scholarly journals Homomorphisms between Algebras of Holomorphic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Verónica Dimant ◽  
Domingo García ◽  
Manuel Maestre ◽  
Pablo Sevilla-Peris
Keyword(s):  
Unit Ball ◽  
Banach Spaces ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Ball ◽  
Generalized Spectrum ◽  
Algebras Of Holomorphic Functions ◽  
Algebra Homomorphisms ◽  
Bounded Holomorphic

For two complex Banach spacesXandY, in this paper, we study the generalized spectrumℳb(X,Y)of all nonzero algebra homomorphisms fromℋb(X), the algebra of all bounded type entire functions onX, intoℋb(Y). We endowℳb(X,Y)with a structure of Riemann domain overℒ(X*,Y*)wheneverXis symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the setℳb,∞(X,BY)of all nonzero algebra homomorphisms fromℋb(X)intoℋ∞(BY)of bounded holomorphic functions on the open unit ball ofYandℳ∞(BX,BY)of all nonzero algebra homomorphisms fromℋ∞(BX)intoℋ∞(BY).

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 Multiplicative Isometries onF-Algebras of Holomorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Osamu Hatori ◽  
Yasuo Iida ◽  
Stevo Stević ◽  
Sei-Ichiro Ueki
Keyword(s):  
Unit Ball ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Ball ◽  
Smirnov Class ◽  
Unit Polydisk ◽  
Algebras Of Holomorphic Functions

We study multiplicative isometries on the followingF-algebras of holomorphic functions: Smirnov classN*(X), Privalov classNp(X), Bergman-Privalov classANαp(X),and ZygmundF-algebraNlogβN(X),whereXis the open unit ball𝔹nor the open unit polydisk𝔻ninℂn.

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**.

Some Extreme Rays of the Positive Pluriharmonic Functions

1979 ◽  
Vol 31 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Ball ◽  
Extreme Point ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Compact Open Topology ◽  
Open Unit Ball ◽  
Pluriharmonic Functions ◽  
Image Position ◽  
Extreme Rays

1.1. We will denote by B the open unit ball in Cn, and we will denote by H(B) the class of all holomorphic functions on B. LetThus N(B) is convex (and compact in the compact open topology). We think that the structure of N(B) is of interest and importance. Thus we proved in [1] that if(1.1)if(1.2)and if n≧ 2, then g is an extreme point of N(B). We will denote by E(B) the class of all extreme points of N(B). If n = 1 and if (1.2) holds, then as is well known g ∈ E(B) if and only if(1.3)

The Maximal Ideal Space of H∞ + C on the ball in Cn

1979 ◽  
Vol 31 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Gerard Mcdonald
Keyword(s):  
Maximal Ideal ◽  
Banach Algebras ◽  
Holomorphic Functions ◽  
Maximal Ideal Space ◽  
Open Unit ◽  
Ideal Space ◽  
Open Unit Ball ◽  
Closed Subalgebra ◽  
Image Position ◽  
Bounded Holomorphic

Let S denote the unit sphere in Cn, B the (open) unit ball in Cn and H∞(B) the collection of all bounded holomorphic functions on B. For f ∈ H∞(B) the limitsexist for almost every ζ in S, and the map ƒ → ƒ* defines an isometric isomorphism from H∞(B) onto a closed subalgebra of L∞(S), denoted H∞(S). (The only measure on S we will refer to in this paper is the Lebesgue measure, dσ, generated by Euclidean surface area.) Rudin has shown in [4] that the spaces H∞(B) + C(B) and H∞(S) + C(S) are Banach algebras in the sup norm. In this paper we will show that the maximal ideal space of H∞(B) + C(B), Σ (H∞(B) + C(B)), is naturally homeomorphic to Σ (H∞(B)) and that Z (H∞(S) + C(S)) is naturally homeomorphic to Σ (H∞(S))\B.

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.

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 ↗ ∞.

DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

2021 ◽  
pp. 1-29
Author(s):  
ALEXANDER BRUDNYI
Keyword(s):  
Disjoint Union ◽  
A Priori ◽  
Holomorphic Functions ◽  
Stable Rank ◽  
Open Unit ◽  
Uniform Estimates ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Similar Approximation ◽  
Bounded Holomorphic

Abstract Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

scholarly journals The group of isometries on Hardy spaces of the n-ball and the polydisc

1980 ◽  
Vol 21 (2) ◽  
pp. 199-204 ◽  
Author(s):  
Earl Berkson ◽  
Horacio Porta
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Inner Product ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Group Of Isometries

Let C be the complex plane, and U the disc |Z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn;. Bn will be the open unit ball {z ∈ Cn:|z| < 1}, and Un will be the unit polydisc in Cn. For l ≤ p < ∞, p ≠ 2, Gp(Bn) (resp., Gp(Un)) will denote the group of all isometries of Hp(Bn) (resp., Hp(Un)) onto itself, where Hp(Bn) and HP(Un) are the usual Hardy spaces.

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