Density conditions in Fréchet and (DF)-spaces

Sin resume

Spectra of weighted algebras of holomorphic functions

We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach-Stone type question is addressed.Comment: 25 pages Corrected typo

Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

The final publication is available at Springer via http://dx.doi.org/10.1007/s00013-012-0458-zWe investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H∞. Applications for composition operators on weighted Bloch spaces are given. © 2012 Springer Basel.1. The authors are thankful to the referee for pointing to us the references [15] and [16] and their relevance in the present research. 2. The research of Bonet was partially supported by MICINN and FEDER Project MTM2010-15200 and by GV project Prometeo/2008/101 and project ACOMP/2012/090.Bonet Solves, JA.; Lindström, M.; Wolf, E. (2012). Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions. Archiv der Mathematik. 99(6):537-546. https://doi.org/10.1007/s00013-012-0458-zS537546996Bierstedt K.D., Bonet J., Galbis A.: Weighted spaces of holomorphic functions on bounded 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)J. Bonet, P. Domański, and M. Lindström, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Canad, Math. Bull. 42, no. 2, (1999), 139–148Bonet J. et al.: Composition operators between weighted Banach spaces of analytic functions. J. Austral. Math. Soc. Ser. A 64, 101–118 (1998)Bonet J., Lindström M, Wolf E.: Isometric weighted composition operators on weighted Banach spaces of type H ∞. Proc. Amer. Math. Soc. 136, 4267–4273 (2008)Bonet J, Wolf E.: A note on weighted spaces of holomorphic functions. Archiv Math. 81, 650–654 (2003)Contreras M.D, Hernández-Díaz A.G.: Weighted composition operators in weighted banach spaces of analytic functions. J. Austral. Math. Soc. Ser. A 69, 41–60 (2000)Cowen C., MacCluer B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)J. Diestel, Geometry of Banach Spaces. Selected Topics, Lecture Notes in Math. vol. 485, Springer, Berlin, 1975.Hammond C.: On the norm of a composition operator with linear fractional symbol. Acta Sci. Math. (Szeged) 69, 813–829 (2003)Hosokawa T., Izuchi K., Zheng D.: Isolated points and essential components of composition operators on H ∞. Proc. Amer. Math. Soc. 130, 1765–1773 (2001)Hosokava T., Ohno S.: Topological strusctures of the sets of composition operatorson the Bloch spaces. J. Math. anal. Appl. 303, 499–508 (2005)Lusky W.: On the isomorphy classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–45 (2006)Martín M.: Norm-attaining composition operators on the Bloch spaces. J. Math. Anal. Appl. 369, 15–21 (2010)A. Montes-Rodríguez, The Pick-Schwarz lemma and composition operators on Bloch spaces, International Workshop on Operator Theory (Cefalu, 1997), Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 167–170.Montes-Rodríguez A.: The essential norm of a composition operator on Bloch spaces. Pacific J. Math. 188, 339–351 (1999)Montes-Rodríguez A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. London Math. Soc. 61, 872–884 (2000)J.H. Shapiro, Composition Operators and Classical Function Theory, Springer, 1993.K. Zhu, Operator Theory in Function Spaces, Second Edition. Amer. Math. Soc., 2007

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 realcompact topological vector spaces

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NN201 2740 33 and for the both authors by the project MTM2008-01502 of the Spanish Ministry of Science and Innovation.Kakol, JM.; López Pellicer, M. (2011). On realcompact topological vector spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 105(1):39-70. https://doi.org/10.1007/s13398-011-0003-0S39701051Argyros S., Mercourakis S.: On weakly Lindelöf Banach spaces. Rocky Mountain J. Math. 23(2), 395–446 (1993). doi: 10.1216/rmjm/1181072569Arkhangel’skii, A. V.: Topological Function Spaces, Mathematics and its Applications, vol. 78, Kluwer, Dordrecht (1992)Batt J., Hiermeyer W.: On compactness in L p (μ, X) in the weak topology and in the topology σ(L p (μ, X), L p (μ,X′)). Math. Z. 182, 409–423 (1983)Baumgartner J.E., van Douwen E.K.: Strong realcompactness and weakly measurable cardinals. Topol. 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Complut. 2(Supplementary Issue), 179–199 (1989)Pérez Carreras P., Bonet J.: Barrelled Locally Convex Spaces, Mathematics Studies 131. North-Holland, Amsterdam (1987)Pfister H.H.: Bemerkungen zum Satz über die separabilität der Fréchet-Montel Raüme. Arch. Math. (Basel) 27, 86–92 (1976). doi: 10.1007/BF01224645Robertson N.: The metrisability of precompact sets. Bull. Aust. Math. Soc. 43(1), 131–135 (1991). doi: 10.1017/S0004972700028847Rogers C.A., Jayne J.E., Dellacherie C., Topsøe F., Hoffman-Jørgensen J., Martin D.A., Kechris A.S., Stone A.H.: Analytic Sets. Academic Press, London (1980)Saxon S.A.: Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology. Math. Ann. 197(2), 87–106 (1972). doi: 10.1007/BF01419586Schawartz L.: Radom Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press, Oxford (1973)Schlüchtermann G., Wheller R.F.: On strongly WCG Banach spaces. Math. 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PLB-spaces of holomorphic functions with logarithmic growth conditions

Countable projective limits of countable inductive limits, called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen, Bierstedt and Bonet. In a previous article, the author extended their investigation to the case of holomorphic functions and characterized when spaces over the unit disc w.r.t. weights whose decay, roughly speaking, is neither faster nor slower than that of a polynomial are ultrabornological or barrelled. In this note, we prove a similar characterization for the case of weights which tend to zero logarithmically.Comment: Version of November 23, 2011. 9 page

(DF)-Spaces of type CB(X,E)CB(X, E) and CV‾(X,E)C\overline{V}(X, E)

Some locally convex properties of the spaces $CB( X, E)$ of the bounded continuous functions on a completely regular Hausdorff space X with values in a (DF-space) E are studied and applied to the (DF)-spaces of type $C\bar{V}(X,E)$ (e.g., see [S]).The following are our main results: 1.$CB(X,E)$ is a (DF)-space if and only if E is a (DF)-space. 2.For a (DF)-space E, $CB(X,E)$ is quasi barrelled if and only if either (i)X is pseudocompact and E is quasibarrelled or (ii) X is not pseudocompact and the bounded subsets of E are metrizaable. 3. If $\mathcal V ⊂ C(X)$ and if each $\bar{v}∈\bar{V}$ is dominated by some $\tilde{v}∈ \bar{V}∩ C(X)$, then $C\bar{V}(X,E)$ (resp., $C\bar{V}(X)⨂_\varepsilon E$) is a (DF)-space if and only if E is a (DF)-space. 4. Let X be a locally compact and σ-compact space, $\mathcal V ⊂ C(X)$ and E a (DF)-space. Then $C\bar{V}(X,E)$ is quasibarrelled if and only if (i) E is quasibarrelled and $\mathcal V$ satisfies condition $( M, K)$ or (ii) the bounded subsets of E are metrizable and $\mathcal V$ satisfies condition (D)