scholarly journals On approximation of homomorphisms of algebras of entire functions on Banach spaces

2019 ◽  
Vol 11 (1) ◽  
pp. 158-162
Author(s):  
H.M. Pryimak
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Commutative Banach Algebra ◽  
Uniformly Continuous ◽  
Positive Results

It is known due to R. Aron, B. Cole and T. Gamelin that every complex homomorphism of the algebra of entire functions of bounded type on a Banach space $X$ can be approximated in some sense by a net of point valued homomorphism. In this paper we consider the question about a generalization of this result for the case of homomorphisms to any commutative Banach algebra $A.$ We obtained some positive results if $A$ is the algebra of uniformly continuous analytic functions on the unit ball of $X.$

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 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 Homomorphisms and functional calculus in algebras of entire functions on Banach spaces

2015 ◽  
Vol 7 (1) ◽  
pp. 108-113 ◽  
Author(s):  
H.M. Pryimak
Keyword(s):  
Banach Algebra ◽  
Banach Spaces ◽  
Functional Calculus ◽  
Entire Functions ◽  
Fixed Number ◽  
Commutative Banach Algebra ◽  
Homogeneous Polynomials

In the paper the homomorphisms of algebras of entire functions on Banach spaces to a commutative Banach algebra are studied. In particular, it is proposed a method of constructing of homomorphisms vanishing on homogeneous polynomials of degree less or equal than a fixed number $n$.

scholarly journals Algebras of entire functions containing functions of unbounded type on a Banach space

2021 ◽  
Vol 13 (2) ◽  
pp. 426-432
Author(s):  
A. Zagorodnyuk ◽  
A. Hihliuk
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Entire Functions ◽  
Uniformly Continuous

In this paper, we consider algebras of entire analytic functions which are bounded on a prescribed family of balls in a Banach space. We investigate the structures of such algebras and describe their spectra in terms of spectra of algebras of uniformly continuous analytic functions. Some partial examples are considered. In particular, we have complete descriptions of the spectra for the case of Tsirelson space and for $c_0$.

scholarly journals Note on bases in algebras of analytic functions on Banach spaces

2019 ◽  
Vol 11 (1) ◽  
pp. 42-47 ◽  
Author(s):  
I.V. Chernega ◽  
A.V. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Symmetric Polynomials ◽  
Homogeneous Polynomials ◽  
Absolute Basis

Let $\{P_n\}_{n=0}^\infty$ be a sequenceof continuous algebraically independent  homogeneous polynomials on a complex Banach space $X.$ We consider the following question: Under which conditions polynomials $\{P_1^{k_1}\cdots P_n^{k_n}\}$ form a Schauder (perhaps absolute) basis in the minimal subalgebra of entire functions of bounded type on $X$ which contains the sequence $\{P_n\}_{n=0}^\infty$? In the paper we study the following examples: when $P_n$ are coordinate functionals on $c_0,$ and when $P_n$ are symmetric polynomials on $\ell_1$ and on $L_\infty[0,1].$ We can see that for some cases $\{P_1^{k_1}\cdots P_n^{k_n}\}$ is a Schauder basis which is not absolute but for some cases it is absolute.

scholarly journals Uniqueness of the maximal ideal of operators on the ℓp-sum of ℓ∞n (n ∈ ) for 1 < p < ∞

2016 ◽  
Vol 160 (3) ◽  
pp. 413-421 ◽  
Author(s):  
TOMASZ KANIA ◽  
NIELS JAKOB LAUSTSEN
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
American Mathematical Society ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Unique Maximal Ideal

AbstractA recent result of Leung (Proceedings of the American Mathematical Society, 2015) states that the Banach algebra ℬ(X) of bounded, linear operators on the Banach space X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓ1 contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓp and X = (⊕n∈$\mathbb{N}$ ℓ1n)ℓp whenever p ∈ (1, ∞).

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

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**.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

scholarly journals Homological characterizations of the approximation property for Banach spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Yu. V. Selivanov
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Cohomology Group ◽  
Approximation Property ◽  
Three Dimensional ◽  
Nuclear Operators ◽  
Finite Dimensionality

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:(i) biprojectivity of N(E);(ii) biflatness of N(E);(iii) homological finite-dimensionality of N(E);(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

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