Vector-valued holomorphic and harmonic functions

Wolfgang Arendt

Concrete Operators (2016)

  • Volume: 3, Issue: 1, page 68-76
  • ISSN: 2299-3282

Abstract

top
Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.

How to cite

top

Wolfgang Arendt. "Vector-valued holomorphic and harmonic functions." Concrete Operators 3.1 (2016): 68-76. <http://eudml.org/doc/277097>.

@article{WolfgangArendt2016,
abstract = {Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.},
author = {Wolfgang Arendt},
journal = {Concrete Operators},
keywords = {Holomorphic functions; Banach space; Harmonic functions; Dirichlet problem; Vitali’s Theorem; holomorphic functions; harmonic functions; Vitali's theorem},
language = {eng},
number = {1},
pages = {68-76},
title = {Vector-valued holomorphic and harmonic functions},
url = {http://eudml.org/doc/277097},
volume = {3},
year = {2016},
}

TY - JOUR
AU - Wolfgang Arendt
TI - Vector-valued holomorphic and harmonic functions
JO - Concrete Operators
PY - 2016
VL - 3
IS - 1
SP - 68
EP - 76
AB - Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.
LA - eng
KW - Holomorphic functions; Banach space; Harmonic functions; Dirichlet problem; Vitali’s Theorem; holomorphic functions; harmonic functions; Vitali's theorem
UR - http://eudml.org/doc/277097
ER -

References

top
  1. [1] H. Amann: Elliptic operators with infinite-dimensional state space. J. Evol. Equ 1 (2001), 143–188.  Zbl1018.35023
  2. [2] W. Arendt, C. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace Transforms and Cauchy Problems. Second edition. Birkhäuser Basel (2011)  Zbl1226.34002
  3. [3] W. Arendt, A.F.M. ter Elst: From forms to semigroups. Oper. Theory Adv. Appl. 221, 47–69.  Zbl1272.47003
  4. [4] W. Arendt, N. Nikolski: Vector-valued holomorphic functions revisited. Math. Z. 234 (2000), no. 4, 777–805.  Zbl0976.46030
  5. [5] W. Arendt, N. Nikolski: Addendum: Vector-valued holomorphic functions revisited. Math. Z. 252 (2006), 687–689.  
  6. [6] S. Axler; P. Bourdon, W. Ramey: Harmonic Function Theory. Springer, Berlin 1992.  Zbl0765.31001
  7. [7] J. Bonet, L. Frerick, E. Jordá: Extension of vector-valued holomorphic and harmonic functions. Studia Math. 183 (2007), 225–248.  Zbl1141.46017
  8. [8] J. Diestel, J.J. Uhl: Vector Measures. Amer. Math. Soc. Providence 1977.  
  9. [9] K.-G. Grosse-Erdmann: The Borel-Ohada theorem revisited. Habilitationsschrift Hagen 1992.  
  10. [10] K.-G. Grosse-Erdmann: A weak criterion for vector-valued holomorphy. Math. Proc. Cambridge Philos. Soc. 136 (2004), 399–411.  Zbl1055.46026
  11. [11] T. Kato: Perturbation Theory for Linear Operators. Springer, Berlin 1995.  Zbl0836.47009
  12. [12] M. Yu. Kokwin: Sets of Uniqueness for Harmonic and Analytic Functions and diverse Problems for Wave equations. Math. Notes. 97 (2015), 376–383.  
  13. [13] T. Ransford: Potential Theory in the Complex Plane. London Math. Soc., Cambridge University Press 1995.  Zbl0828.31001
  14. [14] R. Remmert: Funktionentheorie 2. Springer, Berlin 1992.  
  15. [15] A. Tonolo: Commemorazione di Giuseppe Vitali. Rendiconti Sem. Matem. Univ. Padova 3 (1932), 37–81.  
  16. [16] V. Wrobel: Analytic functions into Banach spaces and a new characterisation of isomorphic embeddings.. Proc. Amer. Math. Soc. (1982), 539–543.  Zbl0498.30004

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.