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 Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

scholarly journals On the M-hypercyclicity of cosine function on Banach spaces

2019 ◽  
Vol 38 (3) ◽  
pp. 133-140
Author(s):  
Abdelaziz Tajmouati ◽  
Abdeslam El Bakkali ◽  
Ahmed Toukmati
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Infinite Dimensional ◽  
Uniformly Continuous

In this paper we introduce and study the M-hypercyclicity of strongly continuous cosine function on separable complex Banach space, and we give the criteria for cosine function to be M-hypercyclic. We also prove that every separable infinite dimensional complex Banach space admits a uniformly continuous cosine function.

scholarly journals Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 133
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Analytic Function ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Main Question ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Taylor Polynomials

In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.

scholarly journals On a local degree for a class of multi-valued vector fields in infinite dimensional Banach spaces

1996 ◽  
Vol 1 (4) ◽  
pp. 381-396 ◽  
Author(s):  
N. M. Benkafadar ◽  
B. D. Gel'man
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Fields ◽  
Existence Of Solutions ◽  
Index Zero ◽  
Compact Mapping ◽  
Infinite Dimensional ◽  
Upper Semicontinuous ◽  
Local Degree

This paper is devoted to the development of a local degree for multi-valued vector fields of the formf−F. Here,fis a single-valued, proper, nonlinear, Fredholm,C1-mapping of index zero andFis a multi-valued upper semicontinuous, admissible, compact mapping with compact images. The mappingsfandFare acting from a subset of a Banach spaceEinto another Banach spaceE1. This local degree is used to investigate the existence of solutions of a certain class of operator inclusions.

scholarly journals ON A CHARACTERISTIC PROPERTY OF FINITE-DIMENSIONAL BANACH SPACES

2011 ◽  
Vol 53 (3) ◽  
pp. 443-449 ◽  
Author(s):  
ANTONÍN SLAVÍK
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Characteristic Property ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Counter Example ◽  
Finite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Dvoretzky’S Theorem ◽  
The Given

AbstractThis paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

scholarly journals Nonwandering operators in Banach space

2005 ◽  
Vol 2005 (24) ◽  
pp. 3895-3908 ◽  
Author(s):  
Lixin Tian ◽  
Jiangbo Zhou ◽  
Xun Liu ◽  
Guangsheng Zhong
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sequence Space ◽  
Separable Banach Space ◽  
Infinite Dimensional ◽  
Banach Sequence Space ◽  
Background Space ◽  
Physical Background ◽  
Structurally Stable ◽  
Banach Sequence

We introduce nonwandering operators in infinite-dimensional separable Banach space. They are new linear chaotic operators and are relative to hypercylic operators, but different from them. Firstly, we show some examples for nonwandering operators in some typical infinite-dimensional Banach spaces, including Banach sequence space and physical background space. Then we present some properties of nonwandering operators and the spectra decomposition of invertible nonwandering operators. Finally, we obtain that invertible nonwandering operators are locally structurally stable.

On the Existence of Asymptotic-lp Structures in Banach Spaces

2007 ◽  
Vol 50 (4) ◽  
pp. 619-631 ◽  
Author(s):  
Adi Tcaciuc
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Infinite Dimensional

AbstractIt is shown that if a Banach space is saturated with infinite dimensional subspaces in which all “special” n-tuples of vectors are equivalent with constants independent of n-tuples and of n, then the space contains asymptotic-lp subspaces for some 1 ≤ p ≤ ∞. This extends a result by Figiel, Frankiewicz, Komorowski and Ryll-Nardzewski.

On Spaces of Compact Operators in Non-Archimedean Banach Spaces

1989 ◽  
Vol 32 (4) ◽  
pp. 450-458
Author(s):  
Takemitsu Kiyosawa
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convergent Sequence ◽  
Compact Operators ◽  
Linear Operators ◽  
Continuous Linear ◽  
Infinite Dimensional ◽  
Valued Field ◽  
Infinite Dimensional Banach Space ◽  
Continuous Linear Operators

AbstractLet K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.

scholarly journals Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)

1986 ◽  
Vol 29 (2) ◽  
pp. 271-282 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Positive Cone ◽  
Infinite Dimensional ◽  
Strongly Exposed Points ◽  
Nikodym Property ◽  
Choquet Theory ◽  
Ordered Banach Spaces

The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.

THE BOUNDED APPROXIMATION PROPERTY FOR THE WEIGHTED SPACES OF HOLOMORPHIC MAPPINGS ON BANACH SPACES

2017 ◽  
Vol 60 (2) ◽  
pp. 307-320 ◽  
Author(s):  
MANJUL GUPTA ◽  
DEEPIKA BAWEJA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Open Subset ◽  
Weighted Space ◽  
Approximation Property ◽  
Holomorphic Mappings ◽  
Countable Family ◽  
Weighted Spaces ◽  
Metric Approximation

AbstractIn this paper, we study the bounded approximation property for the weighted space$\mathcal{HV}$(U) of holomorphic mappings defined on a balanced open subsetUof a Banach spaceEand its predual$\mathcal{GV}$(U), where$\mathcal{V}$is a countable family of weights. After obtaining an$\mathcal{S}$-absolute decomposition for the space$\mathcal{GV}$(U), we show thatEhas the bounded approximation property if and only if$\mathcal{GV}$(U) has. In case$\mathcal{V}$consists of a single weightv, an analogous characterization for the metric approximation property for a Banach spaceEhas been obtained in terms of the metric approximation property for the space$\mathcal{G}_v$(U).

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