Holomorphic invariance on bounded symmetric domains

Abstract

We establish domains of invariance for holomorphic self-maps of a bounded symmetric domain of arbitrary dimension, generalising results of Wol¨and Julia in the open unit disc h of C and well known results on the Hilbert ball. The holomorphically invariant domains are analogues of the Poincare  balls and their limiting horocycles in h. The results appear to be new even in the ®nite dimensional (non-Hilbert space) case. completely describe our invariant domains as the limit in a natural sense of a sequence of Kobayashi balls. Our principal result presented in Section 3 is a generalization of Wol¨'s classical theorem, on invariant domains of a ®xed point free holomorphic self map of h, to compact holomorphic ®xed point free maps on a bounded symmetric domain, giving in particular an exact analogue of Wol¨'s theorem in the ®nite dimensional case. This result, surprisingly, appears to be new even in ®nite dimensions (apart from the Hilbert ball) andBrought to you by | University of Iowa Libraries Authenticated Download Date | 5/31/15 6:48 AM 1.1. De®nition. A JB à -triple is a complex Banach space Z with a real trilinear mapping fÁ Y Á Y Ág X Z  Z  Z 3 Z satisfying:(i) fxY yY zg is complex linear and symmetric in the outer variables x and z, and is complex anti-linear in y.(ii) The map z U 3 fxY xY zg, denoted x a x, is Hermitian, sx a x 0 and kx a xk kxk 2 for all x e Z, where s denotes the spectrum.(iii) The product satis®es the following``triple identity'' faY bY fxY yY zgg ffaY bY xgY yY zg À fxY fbY aY ygY zg fxY yY faY bY zggXwhere y à denotes the usual adjoint of y.(ii) CX , the continuous C-valued functions on a compact Hausdor¨space X, is a JB à -triple for the product fxY yY zg xyz. By a deep result of Kaup [18] every bounded symmetric domain is biholomorphically equivalent to the open unit ball of a JB à -triple and vice versa. Mellon, Holomorphic invariance on bounded symmetric domains200 Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/31/15 6:48 AM

Angular derivatives on bounded symmetric domains

“…The following Julia type lemma for a holomorphic function between bounded symmetric domains B and B ′ , contained in JB * -triples Z and Z ′ , is of a type first proved in [15].…”

“…The following Schwarz-Pick type result holds [15,Corollary 3.3] as a consequence of the Schwarz lemma,…”

“…In the notation of [15], and using [15, Corollary 3.2], we have that w belongs to the Kobayashi ball D α k e,r k for all k large where…”

“…We know from Proposition 3.1 that x n and e are elements of ∂E γn (e). It follows from [15,Proposition 3.15] that E γn (e) has the alternative description…”

“…A principal reason for this sparsity has been the lack of suitable analogues of either the classical Julia's lemma or Wolff's theorem for bounded symmetric domains. These were recently provided in [15] and play a crucial role in achieving angular limit and angular derivative results, in particular motivating the definition of angular approach region for bounded symmetric domains. The existence of angular limits over these regions is then reduced by an extension of the classical Lindelöf-Cirka principle (cf.…”