Classical operators on weighted Banach spaces of entire functions

Beltrán, María; Bonet, José; Fernández, Carmen

doi:10.1090/S0002-9939-2013-11685-0

Classical operators on weighted Banach spaces of entire functions
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by María J. Beltrán, José Bonet and Carmen Fernández

Proc. Amer. Math. Soc. 141 (2013), 4293-4303

DOI: https://doi.org/10.1090/S0002-9939-2013-11685-0

Published electronically: August 9, 2013

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Abstract:

We study the operators of differentiation and of integration and the Hardy operator on weighted Banach spaces of entire functions. We estimate the norm of the operators, study the spectrum, and analyze when they are surjective, power bounded, hypercyclic, and (uniformly) mean ergodic.

References

A.G. Arvanitidis, A.G. Siskakis, Cesàro operators on the Hardy spaces of the half-plane, arXiv:1006.1520v1.
Aharon Atzmon and Bruno Brive, Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions, Bergman spaces and related topics in complex analysis, Contemp. Math., vol. 404, Amer. Math. Soc., Providence, RI, 2006, pp. 27–39. MR 2244002, DOI 10.1090/conm/404/07632
Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318, DOI 10.1017/CBO9780511581113
Klaus D. Bierstedt, José Bonet, and Antonio Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40 (1993), no. 2, 271–297. MR 1226832, DOI 10.1307/mmj/1029004753
Klaus D. Bierstedt, José Bonet, and Jari Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), no. 2, 137–168. MR 1488148, DOI 10.4064/sm-127-2-137-168
K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 54 (1993), no. 1, 70–79. MR 1195659, DOI 10.1017/S1446788700036983
Oscar Blasco and Antonio Galbis, On Taylor coefficients of entire functions integrable against exponential weights, Math. Nachr. 223 (2001), 5–21. MR 1817845, DOI 10.1002/1522-2616(200103)223:1<5::AID-MANA5>3.3.CO;2-F
José Bonet, Dynamics of the differentiation operator on weighted spaces of entire functions, Math. Z. 261 (2009), no. 3, 649–657. MR 2471093, DOI 10.1007/s00209-008-0347-0
José Bonet and Antonio Bonilla, Chaos of the differentiation operator on weighted Banach spaces of entire functions, Complex Anal. Oper. Theory 7 (2013), no. 1, 33–42. MR 3010787, DOI 10.1007/s11785-011-0134-5
José Bonet and Werner J. Ricker, Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions, Arch. Math. (Basel) 92 (2009), no. 5, 428–437. MR 2506944, DOI 10.1007/s00013-009-3061-1
Antonio Galbis, Weighted Banach spaces of entire functions, Arch. Math. (Basel) 62 (1994), no. 1, 58–64. MR 1249586, DOI 10.1007/BF01200439
Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812, DOI 10.1007/978-1-4471-2170-1
Anahit Harutyunyan and Wolfgang Lusky, On the boundedness of the differentiation operator between weighted spaces of holomorphic functions, Studia Math. 184 (2008), no. 3, 233–247. MR 2369141, DOI 10.4064/sm184-3-3
Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257, DOI 10.1007/978-3-322-90559-8
Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
Michael Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337–340. MR 417821, DOI 10.1090/S0002-9939-1974-0417821-6
Heinrich P. Lotz, Tauberian theorems for operators on $L^{\infty }$ and similar spaces, Functional analysis: surveys and recent results, III (Paderborn, 1983) North-Holland Math. Stud., vol. 90, North-Holland, Amsterdam, 1984, pp. 117–133. MR 761376, DOI 10.1016/S0304-0208(08)71470-1
Heinrich P. Lotz, Uniform convergence of operators on $L^\infty$ and similar spaces, Math. Z. 190 (1985), no. 2, 207–220. MR 797538, DOI 10.1007/BF01160459
Wolfgang Lusky, On the Fourier series of unbounded harmonic functions, J. London Math. Soc. (2) 61 (2000), no. 2, 568–580. MR 1760680, DOI 10.1112/S0024610799008443
Wolfgang Lusky, On the isomorphism classes of weighted spaces of harmonic and holomorphic functions, Studia Math. 175 (2006), no. 1, 19–45. MR 2261698, DOI 10.4064/sm175-1-2
Reinhold Meise and Dietmar Vogt, Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2, The Clarendon Press, Oxford University Press, New York, 1997. Translated from the German by M. S. Ramanujan and revised by the authors. MR 1483073
Alfonso Montes-Rodríguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. London Math. Soc. (2) 61 (2000), no. 3, 872–884. MR 1766111, DOI 10.1112/S0024610700008875
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062