scholarly journals Infinite products of holomorphic mappings

2005 ◽  
Vol 2005 (4) ◽  
pp. 327-341 ◽  
Author(s):  
Monika Budzyńska ◽  
Simeon Reich
Keyword(s):  
Banach Space ◽  
Convex Domain ◽  
Complex Banach Space ◽  
Holomorphic Mappings ◽  
Convergence Properties ◽  
Infinite Products ◽  
Norm Closure

LetXbe a complex Banach space,𝒩a norming set forX, andD⊂Xa bounded, closed, and convex domain such that its norm closureD¯is compact inσ(X,𝒩). Let∅≠C⊂Dlie strictly insideD. We study convergence properties of infinite products of those self-mappings ofCwhich can be extended to holomorphic self-mappings ofD. Endowing the space of sequences of such mappings with an appropriate metric, we show that the subset consisting of all the sequences with divergent infinite products isσ-porous.

scholarly journals Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Jianfei Wang
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Arbitrary Order ◽  
Complex Banach Space ◽  
Holomorphic Mappings ◽  
Partial Derivatives ◽  
Carathéodory Metric ◽  
Homogeneous Ball ◽  
Derivatives Of

LetBXbe the unit ball in a complex Banach spaceX. AssumeBXis homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ballBntoBXassociated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

scholarly journals ℂ-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
Lars Filipsson
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Domain ◽  
Holomorphic Mappings ◽  
Convex Domains ◽  
Linear Convexity ◽  
Infinite Dimensional ◽  
Banach Theorem ◽  
Complex Hyperplane

We investigate the concepts of linear convexity andℂ-convexity in complex Banach spaces. The main result is that anyℂ-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given aℂ-convex domainΩin the Banach spaceXand a pointp∉Ω, there is a complex hyperplane throughpthat does not intersectΩ. We also prove that linearly convex domains are holomorphically convex, and that Kergin interpolation can be performed on holomorphic mappings defined inℂ-convex domains.

scholarly journals The Composition Operation on Spaces of Holomorphic Mappings

2020 ◽  
Vol 71 (2) ◽  
pp. 557-572
Author(s):  
María D Acosta ◽  
Pablo Galindo ◽  
Luiza A Moraes
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Holomorphic Mappings ◽  
Fréchet Space ◽  
Frechet Space ◽  
Linear Subspaces ◽  
Left And Right ◽  
Relationship Of ◽  
The Relationship ◽  
Bounded Sets

Abstract We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

scholarly journals Some properties of shift operators on algebras generated by $*$-polynomials

2018 ◽  
Vol 10 (1) ◽  
pp. 206-212
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Holomorphic Functions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Countable System ◽  
Shift Operators ◽  
Underlying Space ◽  
Algebra Of Functions ◽  
Algebras Of Holomorphic Functions ◽  
Proper Subalgebra

A $*$-polynomial is a function on a complex Banach space $X,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for non-negative integers $p$ and $q,$ a $(p,q)$-polynomial is a function on $X,$ which is the restriction to the diagonal of some mapping, defined on the Cartesian power $X^{p+q},$ which is linear with respect to every of its first $p$ arguments and antilinear with respect to every of its other $q$ arguments. The set of all continuous $*$-polynomials on $X$ form an algebra, which contains the algebra of all continuous polynomials on $X$ as a proper subalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions than algebras of holomorphic functions. On the other hand, due to the similarity of structures of $*$-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces. We investigate the Frechet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous $*$-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.

scholarly journals Characteristic Functions and Borel Exceptional Values ofE-Valued Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Zhaojun Wu ◽  
Zuxing Xuan
Keyword(s):  
Banach Space ◽  
Meromorphic Functions ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Characteristic Functions ◽  
Infinite Dimensional ◽  
Borel Exceptional Values

The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values ofE-valued meromorphic functions from theℂR={z:|z|<R},  0<R≤+∞to an infinite-dimensional complex Banach spaceEwith a Schauder basis. Results obtained extend the relative results by Xuan, Wu and Yang, Bhoosnurmath, and Pujari.

Non-Linear A-Proper Mappings of the Analytic Type

1973 ◽  
Vol 25 (3) ◽  
pp. 468-474 ◽  
Author(s):  
A. J. B. Potter
Keyword(s):  
Banach Space ◽  
Open Subset ◽  
Complex Variable ◽  
Complex Banach Space ◽  
Non Linear ◽  
Single Complex

Let Y be a complex Banach space, U an open subset of Y, f a mapping of U into Y. Then f is said to be complex analytic if for each pair of elements x and y of Y with x in U, the function f(x + ξy) of the single complex variable ξ is analytic in ξ on some neighbourhood of the origin.

Holomorphic functions with distinguished properties on infinite dimensional spaces

2018 ◽  
Vol 70 (3) ◽  
pp. 797-811
Author(s):  
Thiago R Alves ◽  
Geraldo Botelho
Keyword(s):  
Banach Space ◽  
Holomorphic Functions ◽  
Complex Banach Space ◽  
Algebraic Structures ◽  
Infinite Dimensional ◽  
Zero Function ◽  
Bounded Holomorphic

Abstract In this paper, we develop a method to construct holomorphic functions that exist only on infinite dimensional spaces. The following types of holomorphic functions f:U→ℂ on some open subsets U of an infinite dimensional complex Banach space are constructed: (1) f is bounded holomorphic on U and is continuously, but not uniformly continuously extended to U¯; (2) f is continuous on U¯ and holomorphic of bounded type on U, but f is unbounded on U; (3) f is holomorphic of bounded type on U and f cannot be continuously extended to U¯. The technique we develop is powerful enough to provide, in the cases (2) and (3) above, large algebraic structures formed by such functions (up to the zero function, of course).

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