Factorization of weakly continuous holomorphic mappings

Manuel González; Joaqín Gutiérrez

Studia Mathematica (1996)

  • Volume: 118, Issue: 2, page 117-133
  • ISSN: 0039-3223

Abstract

top
We prove a basic property of continuous multilinear mappings between topological vector spaces, from which we derive an easy proof of the fact that a multilinear mapping (and a polynomial) between topological vector spaces is weakly continuous on weakly bounded sets if and only if it is weakly uniformly} continuous on weakly bounded sets. This result was obtained in 1983 by Aron, Hervés and Valdivia for polynomials between Banach spaces, and it also holds if the weak topology is replaced by a coarser one. However, we show that it need not be true for a stronger topology, thus answering a question raised by Aron. As an application of the first result, we prove that a holomorphic mapping ƒ between complex Banach spaces is weakly uniformly continuous on bounded subsets if and only if it admits a factorization of the form f = g∘S, where S is a compact operator and g a holomorphic mapping.

How to cite

top

González, Manuel, and Gutiérrez, Joaqín. "Factorization of weakly continuous holomorphic mappings." Studia Mathematica 118.2 (1996): 117-133. <http://eudml.org/doc/216267>.

@article{González1996,
abstract = {We prove a basic property of continuous multilinear mappings between topological vector spaces, from which we derive an easy proof of the fact that a multilinear mapping (and a polynomial) between topological vector spaces is weakly continuous on weakly bounded sets if and only if it is weakly uniformly\} continuous on weakly bounded sets. This result was obtained in 1983 by Aron, Hervés and Valdivia for polynomials between Banach spaces, and it also holds if the weak topology is replaced by a coarser one. However, we show that it need not be true for a stronger topology, thus answering a question raised by Aron. As an application of the first result, we prove that a holomorphic mapping ƒ between complex Banach spaces is weakly uniformly continuous on bounded subsets if and only if it admits a factorization of the form f = g∘S, where S is a compact operator and g a holomorphic mapping.},
author = {González, Manuel, Gutiérrez, Joaqín},
journal = {Studia Mathematica},
keywords = {weakly continuous holomorphic mapping; factorization of holomorphic mappings; polynomial; weakly continuous multilinear mapping; continuous multilinear mappings; weakly continuous on weakly bounded sets; weakly uniformly continuous},
language = {eng},
number = {2},
pages = {117-133},
title = {Factorization of weakly continuous holomorphic mappings},
url = {http://eudml.org/doc/216267},
volume = {118},
year = {1996},
}

TY - JOUR
AU - González, Manuel
AU - Gutiérrez, Joaqín
TI - Factorization of weakly continuous holomorphic mappings
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 2
SP - 117
EP - 133
AB - We prove a basic property of continuous multilinear mappings between topological vector spaces, from which we derive an easy proof of the fact that a multilinear mapping (and a polynomial) between topological vector spaces is weakly continuous on weakly bounded sets if and only if it is weakly uniformly} continuous on weakly bounded sets. This result was obtained in 1983 by Aron, Hervés and Valdivia for polynomials between Banach spaces, and it also holds if the weak topology is replaced by a coarser one. However, we show that it need not be true for a stronger topology, thus answering a question raised by Aron. As an application of the first result, we prove that a holomorphic mapping ƒ between complex Banach spaces is weakly uniformly continuous on bounded subsets if and only if it admits a factorization of the form f = g∘S, where S is a compact operator and g a holomorphic mapping.
LA - eng
KW - weakly continuous holomorphic mapping; factorization of holomorphic mappings; polynomial; weakly continuous multilinear mapping; continuous multilinear mappings; weakly continuous on weakly bounded sets; weakly uniformly continuous
UR - http://eudml.org/doc/216267
ER -

References

top
  1. [1] R. M. Aron, Y. S. Choi and J. G. Llavona, Estimates by polynomials, Bull. Austral. Math. Soc. 52 (1995), 475-486. 
  2. [2] R. M. Aron, J. Gómez and J. G. Llavona, Homomorphisms between algebras of differentiable functions in infinite dimensions, Michigan Math. J. 35 (1988), 163-178. Zbl0709.46010
  3. [3] R. M. Aron, C. Hervés and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), 189-204. Zbl0517.46019
  4. [4] R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195-216. Zbl0413.41022
  5. [5] R. M. Aron and M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Funct. Anal. 21 (1976), 7-30. Zbl0328.46046
  6. [6] A. Braunsz and H. Junek, Bilinear mappings and operator ideals, Rend. Circ. Mat. Palermo Suppl. (2) 10 (1985), 25-35. Zbl0621.47043
  7. [7] S. Dineen, Complex Analysis in Locally Convex Spaces, Math. Stud. 57, North-Holland, Amsterdam, 1981. Zbl0484.46044
  8. [8] S. Dineen, Entire functions on c 0 , J. Funct. Anal. 52 (1983), 205-218. Zbl0538.46032
  9. [9] S. Dineen, Infinite Dimensional Complex Analysis, book in preparation. Zbl0937.46040
  10. [10] M. González and J. M. Gutiérrez, The compact weak topology on a Banach space, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 367-379. Zbl0786.46018
  11. [11] M. González and J. M. Gutiérrez, Weakly continuous mappings on Banach spaces with the Dunford-Pettis property, J. Math. Anal. Appl. 173 (1993), 470-482. Zbl0785.46021
  12. [12] S. Heinrich, Closed operator ideals and interpolation, J. Funct. Anal. 35 (1980), 397-411. Zbl0439.47029
  13. [13] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981. Zbl0466.46001
  14. [14] J. L. Kelley, General Topology, Grad. Texts in Math. 27, Springer, Berlin, 1955. 
  15. [15] J. G. Llavona, Approximation of Continuously Differentiable Functions, Math. Stud. 130, North-Holland, Amsterdam, 1986. Zbl0642.41001
  16. [16] L. A. Moraes, Extension of holomorphic mappings from E to E'', Proc. Amer. Math. Soc. 118 (1993), 455-461. Zbl0796.46033
  17. [17] J. Mujica, Complex Analysis in Banach Spaces, Math. Stud. 120, North-Holland, Amsterdam, 1986. Zbl0586.46040
  18. [18] H. Porta, Compactly determined locally convex topologies, Math. Ann. 196 (1972), 91-100. Zbl0219.46004
  19. [19] R. A. Ryan, Weakly compact holomorphic mappings on Banach spaces, Pacific J. Math. 131 (1988), 179-190. Zbl0605.46038
  20. [20] J. H. Webb, Sequential convergence in locally convex spaces, Proc. Cambridge Philos. Soc. 64 (1968), 341-364. Zbl0157.20202

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.