- Mellon, Pauline

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# Mellon, Pauline

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Mellon, Pauline

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- PublicationDynamics of biholomorphic self-maps on bounded symmetric domainsLet 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=gaâˆ˜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 aâˆ¥aâˆ¥). 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.
289ScopusÂ© Citations 3 - PublicationThe density property for JB*-triples(Polskiej Akademii Nauk, Instytut Matematczny (Polish Academy of Sciences, Institute of Mathematics), 1999)
; ; We obtain conditions on a JBâˆ—-algebra X so that the canonical embedding of X into its associated quasi-invertible manifold has dense range. We prove that if a JBâˆ— -triple has this density condition then the quasi-invertible manifold is homogeneous for biholomorphic mappings. Explicit formulae for the biholomorphic mappings are also given.103 - PublicationAngular derivatives on bounded symmetric domainsIn this paper we generalise the classical Julia-Wolff-CarathÃ©odory theorem to holomorphic functions defined on bounded symmetric domains.
266ScopusÂ© Citations 3 - PublicationA Wolff Theorem for finite rank bounded symmetric domainsWe 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 JBâˆ—-triple and f:Bâ†’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 cÎ»+TÎ»(B), for cÎ»âˆˆBand TÎ»an invertible linear map. These are exact infinite dimensional analogues of the invariant discs in Î”, 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 Câˆ—-algebras and JBâˆ—-algebras in finite dimensional analogues of the invariant discs in Î”, 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.
438ScopusÂ© Citations 1 - PublicationAnalytic aspects of evolution algebrasWe 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=A1âŠ•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,â€¦,en}, then LB is given by Le, where e=âˆ‘iei and La is the multiplication operator La(b)=ab, for bâˆˆA. We establish necessary and sufficient conditions for convergence of (Lna(b))n, for all bâˆˆA, in terms of the multiplicative spectrum Ïƒm(a) of a. Namely, (Lna(b))n converges, for all bâˆˆA, if and only if Ïƒm(a)âŠ†Î”âˆª{1} and Î½(1,a)â‰¤1, where Î½(1,a) denotes the index of 1 in the spectrum of La.
200ScopusÂ© Citations 6 - PublicationA Schwarz lemma and composition operatorsWe give an alternative description of the CarathÃ©odory 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 Hâˆž(Î”) in [12] and for Hâˆž(B) , B the unit ball of a Hilbert space or commutative C*-algebra in [2].
377ScopusÂ© Citations 3