The Cesàro operator in growth Banach spaces of analytic functions

[EN] The CesA ro operator C, when acting in the classical growth Banach spaces and , for , of analytic functions on , 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 which C maps into (resp. into ); this optimal domain space always contains (resp. ) as a proper subspace.The research of the first two authors was partially supported by the projects MTM2013-43540-P and GVA Prometeo II/2013/013.Albanese, A.; Bonet Solves, JA.; Ricker, WJ. (2016). The Cesàro operator in growth Banach spaces of analytic functions. Integral Equations and Operator Theory. 86(1):97-112. https://doi.org/10.1007/s00020-016-2316-zS97112861Albanese 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

On Some Classical Operators of Variable Order in Variable Exponent Spaces

We give a survey of a selection of recent results on weighted and non-weighted estimations of classical operators of Harmonic Analysis in variable exponent Lebesgue, Morrey and Hölder spaces, based on the talk presented at International Conference Analysis, PDEs and Applications on the occasion of the 70th birthday of Vladimir Maz’ya, Rome, June 30–July 3, 2008. We touch both the Euclidean case and the general setting within the frameworks of quasimetric measure spaces. Some of the presented results are new