Now showing 1 - 6 of 6
  • Publication
    Dynamics of biholomorphic self-maps on bounded symmetric domains
    (Royal Danish Library, 2015)
    Let g be a fixed-point free biholomorphic self-map of a bounded symmetric domain B. It is known that the sequence of iterates (gn) may not always converge locally uniformly on B even, for example, if B is an infinite dimensional Hilbert ball. However, g=ga∘T, for a linear isometry T, a=g(0) and a transvection ga, and we show that it is possible to determine the dynamics of ga. We prove that the sequence of iterates (gna) converges locally uniformly on B if, and only if, a is regular, in which case, the limit is a holomorphic map of B onto a boundary component (surprisingly though, generally not the boundary component of a∥a∥). We prove (gna) converges to a constant for all non-zero a if, and only if, B is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.
      289Scopus© Citations 3
  • Publication
    The density property for JB*-triples
    (Polskiej Akademii Nauk, Instytut Matematczny (Polish Academy of Sciences, Institute of Mathematics), 1999) ; ;
    We obtain conditions on a JB∗-algebra X so that the canonical embedding of X into its associated quasi-invertible manifold has dense range. We prove that if a JB∗ -triple has this density condition then the quasi-invertible manifold is homogeneous for biholomorphic mappings. Explicit formulae for the biholomorphic mappings are also given.
  • Publication
    Angular derivatives on bounded symmetric domains
    (Springer, 2003-03) ;
    In this paper we generalise the classical Julia-Wolff-Carathéodory theorem to holomorphic functions defined on bounded symmetric domains.
      266Scopus© Citations 3
  • Publication
    A Wolff Theorem for finite rank bounded symmetric domains
    (Elsevier, 2017-12-01)
    We present a Wolff Theorem for all infinite dimensional bounded symmetric domains of finite rank. Namely, if Bis the open unit ball of any finite rank JB∗-triple and f:B→Bis a compact holomorphic map with no fixed point in B, we prove convex f-invariant subdomains of B(of all sizes and at all points) exist in the form of simple operator balls cλ+Tλ(B), for cλ∈Band Tλan invertible linear map. These are exact infinite dimensional analogues of the invariant discs in Δ, the invariant ellipsoids in the Hilbert ball and invariant domains in finite dimensional triples. Results are new for rank >2, even for classical spaces such as C∗-algebras and JB∗-algebras in finite dimensional analogues of the invariant discs in Δ, the invariant ellipsoids in the Hilbert ball and invariant domains in finite dimensional triples. Results are new for rank > 2, even for classical spaces such as C*-algebras and JB*-algebras.
      438Scopus© Citations 1
  • Publication
    Analytic aspects of evolution algebras
    (Duke University Press, 2019-11-28) ;
    We prove that every evolution algebra A is a normed algebra, for an l1-norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra A is a Banach algebra if and only if A=A1⊕A0, where A1 is finite-dimensional and A0 is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator LB of A with respect to a natural basis B, and we show that LB need not be continuous. Moreover, if A is finite-dimensional and B={e1,…,en}, then LB is given by Le, where e=∑iei and La is the multiplication operator La(b)=ab, for b∈A. We establish necessary and sufficient conditions for convergence of (Lna(b))n, for all b∈A, in terms of the multiplicative spectrum σm(a) of a. Namely, (Lna(b))n converges, for all b∈A, if and only if σm(a)⊆Δ∪{1} and ν(1,a)≤1, where ν(1,a) denotes the index of 1 in the spectrum of La.
      200Scopus© Citations 6
  • Publication
    A Schwarz lemma and composition operators
    (Springer, 2004-04) ;
    We give an alternative description of the Carathéodory pseudodistance on a domain D in an arbitrary complex Banach space. This gives a Schwarz lemma for holomorphic maps of the domain.We specialise to the case of a bounded symmetric domain and obtain some applications. In particular, we give the connected components of the space of composition operators with symbol in a bounded symmetric domain. This generalises results for the space of composition operators on H∞(Δ) in [12] and for H∞(B) , B the unit ball of a Hilbert space or commutative C*-algebra in [2].
      377Scopus© Citations 3