# Countably Generated Algebras of Analytic Functions on Banach Spaces

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Definitions and Preliminary Results

## 3. Subalgebras of Polynomials and Semigroups of Symmetry

**Proposition**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Proposition**

**2.**

**Corollary**

**1.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

## 4. Algebras Generated by Sequences of Polynomials

- Under which conditions can the algebraic isomorphism$${J}_{0}={J}_{0}^{PQ}:{\mathcal{P}}_{\mathbf{P}}\left(X\right)\to {\mathcal{P}}_{\mathbf{Q}}\left(Y\right)$$$${J}_{0}:{P}_{n}\mapsto {Q}_{n}$$

**Example**

**9.**

**Example**

**10.**

**Example**

**11.**

**Definition**

**1.**

**Theorem**

**2.**

- 1.
- ${M}_{b\mathbf{P}}\left(X\right)$ consists of point-evaluation functionals.
- 2.
- ${H}_{b\mathbf{P}}\left(X\right)={H}_{\mathbf{P}}\left(X\right).$

**Proof.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Definition**

**2.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Example**

**12.**

**Example**

**13.**

**Example**

**14.**

**Example**

**15.**

**Proposition**

**4.**

**Proof.**

## 5. The Equivalence of Algebraic Bases

**Lemma**

**1.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 6. Derivations in Countable Generated Algebras

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

- 1.
- For every $j\in \{1,\dots ,n\}$ the mapping ${Q}_{j}:X\to \mathbb{C}$ is a continuous ${d}_{j}$-homogeneous polynomial, where ${d}_{j}\in N$;
- 2.
- The set of polynomials $\{{Q}_{1},{Q}_{2},\dots ,{Q}_{n}\}$ is algebraically independent;
- 3.
- There exists a constant $C>0$ such that for every vector $z=({z}_{1},\dots ,{z}_{n})\in {\mathbb{C}}^{n}$, there exists an element ${x}_{z}\in X$ such that $\parallel {x}_{z}\parallel {\le C\parallel z\parallel}_{\infty}$ and ${Q}_{j}\left({x}_{z}\right)={z}_{j}$ for every $j\in \{1,\dots ,n\},$ where ${\parallel z\parallel}_{\infty}=max\{|{z}_{1}|,\dots ,|{z}_{n}|\}.$

**Definition**

**3.**

**Proposition**

**5.**

**Proof.**

**Example**

**16.**

- 1.
- From Proposition 5, it follows that the algebras ${H}_{bs}\left({\ell}_{p}\right)$, as in Example 3, ${H}_{bs}\left({L}_{\infty}[0,1]\right)$, as in Example 10, and ${H}_{bI}\left(X\right)$, as in Example 8, and algebras of supersymmetric analytic functions of bounded type on ${\ell}_{p},$ $1\le p<\infty $ satisfy the conditions of Theorem 8.
- 2.
- Let $X={\ell}_{p},$ $1\le p\le \infty $ or $X={c}_{0},$ and ${P}_{n}\left(x\right)={x}_{1}{x}_{2}\cdots {x}_{n}.$ Then, for the algebra ${H}_{b\mathbf{P}}\left(X\right)$, we cannot apply Proposition 5 because the polynomial mapping$$({x}_{1},{x}_{2}\dots ,{x}_{n})\mapsto ({x}_{1},{x}_{1}{x}_{2},\dots ,{x}_{1}{x}_{2}\cdots {x}_{n})$$

**Example**

**17.**

**Example**

**18.**

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Novosad, Z.; Vasylyshyn, S.; Zagorodnyuk, A.
Countably Generated Algebras of Analytic Functions on Banach Spaces. *Axioms* **2023**, *12*, 798.
https://doi.org/10.3390/axioms12080798

**AMA Style**

Novosad Z, Vasylyshyn S, Zagorodnyuk A.
Countably Generated Algebras of Analytic Functions on Banach Spaces. *Axioms*. 2023; 12(8):798.
https://doi.org/10.3390/axioms12080798

**Chicago/Turabian Style**

Novosad, Zoriana, Svitlana Vasylyshyn, and Andriy Zagorodnyuk.
2023. "Countably Generated Algebras of Analytic Functions on Banach Spaces" *Axioms* 12, no. 8: 798.
https://doi.org/10.3390/axioms12080798