Abstract
The spectrum of the Cesàro operator \(\mathsf {C}\), which is always continuous (but never compact) when acting on the classical Korenblum space and other related weighted Fréchet or (LB) spaces of analytic functions on the open unit disc, is completely determined. It turns out that such spaces are always Schwartz but, with the exception of the Korenblum space, never nuclear. Some consequences concerning the mean ergodicity of \(\mathsf {C}\) are deduced.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)
Albanese, A.A., Bonet, J., Ricker, W.J.: Montel resolvents and uniformly mean ergodic semigroups of linear operators. Quaest. Math. 36, 253–290 (2013)
Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in growth Banach spaces of analytic functions. Integral Equ. Oper. Theory 86, 97–112 (2016)
Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in the Fréchet spaces \(\ell ^{p+}\) and \(L^{p-}\). Glasgow Math. J. 59, 273–287 (2017)
Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator on power series spaces. Stud. Math. doi:10.4064/sm8590-2-2017
Aleman, A.: A class of integral operators on spaces of analytic functions, In: Proceedings 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., Peláez, J.A.: Spectra of integration operators and weighted square functions. Indiana Univ. Math. J. 61, 1–19 (2012)
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)
Barrett, D.E.: Duality between \(A^\infty \) and \(A^{- \infty }\) on domains with nondegenerate corners, Multivariable operator theory (Seattle, WA, 1993), pp. 77–87, Contemporary Math. Vol. 185, Amer. Math. Soc., Providence (1995)
Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on bounded domains. Mich. Math. J. 40, 271–297 (1993)
Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127, 137–168 (1998)
Bierstedt, K.D., Meise, R., Summers, W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272, 107–160 (1982)
Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. (Ser. A) 54, 70–79 (1993)
Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. (Ser. A) 64, 101–118 (1998)
Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)
Domenig, T.: Composition operators on weighted Bergman spaces and Hardy spaces. Atomic Decompositions and Diagonal Operators, Ph.D. Thesis, University of Zürich (1997). [Zbl 0909.47025]
Domenig, T.: Composition operators belonging to operator ideals. J. Math. Anal. Appl. 237, 327–349 (1999)
Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory. 2nd Printing. Wiley Interscience Publ., New York (1964)
Edwards, R.E.: Functional Analysis. Theory and Applications. Holt, Rinehart and Winston, New York, Chicago San Francisco (1965)
Grothendieck, A.: Topological Vector Spaces. Gordon and Breach, London (1973)
Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Graduate Texts in Mathematics, vol. 199. Springer, New York (2000)
Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)
Korenblum, B.: An extension of the Nevanlinna theory. Acta Math. 135, 187–219 (1975)
Krengel, U.: Ergodic Theorems. de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter Co., Berlin (1985)
Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175(1), 19–40 (2006)
Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)
Melikhov, S.N.: (DFS )-spaces of holomorphic functions invariant under differentiation. J. Math. Anal. Appl. 297, 577–586 (2004)
Persson, A.-M.: On the spectrum of the Cesàro operator on spaces of analytic functions. J. Math. Anal Appl. 340, 1180–1203 (2008)
Pietsch, A.: Nuclear Locally Convex Spaces. Springer, Berlin (1972)
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: Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Serc. Publ., Seville, pp. 51–68 (2006)
Acknowledgements
The research of the first two authors was partially supported by the projects MTM2013-43540-P and MTM2016-76647-P. The second author gratefully acknowledges the support of the Alexander von Humboldt Foundation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Albanese, A.A., Bonet, J. & Ricker, W.J. The Cesàro operator on Korenblum type spaces of analytic functions. Collect. Math. 69, 263–281 (2018). https://doi.org/10.1007/s13348-017-0205-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-017-0205-7
Keywords
- Cesàro operator
- Weighted spaces of analytic functions
- Spectrum
- Fréchet spaces
- (LB)-spaces
- Mean ergodicity