Trace-order and a distortion theorem for linearly invariant families on the unit ball of a finite dimensional JB∗ -triple

Abstract

a b s t r a c tWe give a distortion theorem for linearly invariant families on the unit ball B of a finite dimensional JB * -triple X by using the trace-order. The exponents in the distortion bounds depend on the Bergman metric at 0. Further, we introduce a new definition for the traceorder of a linearly invariant family on B, based on a Jacobian argument. We also construct an example of a linearly invariant family on B which has minimum trace-order and is not a subset of the normalized convex mappings of B for dim X ≥ 2. Finally, we prove a regularity theorem for linearly invariant families on B. All four types of classical Cartan domains are the open unit balls of JB * -triples, and the same holds for any finite product of these domains. Thus the unit balls of JB * -triples are natural generalizations of the unit disc in C and we have a setting in which a large number of bounded symmetric homogeneous domains may be studied simultaneously.

Distortion of locally biholomorphic Bloch mappings on bounded symmetric domains

“…Our results also generalize simultaneously other results on Bonk's distortion theorem for locally univalent Bloch functions in one complex variable in [3,21], and those for locally biholomorphic Bloch mappings in several complex variables in [25]. We refer to [7,11,12,13] for other distortion theorems for normalized locally biholomorphic mappings on unit balls of finite dimensional JB * -triples.…”

“…The novelty of our approach is the use of Jordan theory. Indeed, the exponent of the distortion bound depends on the 'diameter' 2c(B X ) of the ball B X with respect to the Bergman metric at 0, defined by Hamada, Honda and Kohr in [12]. The constant c(B X ) depends on the Jordan structure of the underlying JB*-triple X.…”

“…Given a tripotent v ∈ X, the possible eigenvalues of the box operator v v are 0, 1/2 or 1, which induces the following eigenspace decomposition of X: We recall that the constant c(B X ), introduced in [12], is defined by…”

Distortion theorems, Lipschitz continuity and their applications for Bloch type mappings on bounded symmetric domains in C^n

“…In the rest of this paper, let B X be the unit ball of an n-dimensional JB * -triple X = (C n , • X ). Also, let h 0 be the Bergman metric on B X at 0, and let (see [11])…”

“…(ii) When B X is the Euclidean unit ball B n in C n , then c(B n ) = (n + 1)/2 ( [11]) and det B(z, z) = (1 − z 2 ) n+1 for all z ∈ B n ( [10], see also [13,Lemma 2.2]). Therefore, if B X is the Euclidean unit ball B n in C n , then we have f 0,α = f P(X,α) .…”