Lecture Details
Name: Mary Lilian Lourenço (University of Sao Paulo)Speaking: Fri 10th May 11:30 - 11:55Title: Biholomorphic Mappings on Banach SpacesAbstract:
The H. Cartan theorem stating that a holomorphic self-map of a bounded domain in $\mathbb{C}^n$ with a fixed point at which the derivative is the identity has to be the identity was widened by Cima et al. [2] to separable Hilbert spaces and then to separable dual Banach spaces in [1]. We will present an infinite-dimensional version of Cartan theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. The main assumption is that the derivative operator is power bounded that we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.
Joint work with H. Carrión and P. Galindo.
References
[1] H. Carrión, P. Galindo and M. L. Lourenço, Biholomorphic functions in dual Banach spaces, Complex Anal. Oper. Theory, (7) (2013),107-114.
[2] J. A. Cima, I. Graham, K. T. Kim and S. G. Krantz, The Caratheodory-Cartan-Kaup-Wu theorem on an infinite dimensional Hilbert space, Nagoya Math. J., Vol. 185 ( 2007), 17-30.