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.
References
[1] H. Amann: Elliptic operators with infinite-dimensional state space. J. Evol. Equ 1 (2001), 143–188. 10.1007/PL00001367Search in Google Scholar
[2] W. Arendt, C. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace Transforms and Cauchy Problems. Second edition. Birkhäuser Basel (2011) 10.1007/978-3-0348-0087-7Search in Google Scholar
[3] W. Arendt, A.F.M. ter Elst: From forms to semigroups. Oper. Theory Adv. Appl. 221, 47–69. 10.1007/978-3-0348-0297-0_4Search in Google Scholar
[4] W. Arendt, N. Nikolski: Vector-valued holomorphic functions revisited. Math. Z. 234 (2000), no. 4, 777–805. Search in Google Scholar
[5] W. Arendt, N. Nikolski: Addendum: Vector-valued holomorphic functions revisited. Math. Z. 252 (2006), 687–689. Search in Google Scholar
[6] S. Axler; P. Bourdon, W. Ramey: Harmonic Function Theory. Springer, Berlin 1992. 10.1007/b97238Search in Google Scholar
[7] J. Bonet, L. Frerick, E. Jordá: Extension of vector-valued holomorphic and harmonic functions. Studia Math. 183 (2007), 225–248. Search in Google Scholar
[8] J. Diestel, J.J. Uhl: Vector Measures. Amer. Math. Soc. Providence 1977. 10.1090/surv/015Search in Google Scholar
[9] K.-G. Grosse-Erdmann: The Borel-Ohada theorem revisited. Habilitationsschrift Hagen 1992. Search in Google Scholar
[10] K.-G. Grosse-Erdmann: A weak criterion for vector-valued holomorphy. Math. Proc. Cambridge Philos. Soc. 136 (2004), 399–411. 10.1017/S0305004103007254Search in Google Scholar
[11] T. Kato: Perturbation Theory for Linear Operators. Springer, Berlin 1995. 10.1007/978-3-642-66282-9Search in Google Scholar
[12] M. Yu. Kokwin: Sets of Uniqueness for Harmonic and Analytic Functions and diverse Problems for Wave equations. Math. Notes. 97 (2015), 376–383. 10.1134/S0001434615030086Search in Google Scholar
[13] T. Ransford: Potential Theory in the Complex Plane. London Math. Soc., Cambridge University Press 1995. 10.1017/CBO9780511623776Search in Google Scholar
[14] R. Remmert: Funktionentheorie 2. Springer, Berlin 1992. 10.1007/978-3-642-97397-0Search in Google Scholar
[15] A. Tonolo: Commemorazione di Giuseppe Vitali. Rendiconti Sem. Matem. Univ. Padova 3 (1932), 37–81. Search in Google Scholar
[16] V. Wrobel: Analytic functions into Banach spaces and a new characterisation of isomorphic embeddings.. Proc. Amer. Math. Soc. (1982), 539–543. 10.1090/S0002-9939-1982-0660600-XSearch in Google Scholar
© 2016 Wolfgang Arendt
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