Extending biholomorphic automorphisms of the open unit ball of a complex Banach space to the bidual

2019 ◽  
Vol 119A (1) ◽  
pp. 57-63
Author(s):  
Miguel Cabrera García ◽  
Ángel Rodríguez Palacios
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Complex Banach Space ◽  
Open Unit ◽  
Open Unit Ball

scholarly journals The strict topology on spaces of bounded holomorphic functions

1994 ◽  
Vol 49 (2) ◽  
pp. 249-256 ◽  
Author(s):  
Juan Ferrera ◽  
Angeles Prieto
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Strict Topology ◽  
Open Unit Ball ◽  
Approximation By Polynomials ◽  
Bounded Holomorphic

We introduce in this paper the space of bounded holomorphic functions on the open unit ball of a Banach space endowed with the strict topology. Some good properties of this topology are obtained. As applications, we prove some results on approximation by polynomials and a description of the continuous homomorphisms.

scholarly journals Fixed points and periodic points of semiflows of holomorphic maps

2003 ◽  
Vol 2003 (4) ◽  
pp. 217-260
Author(s):  
Edoardo Vesentini
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Complex Banach Space ◽  
Periodic Points ◽  
General Question ◽  
Open Unit ◽  
Holomorphic Maps ◽  
Open Unit Ball ◽  
Spin Factor ◽  
Cartan Factor

Letϕbe a semiflow of holomorphic maps of a bounded domainDin a complex Banach space. The general question arises under which conditions the existence of a periodic orbit ofϕimplies thatϕitself is periodic. An answer is provided, in the first part of this paper, in the case in whichDis the open unit ball of aJ∗-algebra andϕacts isometrically. More precise results are provided when theJ∗-algebra is a Cartan factor of type one or a spin factor. The second part of this paper deals essentially with the discrete semiflowϕgenerated by the iterates of a holomorphic map. It investigates how the existence of fixed points determines the asymptotic behaviour of the semiflow. Some of these results are extended to continuous semiflows.

On interpolation by analytic maps in infinite dimensions

1978 ◽  
Vol 83 (2) ◽  
pp. 243-252 ◽  
Author(s):  
J. Globevnik
Keyword(s):  
Unit Ball ◽  
Complex Banach Space ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Infinite Dimensions ◽  
Open Unit Ball ◽  
Infinite Dimensional ◽  
Analytic Maps ◽  
Peak Interpolation ◽  
Vector Valued Functions

AbstractLet A be the complex Banach algebra of all bounded continuous complex-valued functions on the closed unit ball of a complex Banach space X, analytic on the open unit ball, with sup norm. For a class of spaces X which contains all infinite dimensional complex reflexive spaces we prove the existence of non-compact peak interpolation sets for A. We prove some related interpolation theorems for vector-valued functions and present some applications to the ranges of analytic maps between Banach spaces. We also show that in general peak interpolation sets for A do not exist.

Spectra of composition operators on algebras of analytic functions on Banach spaces

2009 ◽  
Vol 139 (1) ◽  
pp. 107-121 ◽  
Author(s):  
P. Galindo ◽  
T. W. Gamelin ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Fixed Point ◽  
Unit Ball ◽  
Banach Spaces ◽  
Composition Operator ◽  
Composition Operators ◽  
Open Unit ◽  
Open Unit Ball ◽  
Analytic Symbol

Let E be a Banach space, with unit ball BE. We study the spectrum and the essential spectrum of a composition operator on H∞(BE) determined by an analytic symbol with a fixed point in BE. We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space.

scholarly journals The Denjoy–Wolff Theorem in the Open Unit Ball of a Strictly Convex Banach Space

1999 ◽  
Vol 143 (1) ◽  
pp. 111-123 ◽  
Author(s):  
Jaroslaw Kapeluszny ◽  
Tadeusz Kuczumow ◽  
Simeon Reich
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Convex Banach Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

scholarly journals On approximating curves associated with nonexpansive mappings

2011 ◽  
Vol 27 (1) ◽  
pp. 142-147
Author(s):  
FRANCESCA VETRO ◽  
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unit Ball ◽  
Nonexpansive Mappings ◽  
Complex Hilbert Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Holomorphic Map ◽  
Nonexpansive Map ◽  
The Hyperbolic Metric

Let X be a Banach space with metric d. Let T, N : X → X be a strict d-contraction and a d-nonexpansive map, respectively. In this paper we investigate the properties of the approximating curve associated with T and N. Moreover, following [3], we consider the approximating curve associated with a holomorphic map f : B → α B and a ρ-nonexpansive map M : B → B, where B is the open unit ball of a complex Hilbert space H, ρ is the hyperbolic metric defined on B and 0 ≤ α < 1. We give conditions on f and M for this curve to be injective, and we show that this curve is continuous.

Holomorphically embedded discs with rapidly growing area

1999 ◽  
Vol 129 (2) ◽  
pp. 343-349
Author(s):  
Josip Globevnik
Keyword(s):  
Unit Ball ◽  
Open Unit ◽  
Open Unit Ball

It is shown that if V is a closed submanifold of the open unit ball of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ 1. It is also shown that if V is a closed submanifold of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ ∞.

Weak-Star Continuous Analytic Functions

1995 ◽  
Vol 47 (4) ◽  
pp. 673-683 ◽  
Author(s):  
R. M. Aron ◽  
B. J. Cole ◽  
T. W. Gamelin
Keyword(s):  
Unit Ball ◽  
Finite Type ◽  
Analytic Functions ◽  
Approximation Property ◽  
Entire Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Open Unit Ball ◽  
Weakly Continuous ◽  
Weak Star

AbstractLet 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

Extreme Points in H1(R)

1967 ◽  
Vol 19 ◽  
pp. 312-320 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Convex Set ◽  
Complete Solution ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Open Riemann Surface ◽  
Harmonic Majorant ◽  
Closed Convex Set

Let R be an open Riemann surface. ƒ belongs to H1(R) if ƒ is holomorphic on R and if the subharmonic function |ƒ| has a harmonie majorant on R. Let p be in R and define ||ƒ|| to be the value at p of the least harmonic majorant of |ƒ|. ||ƒ|| is a norm on the linear space H1(R), and with this norm H1(R) is a Banach space (7). The unit ball of H1(R) is the closed convex set of all ƒ in H1(R) with ||ƒ|| ⩽ 1. Problem: What are the extreme points of the unit ball of H1(R)? de Leeuw and Rudin have given a complete solution to this problem where R is the open unit disk (1).

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