×

Weighted vector-valued holomorphic functions on Banach spaces. (English) Zbl 1300.46032

The purpose of this paper is to investigate conditions to ensure when a weakly holomorphic vector-valued function defined on a non void open and connected subset \(U\) of an infinite-dimensional Banach space \(X\) satisfying certain estimates is in fact holomorphic, and to give extension results for functions in a Banach space of holomorphic functions defined also on a non void open and connected \(U\) subset of \(X\). These investigations continue recent work by Arendt, Nikolski, Grosse-Erdmann, Frerick, Wengenroth, the author and the reviewer, among others. Extensions of theorems from Frerick, Jordá and Wengenroth in [L. Frerick et al., Math. Nachr. 282, No. 5, 690–696 (2009; Zbl 1177.46024)] are obtained. Linearization of Banach spaces of vector-valued holomorphic functions constitutes an important technique in the paper. Several nice results and examples about different weighted Banach spaces of holomorphic functions defined on subsets of a Banach space \(X\) are given. They clarify the assumptions of the theorems given in the article.

MSC:

46E40 Spaces of vector- and operator-valued functions
46E15 Banach spaces of continuous, differentiable or analytic functions
46G20 Infinite-dimensional holomorphy

Citations:

Zbl 1177.46024
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dunford, N., Uniformity in linear spaces, Transactions of the American Mathematical Society, 44, 2, 305-356 (1938) · Zbl 0019.41604 · doi:10.2307/1989974
[2] Grothendieck, A., Sur certains espaces de fonctions holomorphes. I, Journal für die Reine und Angewandte Mathematik, 192, 35-64 (1953) · Zbl 0051.08704
[3] Bogdanowicz, W. M., Analytic continuation of holomorphic functions with values in a locally convex space, Proceedings of the American Mathematical Society, 22, 660-666 (1969) · Zbl 0182.19201 · doi:10.1090/S0002-9939-1969-0250067-1
[4] Arendt, W.; Nikolski, N., Vector-valued holomorphic functions revisited, Mathematische Zeitschrift, 234, 4, 777-805 (2000) · Zbl 0976.46030 · doi:10.1007/s002090050008
[5] Bonet, J.; Frerick, L.; Jordá, E., Extension of vector-valued holomorphic and harmonic functions, Studia Mathematica, 183, 3, 225-248 (2007) · Zbl 1141.46017 · doi:10.4064/sm183-3-2
[6] Frerick, L.; Jordá, E., Extension of vector-valued functions, Bulletin of the Belgian Mathematical Society. Simon Stevin, 14, 3, 499-507 (2007) · Zbl 1141.46018
[7] Frerick, L.; Jordá, E.; Wengenroth, J., Extension of bounded vector-valued functions, Mathematische Nachrichten, 282, 5, 690-696 (2009) · Zbl 1177.46024 · doi:10.1002/mana.200610764
[8] Grosse-Erdmann, K.-G., A weak criterion for vector-valued holomorphy, Mathematical Proceedings of the Cambridge Philosophical Society, 136, 2, 399-411 (2004) · Zbl 1055.46026 · doi:10.1017/S0305004103007254
[9] Laitila, J.; Tylli, H.-O., Composition operators on vector-valued harmonic functions and Cauchy transforms, Indiana University Mathematics Journal, 55, 2, 719-746 (2006) · Zbl 1119.47022 · doi:10.1512/iumj.2006.55.2785
[10] Beltrán, M. J., Linearization of weighted (LB)-spaces of entire functions on Banach spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales A, 106, 2, 275-286 (2012) · Zbl 1273.46027 · doi:10.1007/s13398-011-0049-z
[11] Carando, D.; Zalduendo, I., Linearization of functions, Mathematische Annalen, 328, 4, 683-700 (2004) · Zbl 1058.46011 · doi:10.1007/s00208-003-0502-1
[12] Mujica, J., Linearization of bounded holomorphic mappings on Banach spaces, Transactions of the American Mathematical Society, 324, 2, 867-887 (1991) · Zbl 0747.46038 · doi:10.2307/2001745
[13] Bierstedt, K.-D., Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. I, Journal für die Reine und Angewandte Mathematik, 259, 186-210 (1973) · Zbl 0252.46039
[14] Bierstedt, K.-D., Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. II, Journal für die Reine und Angewandte Mathematik, 260, 133-146 (1973) · Zbl 0255.46025
[15] Fabian, M.; Habala, P.; Hájek, P.; Montesinos, V.; Zizler, V., Banach Space Theory, xiv+820 (2011), New York, NY, USA: Springer, New York, NY, USA · Zbl 1229.46001 · doi:10.1007/978-1-4419-7515-7
[16] Meise, R.; Vogt, D., Introduction to Functional Analysis. Introduction to Functional Analysis, Oxford Graduate Texts in Mathematics, 2, x+437 (1997), New York, NY, USA: The Clarendon Press Oxford University Press, New York, NY, USA · Zbl 0924.46002
[17] Pérez Carreras, P.; Bonet, J., Barrelled Locally Convex Spaces. Barrelled Locally Convex Spaces, North-Holland Mathematics Studies 131, Notas de Matemtica [Mathematical Notes] 113 (1987), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0614.46001
[18] Dineen, S., Complex Analysis on Infinite-Dimensional Spaces. Complex Analysis on Infinite-Dimensional Spaces, Springer Monographs in Mathematics, xvi+543 (1999), London, UK: Springer, London, UK · Zbl 1034.46504 · doi:10.1007/978-1-4471-0869-6
[19] García, D.; Maestre, M.; Rueda, P., Weighted spaces of holomorphic functions on Banach spaces, Studia Mathematica, 138, 1, 1-24 (2000) · Zbl 0960.46025
[20] Boyd, C.; Lassalle, S., Geometry and analytic boundaries of Marcinkiewicz sequence spaces, The Quarterly Journal of Mathematics, 61, 2, 183-197 (2010) · Zbl 1211.46009 · doi:10.1093/qmath/han037
[21] Globevnik, J., On interpolation by analytic maps in infinite dimensions, Mathematical Proceedings of the Cambridge Philosophical Society, 83, 2, 243-252 (1978) · Zbl 0369.46051 · doi:10.1017/S0305004100054505
[22] Globevnik, J., Boundaries for polydisc algebras in infinite dimensions, Mathematical Proceedings of the Cambridge Philosophical Society, 85, 2, 291-303 (1979) · Zbl 0395.46040 · doi:10.1017/S0305004100055705
[23] Seip, K., Beurling type density theorems in the unit disk, Inventiones Mathematicae, 113, 1, 21-39 (1993) · Zbl 0789.30025 · doi:10.1007/BF01244300
[24] Bonet, J.; Domański, P.; Lindström, M., Weakly compact composition operators on analytic vector-valued function spaces, Annales Academiæ Scientiarum Fennicæ, 26, 1, 233-248 (2001) · Zbl 1075.47506
[25] Ng, K. F., On a theorem of Dixmier, Mathematica Scandinavica, 29, 279-280 (1971) · Zbl 0243.46023
[26] Horváth, J., Topological Vector Spaces and Distributions. Vol. I, xii+449 (1966), London, UK: Addison-Wesley, London, UK · Zbl 0143.15101
[27] Köthe, G., Topological Vector Spaces. II, 237, xii+331 (1979), Berlin, Germany: Springer, Berlin, Germany · Zbl 0417.46001
[28] Bochnak, J.; Siciak, J., Polynomials and multilinear mappings in topological vector spaces, Studia Mathematica, 39, 59-76 (1971) · Zbl 0214.37702
[29] Gramsch, B., Ein Schwach-Stark-Prinzip der Dualitätstheorie lokalkonvexer Räume als Fortsetzungsmethode, Mathematische Zeitschrift, 156, 3, 217-230 (1977) · Zbl 0389.46026 · doi:10.1007/BF01214410
[30] Bonet, J.; Gómez-Collado, M. C.; Jornet, D.; Wolf, E., Operator-weighted composition operators between weighted spaces of vector-valued analytic functions, Annales Academiæ Scientiarum Fennicæ, 37, 2, 319-338 (2012) · Zbl 1263.47027 · doi:10.5186/aasfm.2012.3723
[31] Bierstedt, K. D.; Bonet, J.; Galbis, A., Weighted spaces of holomorphic functions on balanced domains, The Michigan Mathematical Journal, 40, 2, 271-297 (1993) · Zbl 0803.46023 · doi:10.1307/mmj/1029004753
[32] Bierstedt, K. D.; Summers, W. H., Biduals of weighted Banach spaces of analytic functions, Journal of the Australian Mathematical Society A, 54, 1, 70-79 (1993) · Zbl 0801.46021 · doi:10.1017/S1446788700036983
[33] Bonet, J.; Wolf, E., A note on weighted Banach spaces of holomorphic functions, Archiv der Mathematik, 81, 6, 650-654 (2003) · Zbl 1047.46018 · doi:10.1007/s00013-003-0568-8
[34] Aron, R. M.; Schottenloher, M., Compact holomorphic mappings on Banach spaces and the approximation property, Bulletin of the American Mathematical Society, 80, 1245-1249 (1974) · Zbl 0294.46027 · doi:10.1090/S0002-9904-1974-13701-8
[35] Bierstedt, K. D.; Holtmanns, S., Weak holomorphy and other weak properties, Bulletin de la Société Royale des Sciences de Liège, 72, 6, 377-381 (2004) · Zbl 1077.46030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.