Abstract
The Cesàro operator C, when acting in the classical growth Banach spaces \({A^{-\gamma}}\) and \({A_0^{-\gamma}}\), for \({\gamma} > 0\), of analytic functions on \({\mathbb{D}}\), is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is then possible to characterize the mean ergodic and related properties of C acting in these spaces. In addition, we determine the largest Banach space of analytic functions on \({\mathbb{D}}\) which C maps into \({A^{-\gamma}}\) (resp. into \({A_0^{-\gamma}}\)); this optimal domain space always contains \({A^{-\gamma}}\) (resp. \({A_0^{-\gamma}}\)) as a proper subspace.
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References
Albanese A.A., Bonet J., Ricker W.J.: Convergence of arithmetic means of operators in Fréchet spaces. J. Math. Anal. Appl. 401, 160–173 (2013)
Albanese, A.A., Bonet, J.,Ricker, W.J.: The Cesàro operator on power series spaces. Preprint (2016)
Albrecht E., Miller T.L., Neumann M.M.: Spectral properties of generalized Cesàro operators on Hardy and weighted Bergman spaces. Archiv Math. 85, 446–459 (2005)
Aleman A.: A class of integral operators on spaces of analytic functions. In: Proc. of the Winter School in Operator Theory and Complex Analysis, Univ. Málaga Secr. Publ., Málaga, pp. 3–30 (2007)
Aleman A., Constantin O.: Spectra of integration operators on weighted Bergman spaces. J. Anal. Math. 109, 199–231 (2009)
Aleman A., Persson A.-M.: Resolvent estimates and decomposable extensions of generalized Cesàro operators. J. Funct. Anal. 258, 67–98 (2010)
Aleman A., Siskakis A.G.: An integral operator on H p. Complex Var. Theory Appl. 28, 149–158 (1995)
Aleman A., Siskakis A.G.: Integration operators on Bergman spaces. Indiana Univ. Math. J. 46, 337–356 (1997)
Bayart F., Matheron E.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)
Bierstedt K.D., Bonet J., Galbis A.: Weighted spaces of holomorphic functions on balanced domains. Michigan Math. J. 40, 271–297 (1993)
Bierstedt K.D., Bonet J., Taskinen J.: Associated weights and spaces of holomorphic functions. Studia Math. 127, 137–168 (1998)
Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. 54, 70–79 (1993)
Bonet J., Domanski P., Lindström M.: Essential norm and weak compactness on weighted Banach spaces of analytic functions. Can. Math. Bull. 42, 139–148 (1999)
Curbera G.P., Ricker W.J.: Extensions of the classical Cesàro operator on Hardy spaces. Math. Scand. 108, 279–290 (2011)
Danikas N., Siskakis A.: The Cesàro operator on bounded analytic functions. Analysis 13, 295–299 (1993)
Duren P.: Theory of H p Spaces. Academic Press, New York (1970)
Dunford N., Schwartz J.T.:Linear Operators I: General Theory, 2nd Printing. Wiley Interscience Publ., New York (1964)
Grosse-Erdmann K., Peris A.: Linear Chaos. Springer, London (2011)
Harutyunyan A., Lusky W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)
Hedenmalm H., Korenblum B., Zhu K.: Theory of Bergman Spaces. Grad. Texts in Math., vol. 199. Springer, New York (2000)
Katzelson Y., Tzafriri L.: On power bounded operators. J. Funct. Anal. 68, 313–328 (1968)
Krengel U.: Ergodic Theorems. de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter Co., Berlin (1985)
Lin M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 337–340 (1974)
Lusky W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175(1), 19–40 (2006)
Megginson R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)
Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)
Persson A.-M.: On the spectrum of the Cesàro operator on spaces of analytic functions. J. Math. Anal. Appl. 340, 1180–1203 (2008)
Rubel L.A., Shields A.L.: The second dual of certain spaces of analytic functions. J. Aust. Math. Soc. 11, 276–280 (1970)
Shields A.L., Williams D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971)
Siskakis A.: Volterra operators on spaces of analytic functions—a survey. In: Proc. of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Serc. Publ., Seville, pp. 51–68 (2006)
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Albanese, A.A., Bonet, J. & Ricker, W.J. The Cesàro Operator in Growth Banach Spaces of Analytic Functions. Integr. Equ. Oper. Theory 86, 97–112 (2016). https://doi.org/10.1007/s00020-016-2316-z
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DOI: https://doi.org/10.1007/s00020-016-2316-z