Weighted spaces of holomorphic functions on Banach spaces

D. García; M. Maestre; P. Rueda

Studia Mathematica (2000)

  • Volume: 138, Issue: 1, page 1-24
  • ISSN: 0039-3223

Abstract

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We deal with weighted spaces H V 0 ( U ) and HV(U) of holomorphic functions defined on a balanced open subset U of a Banach space X. We give conditions on the weights to ensure that the weighted spaces of m-homogeneous polynomials constitute a Schauder decomposition for them. As an application, we study their reflexivity. We also study the existence of a predual. Several examples are provided.

How to cite

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García, D., Maestre, M., and Rueda, P.. "Weighted spaces of holomorphic functions on Banach spaces." Studia Mathematica 138.1 (2000): 1-24. <http://eudml.org/doc/216687>.

@article{García2000,
abstract = {We deal with weighted spaces $HV_0(U)$ and HV(U) of holomorphic functions defined on a balanced open subset U of a Banach space X. We give conditions on the weights to ensure that the weighted spaces of m-homogeneous polynomials constitute a Schauder decomposition for them. As an application, we study their reflexivity. We also study the existence of a predual. Several examples are provided.},
author = {García, D., Maestre, M., Rueda, P.},
journal = {Studia Mathematica},
keywords = {weighted spaces of holomorphic functions on Banach spaces; seminorm; reflexivity; Schauder decomposition; predual},
language = {eng},
number = {1},
pages = {1-24},
title = {Weighted spaces of holomorphic functions on Banach spaces},
url = {http://eudml.org/doc/216687},
volume = {138},
year = {2000},
}

TY - JOUR
AU - García, D.
AU - Maestre, M.
AU - Rueda, P.
TI - Weighted spaces of holomorphic functions on Banach spaces
JO - Studia Mathematica
PY - 2000
VL - 138
IS - 1
SP - 1
EP - 24
AB - We deal with weighted spaces $HV_0(U)$ and HV(U) of holomorphic functions defined on a balanced open subset U of a Banach space X. We give conditions on the weights to ensure that the weighted spaces of m-homogeneous polynomials constitute a Schauder decomposition for them. As an application, we study their reflexivity. We also study the existence of a predual. Several examples are provided.
LA - eng
KW - weighted spaces of holomorphic functions on Banach spaces; seminorm; reflexivity; Schauder decomposition; predual
UR - http://eudml.org/doc/216687
ER -

References

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  1. [1] R. Alencar, On reflexivity and basis for P ( m E ) , Proc. Roy. Irish Acad. 85A (1985), 131-138. Zbl0594.46043
  2. [2] R. Alencar, R. Aron and S. Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), 407-411. Zbl0536.46015
  3. [3] J. M. Ansemil and S. Ponte, An example of quasinormable Fréchet function space which is not a Schwartz space, in: Functional Analysis, Holomorphy and Approximation theory, S. Machado (ed.), Lecture Notes in Math. 843, Springer, 1981, 1-8. 
  4. [4] R. Aron, B. Cole and T. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51-93. Zbl0717.46031
  5. [5] R. Aron, P. Galindo, D. García and M. Maestre, Regularity and algebras of analytic functions in infinite dimensions, Trans. Amer. Math. Soc. 348 (1996), 543-559. Zbl0844.46024
  6. [6] K. D. Bierstedt and J. Bonet, Biduality in Fréchet and (LB)-spaces, in: Progress in Functional Analysis, K. D. Bierstedt et al. (eds.), North-Holland Math. Stud. 170, North-Holland, Amsterdam, 1992, 113-133. Zbl0804.46007
  7. [7] K. D. Bierstedt, J. Bonet and A. Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40 (1993), 271-297. Zbl0803.46023
  8. [8] K. D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), 137-168. Zbl0934.46027
  9. [9] K. D. Bierstedt, R. G. Meise and W. H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107-160. Zbl0599.46026
  10. [10] K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 54 (1993), 70-79. Zbl0801.46021
  11. [11] O. Blasco and A. Galbis, On Taylor coefficients of entire functions integrable against exponential weights, preprint, Univ. of Valencia, 1998. Zbl0988.46018
  12. [12] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, Berlin, 1984. 
  13. [13] S. Dineen, Complex Analysis in Locally Convex Spaces, North-Holland Math. Stud. 57, North-Holland, Amsterdam, 1981. Zbl0484.46044
  14. [14] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer Monogr. Math., London, 1999. Zbl1034.46504
  15. [15] S. Dineen, Quasinormable spaces of holomorphic functions, Note Mat. 13 (1993), 155-195. Zbl0838.46036
  16. [16] P. Galindo, D. García and M. Maestre, Holomorphic mappings of bounded type on (DF)-spaces, in: Progress in Functional Analysis, K. D. Bierstedt et al. (eds.), North-Holland Math. Stud. 170, North-Holland, Amsterdam, 1992, 135-148. Zbl0795.46035
  17. [17] P. Galindo, D. García and M. Maestre, Holomorphic mappings of bounded type, J. Math. Anal. Appl. 166 (1992), 236-246. Zbl0806.46047
  18. [18] P. Galindo, M. Maestre and P. Rueda, Biduality in spaces of holomorphic functions, Math. Scand., to appear. Zbl0968.46026
  19. [19] A. Grothendieck, Sur les espaces (F) et(DF), Summa Brasil. Math. 3 (1954), no. 6, 57-122. 
  20. [20] N. J. Kalton, Schauder decompositions in locally convex spaces, Proc. Cambridge Philos. Soc. 68 (1970), 377-392. Zbl0196.13505
  21. [21] G. Köthe, Topological Vector Spaces I, Grundlehren Math. Wiss. 159, Springer, New York, 1969. Zbl0179.17001
  22. [22] W. Lusky, On the Fourier series of unbounded harmonic functions, preprint, Univ. Paderborn, 1998. 
  23. [23] M. Matos, On holomorphy in Banach spaces and absolute convergence of Fourier series, Portugal. Math. 45 (1988), 429-450. Zbl0663.46041
  24. [24] J. Mujica, Linearization of bounded holomorphic mappings on Banach spaces, Trans. Amer. Math. Soc. 324 (1991), 867-887. Zbl0747.46038
  25. [25] J. Mujica, Linearization of holomorphic mappings of bounded type, in: Progress in Functional Analysis, K. D. Bierstedt et al. (eds.), North-Holland Math. Stud. 170, North-Holland, Amsterdam, 1992, 149-162. Zbl0795.46036
  26. [26] G. Pólya and G. Szegő, Problems and Theorems in Analysis I, 4th ed., Springer, New York, 1970. Zbl0201.38102
  27. [27] A. Prieto, The bidual of spaces of holomorphic functions in infinitely many variables, Proc. Roy. Irish Acad. 92A (1992), 1-8. Zbl0741.46016
  28. [28] L. A. Rubel and A. L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970), 276-280. Zbl0197.39001
  29. [29] P. Rueda, On the Banach-Dieudonné theorem for spaces of holomorphic functions, Quaestiones Math. 19 (1996), 341-352. Zbl0858.46036
  30. [30] R. Ryan, Application of topological tensor products to infinite dimensional holomorphy, Ph.D. thesis, Trinity College, Dublin, 1980. 
  31. [31] R. Ryan, Holomorphic mappings on ℓ 1 , Trans. Amer. Math. Soc. 302 (1987), 797-811. Zbl0637.46045
  32. [32] K. R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, Belmont, 1981. Zbl0454.26001
  33. [33] W. H. Summer, Dual spaces of weighted spaces, Trans. Amer. Math. Soc. 151 (1970), 323-333. Zbl0203.12401
  34. [34] D. L. Williams, Some Banach spaces of entire functions, Ph.D. thesis, Univ. of Michigan, 1967. 

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