Mouze, Augustin; Munnier, Vincent Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc. (English) Zbl 07373609 Can. Math. Bull. 64, No. 2, 264-281 (2021). Let \(\alpha\in \mathbb R\). The authors consider on the space \(H(\mathbb D)\) of holomorphic functions the weighted shift operators \[f(z)=\sum_{n=0}^\infty a_nz^n\mapsto T_\alpha(f)(z):=a_1+\sum_{k=1}^\infty {\left(1+\frac{1}{k}\right)}^\alpha a_{k+1}z^k.\] It is first observed (by using a criterion due to [F. Bayart and S. Grivaux, Trans. Am. Math. Soc. 358, No. 11, 5083–5117 (2006; Zbl 1115.47005)]) that each \(T_\alpha\) is a frequently hypercyclic operator. The main results in the present paper give optimal growth conditions in the \(L^p\)-norm for a function \(f\in H(\mathbb D)\) to be frequently hypercyclic. For instance, if \(f\) is such a function then, whenever \(2\leq p<\infty\), \[M_p(r,f)\gtrsim \kappa(r,\alpha):= \begin{cases}(1-r)^{\alpha-1/2}& \text{ if }\alpha<1/2\\ \sqrt{|\log(1-r)|}& \text{ if }\alpha=1/2 \\ 1 &\text{ if }\alpha>1/2\end{cases} \] and there do exist such functions with \(M_p(r,f)\lesssim \kappa(r,\alpha)\). Similar estimations are given for \(1\leq p<2\). Reviewer: Raymond Mortini (Metz) Cited in 1 Document MSC: 47A16 Cyclic vectors, hypercyclic and chaotic operators 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 30B30 Boundary behavior of power series in one complex variable; over-convergence Keywords:frequently hypercyclic operator; Taylor shift; boundary behavior Citations:Zbl 1115.47005 PDFBibTeX XMLCite \textit{A. Mouze} and \textit{V. Munnier}, Can. Math. Bull. 64, No. 2, 264--281 (2020; Zbl 07373609) Full Text: DOI References: [1] Bayart, F. and Grivaux, S., Hypercyclicité: le rôle du spectre ponctuel unimodulaire . C. R. Acad. Sci. Paris.338(2004), 703-708. https://doi.org/10.1016/j.crma.2004.02.012 · Zbl 1059.47006 [2] Bayart, F. and Grivaux, S., Frequently hypercyclic operators . Trans. Amer. Math. Soc.358(2006), 5083-5117. https://doi.org/10.1090/S0002-9947-06-04019-0 · Zbl 1115.47005 [3] Bayart, F. and Matheron, E., Dynamics of linear operators . Cambridge Tracts in Mathematics, 179, Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CB09780511581113 · Zbl 1187.47001 [4] Beise, H. P., Meyrath, T., and Müller, J., Mixing Taylor shifts and universal Taylor series . Bull. London Math. Soc.47(2015), 136-142. https://doi.org/10.1112/blms/bdu104 · Zbl 1312.30068 [5] Blasco, O., Bonilla, A., and Grosse Erdmann, K-G., Rate of growth of frequently hypercyclic functions . Proc. Edinb. Math. Soc.53(2010), 39-59. https://doi.org/10.1017/S0013091508000564 · Zbl 1230.47019 [6] Drasin, D., and Saksman, E., Optimal growth of frequently hypercyclic entire functions . J. Funct. Anal.263(2012), 3674-3688. https://doi.org/10.1016/j.jfa.2012.09.007 · Zbl 1315.47007 [7] Duren, P. L., Theory of \(H^p\) spaces.Pure and Applied Mathematics, 38, Academic Press, New York and London, 1970. · Zbl 0215.20203 [8] Edwards, R. E. and Gaudry, G. I., Littlewood-Paley and multiplier theory . Ergebnisse der Mathematik und ihrer Grenzgebiete, 90, Springer-Verlag, Berlin and New York, 1977. · Zbl 0467.42001 [9] Girela, D. and Peláez, J. A., Integral means of analytic functions . Ann. Acad. Sci. Fenn. Math.29(2004), 459-469. · Zbl 1069.30059 [10] Grosse Erdmann, K-G. and Peris, A., Linear chaos. Universitext, Springer, London, 2011. https://doi.org/10.1007/978-1-4471-2170-1 · Zbl 1246.47004 [11] Mouze, A. and Munnier, V., Frequent hypercyclicity of random holomorphic functions for Taylor shifts and optimal growth. J. Anal. Math., to appear. · Zbl 1326.47007 [12] Rudin, W., Some theorems on Fourier coefficients . Proc. Amer. Math. Soc.10(1959), 855-859. https://doi.org/10.2307/2033608 · Zbl 0091.05706 [13] Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional integrals and derivatives. In: Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993. · Zbl 0818.26003 [14] Thelen, M., Frequently hypercyclic Taylor shifts . Comput. Methods Funct. Theory17(2017), 129-138. https://doi.org/10.1007/s40315-016-0173-z · Zbl 1372.30061 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.