Integral formulas for the ? $$\bar \partial $$ -equation and zeros of bounded holomorphic functions in the unit ball

1980 ◽  
Vol 249 (2) ◽  
pp. 163-176 ◽  
Author(s):  
Bo Berndtsson
Keyword(s):  
Unit Ball ◽  
Holomorphic Functions ◽  
Integral Formulas ◽  
Bounded Holomorphic

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 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 Linear combinations of composition operators on $H^{\infty }$ spaces over the unit ball and polydisk

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xingxing Yao
Keyword(s):  
Unit Ball ◽  
Composition Operators ◽  
Holomorphic Functions ◽  
Linear Combinations ◽  
Unit Polydisk ◽  
Bounded Holomorphic

AbstractIn this paper, we characterize completely the compactness of linear combinations of composition operators acting on the space $H^{\infty }(\mathbb{B}_{N})$ H ∞ ( B N ) of bounded holomorphic functions over the unit ball $\mathbb{B}_{N}$ B N from two different aspects. The same problems are also investigated on the space $H^{\infty }(\mathbb{D}^{N})$ H ∞ ( D N ) over the unit polydisk $\mathbb{D}^{N}$ D N .

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 A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

1962 ◽  
Vol 14 ◽  
pp. 334-348 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Blaschke Product ◽  
Common Knowledge ◽  
Holomorphic Functions ◽  
Blaschke Products ◽  
Radial Limit ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Tangential Limits

In this paper, we shall be concerned with bounded, holomorphic functions of the formwhere(1)(2)and(3)B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

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