Symmetric analytic functions on the Cartesian power of the complex Banach space of Lebesgue measurable essentially bounded functions on [0,1]

2021 ◽  
pp. 125977
Author(s):  
Taras Vasylyshyn
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Complex Banach Space ◽  
Cartesian Power ◽  
Bounded Functions

scholarly journals Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power $p\in [1,+\infty)$ functions

2021 ◽  
Vol 13 (2) ◽  
pp. 340-351
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Entire Functions ◽  
Complex Banach Space ◽  
Cartesian Power ◽  
Entire Complex ◽  
Continuous Complex ◽  
Fréchet Algebra ◽  
Algebraic Basis ◽  
Complex Valued

The work is devoted to the study of Fréchet algebras of symmetric (invariant under the composition of every of components of its argument with any measure preserving bijection of the domain of components of the argument) analytic functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ and on the semi-axis. We show that the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,1]$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ is isomorphic to the Fréchet algebra of all analytic entire functions on $\mathbb C^m,$ where $m$ is the cardinality of the algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on this Cartesian power. The analogical result for the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,+\infty)$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the semi-axis $[0,+\infty)$ is proved.

scholarly journals Expansion of analytic functions of an operator in series of Faber polynomials

1997 ◽  
Vol 56 (2) ◽  
pp. 303-318 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Analytic Functions ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Faber Polynomials ◽  
Bounded Linear ◽  
In Series ◽  
Image Position

Let T: B → B be a bounded linear operator on the complex Banach space B and let f(z) be analytic on a domain D containing the spectrum Sp(T) of T. Then f(T) is defined bywhere C is a contour surrounding SP(T) and contained in D.

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 A note on approximation of continuous functions on normed spaces

2020 ◽  
Vol 12 (1) ◽  
pp. 107-110
Author(s):  
M.A. Mytrofanov ◽  
A.V. Ravsky
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Normed Space ◽  
Analytic Functions ◽  
Real Banach Space ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Normed Spaces ◽  
Approximation Of Continuous Functions

Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions, which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from a complex separable normed space, admitting a separating $*$-polynomial, can be uniformly approximated by $*$-analytic functions.

SOME PROPERTIES OF ELEMENTARY SYMMETRIC POLYNOMIALS ON THE CARTESIAN SQUARE OF THE COMPLEX BANACH SPACE L∞ [ 0,1 ]

2018 ◽  
pp. 9-16
Author(s):  
T. V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Symmetric Polynomials ◽  
Bounded Functions ◽  
Elementary Symmetric Polynomials ◽  
Algebraic Basis

We construct the element of the Cartesian square of the complex Banach space L ∞ [ 0,1 ] of all Lebesgue measurable essentially bounded functions on [ 0,1 ] by the predefined values of elementary symmetric polynomials on this element. Results of this work can be applied to the investigation of an algebraic basis of the algebra of continuous symmetric polynomials on the Cartesian square of the complex Banach space L ∞ [ 0,1 ] .

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 Symmetric polynomials on the Cartesian power of the real Banach space $L_\infty[0,1]$

2020 ◽  
Vol 53 (2) ◽  
pp. 192-205
Author(s):  
T. Vasylyshyn ◽  
A. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Analytic Functions ◽  
Symmetric Functions ◽  
Real Banach Space ◽  
Symmetric Polynomials ◽  
The Real ◽  
Cartesian Power ◽  
The Family ◽  
Algebraic Basis

We construct an algebraic basis of the algebra of symmetric (invariant under composition of the variable with any measure preserving bijection of $[0,1]$) continuous polynomials on the $n$th Cartesian power of the real Banachspace $L_^{(\mathbb{R})}\infty[0,1]$ of Lebesgue measurable essentially bounded real valued functions on $[0,1].$ Also we describe the spectrum of the Fr\'{e}chet algebra $A_s(L_^{(\mathbb{R})}\infty[0,1])$ of symmetric real-valued functions on the space $L_^{(\mathbb{R})}\infty[0,1]$, which is the completion of the algebra of symmetric continuous real-valued polynomials on  $L_^{(\mathbb{R})}\infty[0,1]$ with respect to the family of norms of uniform convergence of complexifications of polynomials. We show that $A_s(L_^{(\mathbb{R})}\infty[0,1])$ contains not only analytic functions. Results of the paper can be used for investigations of algebras of symmetric functions on the $n$th Cartesian power of the Banach space $L_^{(\mathbb{R})}\infty[0,1]$.

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 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.

The polynomial Daugavetian index of a complex Banach space

2018 ◽  
Vol 112 (4) ◽  
pp. 407-416
Author(s):  
Elisa R. Santos
Keyword(s):  
Banach Space ◽  
Complex Banach Space
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