}^\infty ({\mathbb {D}})\) and \(M_{\phi }\) is the associated multiplication operator on \({\mathcal {H}}^{2}({\mathbb {D}})\). Then, we give some applications of Cowen–Douglas function on chaos, such as its application on the inverse problem of chaos for \(\phi (T)\), where \(\phi\) is a Cowen–Douglas function and T is the backward shift operator on the Hilbert space \({\mathcal {L}}^2({\mathbb {N}})\).
Similar content being viewed by others
References
Arveson, W.: A Short Course on Spectral Theory. Springer, New York (2002)
Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211(2), 766–793 (2007)
Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)
Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li–Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373, 83–93 (2011)
Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Li–Yorke chaos in linear dynamics. Ergod. Theory Dyn. Syst. 35, 1–23 (2014)
Bernardes Jr., N.C., Peris, A., Rodenas, F.: Set-valued chaos in linear dynamics. Integral Equ. Oper. Theory 88, 451–463 (2017)
Cartan, H.: (Translated by Yu, Jiarong): Theorie Elementaire des Fonctions Analytuques d’une ou Plusieurs Variables Complexes. Higer Education Press, Peking (2008)
Chen, L., Douglas, R.G., Guo, K.: On the double commutant of Cowen–Douglas operators. J. Funct. Anal. 260, 1925–1943 (2011)
Conway, J.B.: A Course in Functional Analysis. Springer, New York (1990)
Cowen, M.J., Douglas, R.G.: Complex geometry and operator theory. Acta Math. 141, 187–261 (1978)
Curto, R.E., Salinas, N.: Generalized Bergman kernels and the Cowen–Douglas theory. Am. J. Math. 106, 447–488 (1984)
Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. The Benjamin/Cummings Publishing Co., Menlo Park (1986)
Douglas, R.G.: Banach Algebra Techniques in Operator Theory. Springer, New York (1998)
Eschmeier, Jörg, Schmitt, J.: Cowen–Douglas operators and dominating sets. J. Oper. Theory 72, 277–290 (2014)
Garnett, J.B.: Bounded Analytic Functions, Revised First edn. Springer, New York (2007)
Grosse-Erdmann, K.-G., Peris, A.: Linear Chaos. Springer, London (2011)
Godefroy, G., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)
Heiatian Naeini, P., Yousefi, B.: On some properties of Cowen–Douglas class of operators. J. Funct. Sp. (2018)
Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Englewood Cliffs (1962)
Hou, B., Cui, P., Cao, Y.: Chaos for Cowen–Douglas operators. Proc. Am. Math. Soc. 138, 929–936 (2010)
Hou, B., Liao, G., Cao, Y.: Dynamics of shift operators. Houst. J. Math. 38, 1225–1239 (2012)
Hou, B., Luo, L.: Li–Yorke chaos for invertible mappings on noncompact spaces. Turk. J. Math. 40, 411–416 (2016)
Hou, B., Luo, L.: Li–Yorke chaos translation set for linear operators. Arch. Math. (Basel) 3, 267–278 (2018)
Hou, B., Tian, G., Shi, L.: Some dynamical properties for linear operators. Ill. J. Math. 53, 857–864 (2010)
Huang, W., Ye, X.: Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topol. Appl. 117, 259–272 (2002)
Jiang, C.: Similarity, reducibility and approximation of the Cowen–Douglas operators. J. Oper. Theory 32, 77–89 (1994)
Jiang, C., He, H.: Quasisimilarity of Cowen–Douglas operators. Sci. China Ser. A 47, 297–310 (2004)
Korányi, A., Misra, G.: A classification of homogeneous operators in the Cowen–Douglas class. Integral Equ. Oper. Theory 63, 595–599 (2009)
Korányi, A., Misra, G.: A classification of homogeneous operators in the Cowen–Douglas class. Adv. Math. 226, 5338–5360 (2011)
Li, T., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82, 985–992 (1975)
Luo, L., Hou, B.: Some remarks on distributional chaos for bounded linear operators. Turk. J. Math. 39, 251–258 (2015)
Luo, L., Hou, B.: Li–Yorke chaos for invertible mappings on compact metric spaces. Arch. Math. (Basel) 108, 65–69 (2017)
Mai, J.: Devaney’s chaos implies existence of s-scrambled sets. Proc. Am. Math. Soc. 132, 2761–2767 (2004)
Martínez-Giménez, F., Oprocha, P., Peris, A.: Distributional chaos for backward shifts. J. Math. Anal. Appl. 351(2), 607–615 (2009)
Martínez-Giménez, F., Oprocha, P., Peris, A.: Distributional chaos for operators with full scrambled sets. Math. Z. 274, 603–612 (2013)
Menet, Q.: Linear chaos and frequent hypercyclicity. Trans. Am. Math. Soc. 369(7), 4977–4994 (2017)
Moothathu, T.K.S.: Constant-norm scrambled sets for hypercyclic operators. J. Math. Anal. Appl. 387, 1219–1220 (2012)
Oprocha, P.: Relations between distributional and Devaney chaos. Chaos 16(3), 033112 (2006)
Schweizer, B., Smítal, J.: Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Am. Math. Soc. 344(2), 737–754 (1994)
Stockman, D.: Li–Yorke chaos in models with backward dynamics. Stud. Nonlinear Dyn. Econom. 20(5), 587–606 (2016)
Wang, L., Yang, Y., Chu, Z., Liao, G.: Weakly mixing implies distributional chaos in a sequence. Mod. Phys. Lett. B 24(14), 1595–1660 (2010)
Wu, X.: Li–Yorke chaos of translation semigroups. J. Differ. Equ. Appl. 20, 49–57 (2014)
Wu, X., Chen, G., Zhu, P.: Invariance of chaos from backward shift on the Köthe sequence space. Nonlinearity 27, 271–288 (2014)
Wu, X., Zhu, P.: Li–Yorke chaos of backward shift operators on Köthe sequence spaces. Topol. Appl. 160, 924–929 (2013)
Yin, Z., He, S., Huang, Y.: On Li–Yorke and distributionally chaotic direct sum operators. Topol. Appl. 239, 35–45 (2018)
Zhang, G., Lin, Y.: Lecture Notes of Functional Analysis, vol. 1. Peking University Press, Peking (2006)
Zhu, K.: Operators in Cowen-Douglas classes. Ill. J. Math. 44, 767–783 (2000)
Acknowledgements
This work is supported by the National Nature Science Foundation of China (Grant No. 11801428). I would like to thank the referee for his/her careful reading of the paper and helpful comments and suggestions. Also, I would like to show my deepest gratitude to Prof. Xiaoman Chen and Yijun Yao for their helpful suggestions in the completion of the paper. Lastly, I shall extend my thanks to Puyu Cui for the helpful technique on the proof (1) of Theorem 3.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Kehe Zhu.
Rights and permissions
About this article
Cite this article
Luo, L. Cowen-Douglas function and its application on chaos. Ann. Funct. Anal. 11, 897–913 (2020). https://doi.org/10.1007/s43034-020-00061-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s43034-020-00061-1