A note on completeness of weighted normed spaces of analytic functions

[EN] Given a non-negative weight v, not necessarily bounded or strictly positive, defined on a domain G in the complex plane, we consider the weighted space H-v(infinity) (G)of all holomorphic functions on G such that the product v vertical bar f vertical bar is bounded in G and study the question of when such a space is complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight v, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.The first author was partially supported by MTM2013-43540-P and MTM2016-76647-P by MINECO/FEDER-EU and GVA Prometeo II/2013/013. The second author was partially supported by the MINECO/FEDER-EU Grant MTM2015-65792-P. Both authors were partially supported by Thematic Research Network MTM2015-69323-REDT, MINECO, Spain.Bonet Solves, JA.; Vukotic, D. (2017). A note on completeness of weighted normed spaces of analytic functions. Results in Mathematics. 72(1-2):263-279. https://doi.org/10.1007/s00025-017-0696-2S263279721-2Arcozzi, N., Björn, A.: Dominating sets for analytic and harmonic functions and completeness of weighted Bergman spaces. Math. Proc. R. Ir. Acad. 102A, 175–192 (2002)Berenstein, C.A., Gay, R.: Complex Variables, An Introduction. Springer, New York (1991)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)Björn, A.: Removable singularities for weighted Bergman spaces. Czechoslov. Math. J. 56, 179–227 (2006)Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., Vogt, D.: Weighted spaces of holomorphic functions and sequence spaces. Note Mat. 17, 87–97 (1997)Conway, J.B.: Functions of One Complex Variable, Second Edition, Graduate Texts in Mathematics, vol. 11. Springer, New York (1978)Gaier, D.: Lectures on Complex Approximation. Birkhäuser, Boston (1987)Grosse-Erdmann, K.-G.: A weak criterion for vector-valued holomorphic functions. Math. Proc. Camb. Philos. Soc. 136, 399–411 (2004)Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland, Amsterdam (1979)Horváth, J.: Topological Vector Spaces and Distributions. Addison-Wesley, Reading (1966)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175, 19–45 (2006)Nakazi, T.: Weighted Bloch spaces which are Banach spaces. Rend. Circ. Mat. Palermo 62, 427–440 (2013)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

The final publication is available at Springer via: http://dx.doi.org/10.1007/s00605-012-0441-6[EN] We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type H∞ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered. © 2012 Springer-Verlag Wien.The research of Bonet was partially supported by MICINN and FEDER Project MTM2010-15200, by GV project Prometeo/2008/101, and by ACOMP/2012/090. The research of Vukotic was partially supported by MICINN grant MTM2009-14694-C02-01, Spain and by the European ESF Network HCAA ("Harmonic and Complex Analysis and Its Applications").Bonet Solves, JA.; Vukotić, D. (2013). Superposition operators between weighted Banach spaces of analytic functions of controlled growth. 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