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POWERS OF COMPOSITION OPERATORS: ASYMPTOTIC BEHAVIOUR ON BERGMAN, DIRICHLET AND BLOCH SPACES

Part of: Special classes of linear operators Entire and meromorphic functions, and related topics Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  27 November 2019

W. ARENDT ,
I. CHALENDAR ,
M. KUMAR  and
S. SRIVASTAVA
W. ARENDT
Affiliation:
Institute of Applied Analysis, University of Ulm, 89069, Ulm, Germany email wolfgang.arendt@uni-ulm.de
I. CHALENDAR*
Affiliation:
Université Paris-Est, LAMA (UMR 8050), UPEM, UPEC, CNRS, F-77454, Marne-la-Vallée, France email isabelle.chalendar@u-pem.fr
M. KUMAR
Affiliation:
Lady Shri Ram College For Women, Department of Mathematics, University of Delhi, Delhi, India email mahekumar81@gmail.com
S. SRIVASTAVA
Affiliation:
Department of Mathematics, University of Delhi, South Campus, Delhi, India email sachi_srivastava@yahoo.com
*
✉isabelle.chalendar@u-pem.fr
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Abstract

We study the asymptotic behaviour of the powers of a composition operator on various Banach spaces of holomorphic functions on the disc, namely, standard weighted Bergman spaces (finite and infinite order), Bloch space, little Bloch space, Bloch-type space and Dirichlet space. Moreover, we give a complete characterization of those composition operators that are similar to an isometry on these various Banach spaces. We conclude by studying the asymptotic behaviour of semigroups of composition operators on these various Banach spaces.

Keywords

composition operatorsweighted Bergman spacesBloch spaceasymptotic behaviourholomorphic semiflows

MSC classification

Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Secondary: 47D03: Groups and semigroups of linear operators 47B33: Composition operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 3 , June 2020 , pp. 289 - 320
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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References

Allen, R. F. and Colonna, F., ‘On the isometric composition operators on the Bloch space in ℂn’, J. Math. Anal. Appl. 355 (2009), 675–688.Google Scholar
Arendt, W., Batty, C. J. K., Hieber, M. and Neubrander, F., Vector-valued Laplace Transforms and Cauchy Problems, 2nd edn, Monographs in Mathematics, 96 (Birkhäuser, Basel, 2011).Google Scholar
Arendt, W., Chalendar, I., Kumar, M. and Srivastava, S., ‘Asymptotic behaviour of the powers of composition operators on Banach spaces of holomorphic functions’, Indiana Math. J. 64(4) (2018), 1571–1595.Google Scholar
Bayart, F., ‘Similarity to an isometry of a composition operator’, Proc. Amer. Math. Soc. 131 (2003), 1789–1791.Google Scholar
Beltrán-Meneu, M. J., Gómez-Collado, M. C., Jordán, E. and Jornet, D., ‘Mean ergodic composition operators on Banach spaces of holomorphic functions’, J. Math. Anal. Appl. 444(2) (2016), 1640–1651.Google Scholar
Bonet, J., Domański, P., Lindström, M. and Taskinen, J., ‘Composition operators between weighted Banach spaces of analytic functions’, J. Aust. Math. Soc. Ser. A 64(1) (1998), 101–118.Google Scholar
Bourdon, P. S. and Shapiro, J., ‘Mean growth of Koenigs eigenfunctions’, J. Amer. Math. Soc. 10 (1997), 299–325.Google Scholar
Carswell, B. and Hammond, C., ‘Composition operators with maximal norm on weighted Bergman spaces’, Proc. Amer. Math. Soc. 134 (2006), 2599–2605.Google Scholar
Chacón, G. A., Chacón, G. R. and Giménez, J., ‘Composition operators on the Dirichlet space and related problems’, Bol. Asoc. Mat. Venez. 13(2) (2006), 155–164.Google Scholar
Chalendar, I. and Partington, J. R., ‘Norm estimates for weighted composition operators on spaces of holomorphic functions’, Complex Anal. Oper. Theory 8(5) (2014), 1087–1095.Google Scholar
Contreras, M. D. and Hernandez-Diaz, A. G., ‘Weighted composition operators in weighted Banach spaces of analytic functions’, J. Aust. Math. Soc. Ser. A 69(1) (2000), 41–60.Google Scholar
Cowen, C. C. and MacCluer, B. D., Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, FL, 1995).Google Scholar
Gallardo-Gutíerrez, E. A. and Montes-Rodríguez, A., ‘Adjoints of linear fractional composition operators on the Dirichlet space’, Math. Ann. 327 (2003), 117–134.Google Scholar
Higdon, W. M., ‘The spectra of composition operators from linear fractional maps acting upon the Dirichlet space’, J. Funct. Anal. 220 (2005), 55–75.Google Scholar
Jovović, M. and MacCluer, B. D., ‘Composition operators on Dirichlet spaces’, Acta Sci. Math. (Szeged) 63(1–2) (1997), 229–247.Google Scholar
Lou, Z., ‘Composition operators on Bloch type spaces’, Analysis 23 (2003), 81–95.Google Scholar
MacCluer, B. D. and Saxe, K., ‘Spectra of composition operators on the Bloch and Bergman spaces’, Israel J. Math. 128 (2002), 325–354.Google Scholar
Madigan, K., ‘Composition operators on analytic Lipschitz spaces’, Proc. Amer. Math. Soc. 119(2) (1993), 465–473.Google Scholar
Madigan, K. and Matheson, A., ‘Compact composition operators on the Bloch space’, Trans. Amer. Math. Soc. 347(7) (1995), 2679–2687.Google Scholar
Martín, M. J. and Vukotić, D., ‘Norms and spectral radii of composition operators acting on the Dirichlet space’, J. Math. Anal. Appl. 304 (2005), 22–32.Google Scholar
Martín, M. J. and Vukotić, D., ‘Isometries of some classical function spaces among the composition operators’, Contemp. Math. 393 (2006), 133–138.Google Scholar
Martín, M. J. and Vukotić, D., ‘Isometries of the Dirichlet space among the composition operators’, Proc. Amer. Math. Soc. 134(6) (2006), 1701–1705.Google Scholar
Martín, M. J. and Vukotić, D., ‘Isometric composition operators on the Bloch space among the composition operators’, Bull. Lond. Math. Soc. 39(1) (2007), 151–155.Google Scholar
Montes-Rodríguez, A., ‘The essential norm of a composition operator on Bloch spaces’, Pacific J. Math. 188 (1999), 339–351.Google Scholar
Montes-Rodríguez, A., ‘Weighted composition operators on weighted Banach spaces of analytic functions’, J. Lond. Math. Soc. 61(2) (2000), 872–884.Google Scholar
Petersen, K., Ergodic Theory (Cambridge University Press, Cambridge, 1983).Google Scholar
Pommerenke, C., Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften, 299 (Springer, Berlin, 1992).Google Scholar
Shapiro, J. H., ‘The essential norm of a composition operator’, Ann. of Math. (2) 125 (1987), 375–404.Google Scholar
Smith, W., ‘Composition operators between Bergman and Hardy spaces’, Trans. Amer. Math. Soc. 348 (1996), 2331–2348.Google Scholar
Vukotić, D., ‘A sharp estimate for A 𝛼 functions in ℂn’, Proc. Amer. Math. Soc. 117 (1993), 753–756.Google Scholar
Xiao, J., ‘Composition operators associated with Bloch-type spaces’, Complex Var. Theory Appl. 46(2) (2001), 109–121.Google Scholar
Xiong, C., ‘Norm of composition operators on the Bloch space’, Bull. Aust. Math. Soc. 70 (2004), 293–299.Google Scholar
Yosida, K., Functional Analysis (Springer, Berlin, Heidelberg, 1965).Google Scholar
Zorboska, N., ‘Isometric composition operators on the Bloch-type spaces’, C. R. Math. Acad. Sci. Soc. R. Can. 29(3) (2007), 91–96.Google Scholar
Zorboska, N., ‘Isometric and closed-range composition operators between Bloch-type spaces’, Int. J. Math. Math. Sci. 15 (2011), Article ID 132541, 15 pages.Google Scholar
Zhu, K., Operator Theory in Function Spaces, 2nd edn, Mathematical Surveys and Monographs, 138 (American Mathematical Society, Providence, RI, 2007).Google Scholar
Zygmund, A., Trigonometric Series, Vol. I (Cambridge University Press, Cambridge, 1959).Google Scholar