scholarly journals Extreme integral polynomials on a complex Banach space

2003 ◽  
Vol 92 (1) ◽  
pp. 129 ◽  
Author(s):  
Seán Dineen
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Space ◽  
Extreme Points ◽  
Complex Banach Space ◽  
The Real

We obtain upper and lower set-theoretic inclusion estimates for the set of extreme points of the unit balls of $\mathcal{P}_{I}({}^{n}\!E)$ and $\mathcal{P}_{N}({}^{n}\!E)$, the spaces of $n$-homogeneous integral and nuclear polynomials, respectively, on a complex Banach space $E$. For certain collections of Banach spaces we fully characterise these extreme points. Our results show a difference between the real and complex space cases.

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 Multipliers on Vector Valued Bergman Spaces

2002 ◽  
Vol 54 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
Oscar Blasco ◽  
José Luis Arregui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bergman Space ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Taylor Coefficients ◽  
Vector Valued

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

scholarly journals Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces

2021 ◽  
Vol 56 (1) ◽  
pp. 106-112
Author(s):  
S.I. Halushchak
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Complex Banach Space ◽  
Homogeneous Polynomials ◽  
Topological Algebras ◽  
Nonlinear Functional

The theory of analytic functions is an important section of nonlinear functional analysis.In many modern investigations topological algebras of analytic functions and spectra of suchalgebras are studied. In this work we investigate the properties of the topological algebras of entire functions,generated by countable sets of homogeneous polynomials on complex Banach spaces. Let $X$ and $Y$ be complex Banach spaces. Let $\mathbb{A}= \{A_1, A_2, \ldots, A_n, \ldots\}$ and $\mathbb{P}=\{P_1, P_2,$ \ldots, $P_n, \ldots \}$ be sequences of continuous algebraically independent homogeneous polynomials on spaces $X$ and $Y$, respectively, such that $\|A_n\|_1=\|P_n\|_1=1$ and $\deg A_n=\deg P_n=n,$ $n\in \mathbb{N}.$ We consider the subalgebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ of the Fr\'{e}chet algebras $H_b(X)$ and $H_b(Y)$ of entire functions of bounded type, generated by the sets $\mathbb{A}$ and $\mathbb{P}$, respectively. It is easy to see that $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ are the Fr\'{e}chet algebras as well. In this paper we investigate conditions of isomorphism of the topological algebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y).$ We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra $H_{bs}(L_{\infty})$ of entire functions of bounded type on $L_{\infty}[0,1]$ which are symmetric, i.e. invariant with respect to measurable bijections of $[0,1]$ that preserve the measure. We prove that$H_{bs}(L_{\infty})$ is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space $\ell_{\infty}.$

scholarly journals On a characterization of compactness and the Abel-Poisson summability of fourier coefficients in Banach spaces

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1061-1068
Author(s):  
Seda Öztürk
Keyword(s):  
Banach Space ◽  
Topological Group ◽  
Integral Representation ◽  
Unit Circle ◽  
Banach Spaces ◽  
Fourier Coefficients ◽  
Linear Representation ◽  
Complex Banach Space ◽  
Continuous Linear

In this paper, for an isometric strongly continuous linear representation denoted by ? of the topological group of the unit circle in complex Banach space, we study an integral representation for Abel-Poisson mean A?r (x) of the Fourier coefficients family of an element x, and it is proved that this family is Abel-Poisson summable to x. Finally, we give some tests which are related to characterizations of relatively compactness of a subset by means of Abel-Poisson operator A?r and ?.

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.

scholarly journals On Generalized Hyers-Ulam Stability of Admissible Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Banach Spaces ◽  
Unit Disk ◽  
Fractional Differential Equations ◽  
Complex Banach Space ◽  
Ulam Stability ◽  
Fractional Operators ◽  
Fractional Differential ◽  
Admissible Functions

We consider the Hyers-Ulam stability for the following fractional differential equations in sense of Srivastava-Owa fractional operators (derivative and integral) defined in the unit disk:Dzβf(z)=G(f(z),Dzαf(z),zf'(z);z),0<α<1<β≤2, in a complex Banach space. Furthermore, a generalization of the admissible functions in complex Banach spaces is imposed, and applications are illustrated.

Extreme points of unbounded, closed and convex sets in Banach spaces

1984 ◽  
Vol 95 (2) ◽  
pp. 319-323 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Convex Subsets

In this paper we examine the existence of extreme points in unbounded, closed and convex subsets K of a Banach space X.

scholarly journals On sequences of integrable functions

1962 ◽  
Vol 2 (3) ◽  
pp. 295-300
Author(s):  
Basil C. Rennie
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Finite Measure ◽  
Complex Banach Space ◽  
The Real ◽  
Integrable Functions ◽  
Necessary And Sufficient ◽  
Sequence Of Functions

Let f1(x), f2(x), … be a sequence of functions belonging to the real or complex Banach space L, (see S. Banach: [1] (The results can be generalised to functions on any space that is the union of countably many sets of finite measure). We are concerned with various properties that such a sequence may have, and in particular with the more important kinds of convergence (strong, weak and pointwise). This article shows what relations connect the various properties considered; for instance that for strong convergence (i.e. ║fn — f║ → 0) it is necessary and sufficient firstly that the sequence should converge weakly (i.e. if g is bounded and measurable then f(fn(x) — f(x))g(x)dx → 0) and secondly that any sub-sequence should contain a sub-sub-sequence converging p.p. to f(x).

EXTREME POINTS FOR COMBINATORIAL BANACH SPACES

2018 ◽  
Vol 61 (2) ◽  
pp. 487-500
Author(s):  
KEVIN BEANLAND ◽  
NOAH DUNCAN ◽  
MICHAEL HOLT ◽  
JAMES QUIGLEY
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Unit Ball ◽  
Banach Spaces ◽  
Extreme Points ◽  
Explicit Construction ◽  
Stability Properties ◽  
Original Space ◽  
Regular Family

AbstractA norm ‖ċ‖ on c00 is called combinatorial if there is a regular family of finite subsets $\mathcal{F}$, so that $\|x\|=\sup_{F \in \mathcal{F}} \sum_{i \in F} |x(i)|$. We prove the set of extreme points of the ball of a combinatorial Banach space is countable. This extends a theorem of Shura and Trautman. The second contribution of this article is to exhibit many new examples of extreme points for the unit ball of dual Tsirelson's original space and give an explicit construction of an uncountable collection of extreme points of the ball of Tsirelson's original space. We also prove some stability properties of the intermediate norms used to define Tsirelson's space and give a lower bound of the stabilization function for these intermediate norms.

GENERALIZED p-FRAME IN SEPARABLE COMPLEX BANACH SPACES

2010 ◽  
Vol 08 (01) ◽  
pp. 133-148 ◽  
Author(s):  
XIANG-CHUN XIAO ◽  
YU-CAN ZHU ◽  
XIAO-MING ZENG
Keyword(s):  
Banach Space ◽  
Functional Analysis ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Riesz Basis ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
The Stability

The concept of g-frame and g-Riesz basis in a complex Hilbert space was introduced by Sun.18 In this paper, we generalize the g-frame and g-Riesz basis in a complex Hilbert space to a complex Banach space. Using operators theory and methods of functional analysis, we give some characterizations of a g-frame or a g-Riesz basis in a complex Banach space. We also give a result about the stability of g-frame in a complex Banach space.

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