Subsymmetric Polynomials on Banach Spaces and Their Applications

Abstract

We investigate algebraic and topological properties of subsymmetric polynomials on finite- and infinite-dimensional spaces. In particular, we focus on the problem of the existence of an algebraic basis in the algebra of subsymmetric polynomials, as well as possible extensions of subsymmetric polynomials and analytic functions to larger spaces. We consider algebras of subsymmetric analytic functions of bounded type and their spectra, and study linear subspaces in the zero-sets of subsymmetric polynomials, as well as subspaces where a subsymmetric polynomial is symmetric. In addition, we propose some possible applications of subsymmetric polynomials in cryptography and in operator theory.

1. Introduction

The concept of symmetric polynomials on a finite-dimensional linear space was very successful in mathematics and important for the development of algebraic geometry, invariant theory and combinatorics (see e.g., [1]). This concept admits a natural generalization for the infinite-dimensional space of eventually finite sequences $c_{00}={\cup }_{n=1}^{\infty }{\mathbb{C}}^n.$ Symmetric polynomials on ${\ell }_p$ for $1\le p<\infty$ were considered first in [2,3]. In particular, it was observed that polynomials

$$P_k(x)=\sum _{i=1}^{\infty }{x}_i^k,k\ge \lceil p\rceil$$

form an algebraic basis in the algebra of symmetric polynomials on ${\ell }_p.$ Here $\lceil p\rceil$ is the ceiling of $p,$ and the Banach space ${\ell }_p$ consists of the vectors $x=(x_1,\dots ,x_i,\dots )$ such that

$$\parallel x\parallel ={\left(\sum _{i=1}^{\infty }{|x_i|}^p\right)}^{1/p}<\infty .$$

In other words, for every symmetric polynomial P of degree n on ${\ell }_p$ there is a unique polynomial $q(t_1,\dots ,t_{n-\lceil p\rceil +1})$ of $n-\lceil p\rceil +1$ variables such that

$$P(x)=q\left(P_{\lceil p\rceil }(x),\dots ,P_n(x)\right),x\in {\ell }_p.$$

Algebras of symmetric analytic functions on ${\ell }_p$ were considered in many works (see, e.g., [4,5,6,7,8] and references therein). Some other generalizations of symmetric polynomials and analytic functions for the various cases of Banach spaces and (semi)groups of symmetry can be found in [9,10,11,12,13,14,15]. Algebras of subsymmetric polynomials and their generators were investigated in [16,17,18,19].

Let X be a complex sequence space $c_{00}$ or ${\ell }_p$ for $1\le p<\infty .$ A function F on X is subsymmetric if it is invariant with respect to the following operators ${\mathcal{C}}_j$ on X:

$${\mathcal{C}}_j:(x_1,\dots ,x_n,\dots )\mapsto (x_1,\dots ,x_{j-1},0,x_j,x_{j+1}\dots ),j\in \mathbb{N}.$$

(1)

That is, $F({\mathcal{C}}_j(x))=F(x)$ for every $j\in \mathbb{N}.$ Clearly, the operators ${\mathcal{C}}_j$ are continuous on ${\ell }_p.$ It is known [19] that the following (so-called standard) subsymmetric polynomials

$$P_{{\alpha }_1,\dots ,{\alpha }_n}(x)=\sum _{i_1<\cdots <i_n}{x}_{i_1}^{{\alpha }_1}\cdots x_{i_n}^{{\alpha }_n},{\alpha }_j\ge \lceil p\rceil$$

form a linear basis in the linear space of subsymmetric polynomials on ${\ell }_p.$ In particular, any m-homogeneous subsymmetric polynomial can be uniquely represented as a finite linear combination of the standard subsymmetric polynomials $P_{{\alpha }_1,\dots ,{\alpha }_n}$ such that ${\alpha }_1+\cdots +{\alpha }_n=m.$ We denote by $\mathfrak{S}$ the minimal semigroup generated by operators ${\mathcal{C}}_j$ and by ${\mathcal{P}}_{\mathfrak{S}}(X)$ the algebra of all subsymmetric polynomials on $X.$

In this paper, we consider properties of subsymmetric polynomials on finite- and infinite-dimensional spaces, the problem of the existence of an algebraic basis in the algebra of subsymmetric polynomials, extensions of subsymmetric polynomials and analytic functions to larger spaces, algebras of subsymmetric analytic functions of bounded type and their spectra, linear subspaces in zero-sets of subsymmetric polynomials, and subspaces where a subsymmetric polynomial is symmetric. In addition, we consider some possible applications of subsymmetric polynomials in cryptography and in operator theory. Algebras of all analytic functions of bounded type on Banach spaces were investigated in [20,21,22,23,24]. Linear subspaces in zero sets of polynomials on complex Banach spaces and related questions were studied in [25,26,27,28,29,30,31].

It is well-known [32] that there is no Hahn–Banach theorem for polynomials in the general case. In [9], the problem of the existence of a symmetry-preserving extension of symmetric polynomials on a Banach space was considered. In this paper, we investigate extensions of subsymmetric polynomials and find some general conditions when the operator of extension is a homomorphism of corresponding algebras. Using this approach, we obtain some advances for the description of the spectrum of algebras of subsymmetric functions on ${\ell }_p.$ These results generalise results in [6,7,20] for the case of subsymmetric functions.

It is known [19] that for a large class of spaces $X,$ in particular for ${\ell }_p,$ $1\le p<\infty ,$ for every polynomial Q on X there is $\epsilon >0$ and an infinite-dimensional subspace $X_0\subset X$ and a subsymmetric polynomial P on $X_0$ such that $|Q(x)-P(x)|<\epsilon$ for every $x\in X_0.$ This fact gives some kind of approximation of polynomials by subsymmetric polynomials. In the paper, we considered linear subspaces of subsymmetric polynomials and some subspaces V such that the restriction of a subsymmetric polynomial to V is symmetric with respect to a linear basis.

Let $\mathcal{P}$ be an algebra of polynomials. Let us recall that a finite or infinite sequence $(Q_1,Q_2,\dots )$ of nonzero polynomials is algebraically dependent if there is a number m and a nontrivial polynomial of m variables $q(t_1,\dots ,t_m)$ such that $q(Q_1,\dots ,Q_m)\equiv 0.$ If the sequence $(Q_i)$ is not algebraically dependent, then it is algebraically independent. An algebraically independent sequence $(Q_i)$ is an algebraic basis in $\mathcal{P}$ if every polynomial in $\mathcal{P}$ can be represented as an algebraic combination of polynomials $Q_i.$ Note that an algebraic basis does not always exist even if $\mathcal{P}$ is a subalgebra of polynomials on a finite-dimensional space. In this case, for any sequence of generators of $\mathcal{P}$, there are algebraic dependencies. The problem of the existence of an algebraic basis in the algebra of subsymmetric polynomials was investigated in [16,17,18,20]. Unfortunately, this problem is still open, and we propose solutions for some partial cases.

In Section 2, we investigate subsymmetric polynomials on finite-dimensional complex spaces ${\mathbb{C}}^M$ and show that the algebra of all subsymmetric polynomials on ${\mathbb{C}}^M$ has no algebraic basis. Also, we propose a general method of calculating algebraic dependencies between symmetric polynomials and a nonsymmetric polynomial. In Section 3, we consider algebraic properties of subsymmetric polynomials on the linear space $c_{00}$ of eventually zero sequences. We described some algebraically independent sequences of subsymmetric polynomials which, however, do not form algebraic bases. Also, we proved that two points, x an y in $c_{00},$ are equivalent up to actions of operators ${\mathcal{C}}_j$ if and only if $P(x)=P(y)$ for every subsymmetric $P.$ Moreover, it is enough to check this equality for polynomials P of some special form. Section 4 is devoted to the extension of the obtained results to continuous subsymmetric polynomials on some Banach spaces containing $c_{00}.$ In particular in Section 4.1 we consider subsymmetric polynomials on ${\ell }_p,$ $1\le p<\infty .$ In Section 4.2, we observed that there exists a natural extending topological homomorphism from the algebra of subsymmetric polynomials on ${\ell }_p$ to the algebra of subsymmetric polynomials on ${\ell }_p(\mathfrak{A})$ for any well-ordered set $\mathfrak{A}.$ This approach is useful for the investigation of algebras of subsymmetric analytic functions of bounded type and their spectra, which we considered in Section 5. In Section 6, we constructed some subspaces of ${\ell }_p$ where subsymmetric polynomials are symmetric with respect to some bases. Also, we considered linear subspaces in zero sets of some subsymmetric polynomials. Section 7 is devoted to some possible applications of the obtained results. In Section 7.1, we propose subsymmetric polynomials for the construction of hash functions that may be of interest in cryptography. In Section 7.2, using subsymmetric polynomials on eigenvalues of a p-trace operator $A,$ we show that A is completely defined by traces of $A^n$ and ${(A\wedge A^2)}^n,$ $n\ge \lceil p\rceil .$

We refer the reader for the general theory of polynomial and analytic mappings on Banach spaces to books [33,34].

2. Subsymmetric Polynomials on ${\mathbb{C}}^{\mathit{M}}$

In this section, we prove that there is no algebraic basis in the algebra of subsymmetric polynomials on a finite-dimensional space.

Subsymmetric polynomials were introduced using operators ${\mathcal{C}}_j$ that are well-defined on an infinite-dimensional vector space. Let $P_{{\alpha }_1,\dots ,{\alpha }_n}^{(M)}$ be the restriction of $P_{{\alpha }_1,\dots ,{\alpha }_n}$ to the M-dimensional subspace ${\mathbb{C}}^M$ of X spanned to $e_1,\dots ,e_M.$ We say that a polynomial P on ${\mathbb{C}}^M$ is subsymmetric if it is a linear combination of polynomials $P_{{\alpha }_1,\dots ,{\alpha }_n}^{(M)}.$

It is well-known in algebraic geometry that every family of nonconstant polynomials $Q_1,Q_2,\dots ,Q_{M+1}$ is algebraically dependent on ${\mathbb{C}}^M.$ From here it follows, in particular, that if some algebra of polynomials on ${\mathbb{C}}^M$ is generated by all symmetric polynomials and a single nonsymmetric polynomial, then it has no algebraic basis because such an algebra must have at least $M+1$ generators that cannot be algebraically independent. The following example delivers an algebraic dependence between a basis of symmetric polynomials and a subsymmetric polynomial on ${\mathbb{C}}^3.$

Example 1.

Let

$${\xi }_1(x)=P_1^{(3)}(x)=x_1+x_2+x_3,$$

$${\xi }_2(x)=P_{1,1}^{(3)}(x)=x_1{x}_2+x_1{x}_3+x_2{x}_3,$$

$${\xi }_3(x)=P_{1,1,1}^{(3)}(x)=x_1{x}_2{x}_3,$$

$${\xi }_4(x)=P_{1,3}^{(3)}(x)=x_1{x}_2^2+x_1{x}_3^2+x_2{x}_3^2.$$

Then, by direct calculations we can check that

$$\begin{array}{cc}\hfill & {\xi }_4^6+(9{\xi }_3-3{\xi }_1{\xi }_2){\xi }_4^5+({\xi }_1^3{\xi }_3+36{\xi }_3^2+3{\xi }_1^2{\xi }_2^2+3{\xi }_2^3-28{\xi }_1{\xi }_2{\xi }_3){\xi }_4^4\hfill \\ \hfill +& 6{\xi }_1^3{\xi }_3^2+81{\xi }_3^3-{\xi }_1^3{\xi }_2^3-6{\xi }_1{\xi }_2^4-2{\xi }_1^4{\xi }_2{\xi }_3+29{\xi }_1^2{\xi }_2^2{\xi }_3-105{\xi }_1{\xi }_2{\xi }_3^2+18{\xi }_2^3{\xi }_3{\xi }_4^3\hfill \\ \hfill +& (108{\xi }_3^4+13{\xi }_1^3{\xi }_3^3+3{\xi }_2^6+3{\xi }_1^2{\xi }_2^5-38{\xi }_1{\xi }_2^4{\xi }_3+49{\xi }_2^3{\xi }_3^2-8{\xi }_1^3{\xi }_2^3{\xi }_3+96{\xi }_1^2{\xi }_2^2{\xi }_3^2+{\xi }_1^5{\xi }_2^2{\xi }_3\hfill \\ \hfill -& 198{\xi }_1{\xi }_2{\xi }_3^3-10{\xi }_1^4{\xi }_2{\xi }_3^2){\xi }_4^2+(81{\xi }_3^5+12{\xi }_1^3{\xi }_3^4-3{\xi }_1{\xi }_2^7+9{\xi }_2^6{\xi }_3+20{\xi }_1^2{\xi }_2^5{\xi }_3-82{\xi }_1{\xi }_2^4{\xi }_3^2\hfill \\ \hfill -& 2{\xi }_1^4{\xi }_2^4{\xi }_3+66{\xi }_2^3{\xi }_3^3-21{\xi }_1^3{\xi }_2^3{\xi }_3^2+135{\xi }_1^2{\xi }_2^2{\xi }_3^3+4{\xi }_1^5{\xi }_2^2{\xi }_3^2-189{\xi }_1{\xi }_2{\xi }_3^4-16{\xi }_1^4{\xi }_2{\xi }_3^3){\xi }_4\hfill \\ \hfill +& (27{\xi }_3^6+4{\xi }_1^3{\xi }_3^5+{\xi }_2^9-10{\xi }_1{\xi }_2^7{\xi }_3+13{\xi }_2^6{\xi }_3^2+{\xi }_1^3{\xi }_2^6{\xi }_3+27{\xi }_1^2{\xi }_2^5{\xi }_3^2-62{\xi }_1{\xi }_2^4{\xi }_3^3-4{\xi }_1^4{\xi }_2^4{\xi }_3^2\hfill \\ \hfill +& 31{\xi }_2^3{\xi }_3^4-18{\xi }_1^3{\xi }_2^3{\xi }_3^3++71{\xi }_1^2{\xi }_2^2{\xi }_3^4+4{\xi }_1^5{\xi }_2^2{\xi }_3^3-72{\xi }_1{\xi }_2{\xi }_3^5-8{\xi }_1^4{\xi }_2{\xi }_3^4)\equiv 0.\hfill \end{array}$$

(2)

Note that in [6] it is proved that polynomials $P_n,$ $n\in \mathbb{N}$ and $P_{1,2}$ are algebraically independent on ${\ell }_1.$ However, the question about the existence of an algebraic basis in the algebra of all subsymmetric polynomials on ${\ell }_p$ or $c_{00}$ is still open.

Now we consider how to find explicitly an algebraic dependence of a non-symmetric polynomial on symmetric polynomials defined on ${\mathbb{C}}^n.$ Let $P(z)=P(z_1,\dots ,z_n)\in \mathcal{P}({\mathbb{C}}^n).$ We denote by $\mathrm{Orb}(P)$ the orbit of P under permutations of the basis vectors:

$$\mathrm{Orb}(P)=\{\sigma (P)(z):=P(z_{\sigma (1)},\dots ,z_{\sigma (n)}):\sigma \in S_n\}$$

and by $|\mathrm{Orb}(P)|$ the cardinality of $\mathrm{Orb}(P).$ Here $S_n$ is the group of permutations of $\{1,\dots ,n\}.$ Clearly, $1\le |\mathrm{Orb}(P)|\le n!$ and if $|\mathrm{Orb}(P)|=1,$ then P is symmetric.

Theorem 1.

For a given polynomial P on ${\mathbb{C}}^n$ there is a polynomial of a complex variable t

$$A(t)=t^m-a_1(·)t^{m-1}+a_2(·)t^{m-2}-\cdots +{(-1)}^m{a}_m(·)$$

of degree $m=|\mathrm{Orb}(P)|,$ where coefficients $a_j(·)$ are symmetric polynomials on ${\mathbb{C}}^n,$ such that $A(Q)\equiv 0$ for every $Q\in \mathrm{Orb}(P).$ In particular, $A(P)\equiv 0.$

Proof. 

Let $\mathrm{Orb}(P)=\{Q_1,\dots ,Q_m\}$ and $Q_1=P.$ We denote by

$$a_k(z)=\sum _{i_1<\cdots <i_k}{Q}_{i_1}(z)\cdots Q_{i_k}(z),1\le k\le m,z\in {\mathbb{C}}^n,$$

the elementary symmetric functions of $\{Q_1(z),\dots ,Q_m(z)\}.$ Since for every $\sigma \in S_n$ the mapping

$$\{Q_1(z),\dots ,Q_m(z)\}\mapsto \{Q_1(\sigma (z)),\dots ,Q_m(\sigma (z))\}$$

is a bijection, $\sigma (a_k)(z)=a_k(z)$ and so $a_k(z)$ is symmetric for every $1\le k\le m.$ By the Vieta Theorem (see e.g., [35], pp. 29–30), for every $z\in {\mathbb{C}}^n$ the numbers $\{Q_1(z),\dots ,Q_m(z)\}$ are zeros of $A(t),$ in particular, $A(P(z))=0,$ $z\in {\mathbb{C}}^n.$ □

From the proof of Theorem 1 it follows that if P is a nonsymmetric polynomial on ${\mathbb{C}}^n,$ then in order to construct the polynomial $A(t),$ we have to find the orbit $\mathrm{Orb}(P)=\{Q_1,\dots ,Q_m\}$ and define $a_k(z)$ as elementary symmetric polynomials of $Q_1,\dots ,Q_m.$ Then $A(P)=0$ is an algebraic dependence between symmetric polynomials and $P.$ In particular, identity (2) can be obtained by this way.

Each polynomial on the 1-dimensional space $\mathbb{C}$ is trivially symmetric (and so subsymmetric). If $M\ge 2,$ then $P_1^{(M)},$ ${\left(P_1^{(M)}\right)}^2,$ and $P_2^{(M)}$ form a linear basis in the linear space of subsymmetric polynomials of degree no greater than $2.$ Thus, each 1-degree and 2-degree subsymmetric polynomial is symmetric. But we have some nonsymmetric 3-degree polynomials. Even if $M=2,$ $P_{1,2}^{(2)}(x)=x_1{x}_2^2$ is a nontrivial subsymmetric polynomial that cannot be generated by symmetric polynomials. Thus, any system of generators of the algebra of subsymmetric polynomials on ${\mathbb{C}}^2$ must have at least 3 elements, and they are always algebraically dependent. In the general case, it is easy to check that the symmetric polynomial $P_k^{(M)},$ $k\le M$ cannot be represented by subsymmetric polynomials of degree less than k (see [18]) and so any system of generators of the algebra of subsymmetric polynomials on ${\mathbb{C}}^M$ must have at least one generator of each degree $1\le k\le M.$ But as we observed, for 3-degree polynomials we need to have one generator more. Thus, the total number of generators is not less than $M+1.$ As we mentioned above, $M+1$ polynomials on ${\mathbb{C}}^M$ are algebraically dependent. Hence, we have the following proposition.

Proposition 1.

For every $M>1,$ the algebra of subsymmetric polynomials on ${\mathbb{C}}^M$ has no algebraic basis.

3. Subsymmetric Polynomials on ${\mathbf{c}}_{\mathit{00}}$

In this section we consider basic algebraic properties of subsymmetric polynomials on the linear space of finite sequences $c_{00}={\cup }_{n=1}^{\infty }{\mathbb{C}}^n.$

Note that Theorem 1 cannot be extended to the infinite-dimensional case because the algebra of symmetric polynomials of infinite many variables is factorial (see e.g., [36]), that is, if a symmetric polynomial P can be represented as a product of two polynomials, $P=P_1{P}_2,$ then both $P_1$ and $P_2$ must be symmetric. We denote by ${\mathcal{P}}_s(c_{00})$ the algebra of symmetric polynomials on $c_{00}$ and by ${\mathcal{P}}_{\mathfrak{S}}(c_{00})$ the algebra of subsymmetric polynomials on $c_{00}.$

Proposition 2.

Let

$$A(t)=t^m-a_1{t}^{m-1}+a_2{t}^{m-2}-\cdots +{(-1)}^m{a}_m$$

be a polynomial with coefficients $a_j\in {\mathcal{P}}_s(c_{00}).$ If P is a polynomial on $c_{00}$ such that $A(P(x))=0$ for every $x\in c_{00},$ then $P\in {\mathcal{P}}_s(c_{00}).$

Proof. 

The identity $A(P)\equiv 0$ implies that $A\left(P(T_{\sigma }(x))\right)=0$ for every $x\in c_{00}$ and permutation $\sigma \in S_{\mathbb{N}},$ where $T_{\sigma }(x)=(x_{\sigma (1)},x_{\sigma (2)},\dots ).$ Let $Q_1,\dots ,Q_m$ be m different elements in $\mathrm{Orb}(P).$ Then for every $x\in c_{00},$ numbers $\{Q_1(x),\dots ,Q_m(x)\}$ are roots of $A(t)(x).$ By the Vieta theorem,

$$a_k(x)=\sum _{i_1<\cdots <i_k}{Q}_{i_1}(x)\cdots Q_{i_k}(x),1\le k\le m.$$

Since each $a_k$ is symmetric and $a_m(x)=Q_1(x)\cdots Q_m(x)$ for every $x\in c_{00},$ the polynomial $Q_1\cdots Q_m$ is symmetric while $Q_1,\dots ,Q_m$ are not. This is a contradiction. Thus, the number of different elements in $\mathrm{Orb}(P)$ is less than $m.$ Let $\mathrm{Orb}(P)$ has r different elements. If $r>1,$ then $Q_1\cdots Q_r$ must be symmetric while the multipliers $Q_1,\dots ,Q_r$ are not, and again we have a contradiction. Thus, $r=1$ and so $P\in {\mathcal{P}}_s(c_{00}).$ □

From the straightforward computations we can see that

$$\begin{array}{c}\hfill P_n(x)P_{{\alpha }_1,\dots ,{\alpha }_m}(x)=P_{n,{\alpha }_1,\dots ,{\alpha }_m}(x)+P_{{\alpha }_1+n,\dots ,{\alpha }_m}(x)+\cdots +P_{{\alpha }_1,\dots ,n,{\alpha }_m}(x)+P_{{\alpha }_1,\dots ,{\alpha }_m+n}(x).\end{array}$$

(3)

Proposition 3.

For every standard subsymmetric polynomial $P_{{\alpha }_1,\dots ,{\alpha }_m}$ on $c_{00},$ the polynomial

$$P_{{\alpha }_{\sigma (1)},\dots ,{\alpha }_{\sigma (m)}}^s=\sum _{\sigma \in S_m}{P}_{{\alpha }_{\sigma (1)},\dots ,{\alpha }_{\sigma (m)}}$$

is symmetric. Here $S_m$ is the group of permutations of the set $\{1,\dots ,m\}.$

Proof. 

It is enough to check that $P_{{\alpha }_{\sigma (1)},\dots ,{\alpha }_{\sigma (m)}}^s$ is invariant with respect to any operator of transposition

$${\sigma }_{ij}:(x_1,\dots ,x_i,\dots ,x_j,\dots )\mapsto (x_1,\dots ,x_j,\dots ,x_i,\dots ).$$

Let us consider the set of terms of $P_{{\alpha }_{\sigma (1)},\dots ,{\alpha }_{\sigma (m)}}^s$ containing coordinates $x_i$ and $x_j,$

$$\left\{x_{i_1}^{{\alpha }_1}\cdots x_i^{{\alpha }_s}\cdots x_j^{{\alpha }_r}\cdots x_{i_m}^{{\alpha }_m}\right\},$$

where $i_k$ are mutually different. From the definition of $P_{{\alpha }_{\sigma (1)},\dots ,{\alpha }_{\sigma (m)}}^s$ it follows that coordinates $x_i$ and $x_j$ may take any places s and r so that $s\ne r.$ Clearly, this set is invariant with respect the interchanging of $x_i$ and $x_j$ and the sum of its elements is invariant as well. Thus, $P_{{\alpha }_{\sigma (1)},\dots ,{\alpha }_{\sigma (m)}}^s$ is invariant with respect to every ${\sigma }_{ij}$ and so is symmetric. □

Example 2.

From (3) it follows that $P_k{P}_m=P_{k,m}+P_{k+m}+P_{m,k}$ and so

$$P_{k,m}^s=P_{k,m}+P_{m,k}=P_k{P}_m-P_{k+m}$$

(4)

is the representation of $P_{k,m}^s$ by the power symmetric polynomials.

From Proposition 3 it follows that if a polynomial $P\in {\mathcal{P}}_{\mathfrak{S}}(c_{00})$ is symmetric with respect to the permutation of the indexes ${\alpha }_1,\dots ,{\alpha }_m$ in its representation as a linear combination of the standard polynomials $P_{{\alpha }_1,\dots ,{\alpha }_m},$ then P is symmetric. However, for an algebraic combination it is not so. For example, polynomial $P=P_{km}{P}_{mk},$ $1\le k<m$ is invariant with respect to the permutation $k\leftrightarrow m$ but not symmetric because a symmetric polynomial cannot be a product of two nonsymmetric polynomials. By the same reasoning, $P_{km}^2+P_{mk}^2=(P_{km}+i{P}_{mk})(P_{km}-i{P}_{mk})$ is not symmetric.

For every $x=(x_1,\dots ,x_m,0,0,\dots )\in c_{00}$ we denote by $\check{x}=(x_m,\dots ,x_1,0,0,\dots ),$ where $x_m$ is the last nonzero coordinate of $x.$ Also, if $y=(y_1,\dots ,y_n,0,0,\dots )\in c_{00},$ then we denote

$$x⊲y=(x_1,\dots ,x_m,y_1,\dots ,y_n,0,0,\dots ),$$

and

$$x•y=(x_1,y_1,x_2,y_2,\dots ).$$

Formally, the definitions of $\check{x}$ and $x⊲y$ are dependent on the m that bounds the number of nonzero coordinates of $x.$ But it is easy to check that for every $P\in {\mathcal{P}}_{\mathfrak{S}}(c_{00}),$

$$P(\check{x})=\underset{k\to \infty }{lim}P(x_k,x_{k-1},\dots ,x_1,0,0,\dots ),$$

(5)

and

$$P(x⊲y)=\underset{k\to \infty }{lim}P(x_1,x_2,\dots ,x_k,y_1,y_2,\dots ).$$

(6)

The right-hand-side parts of (5) and (6) do not depend on m and can be used to extend these definitions for more general spaces (see Section 4).

Let us consider some useful identities. From direct computations (see e.g., [18]) it is possible to check that

$$P_{{\alpha }_1,\dots ,{\alpha }_m}(x⊲y)=\sum _{j=0}^{m+1}{P}_{{\alpha }_1,\dots ,{\alpha }_j}(x)P_{{\alpha }_{j+1},\dots ,{\alpha }_m}(y),x,y\in c_{00}.$$

(7)

Also, obviously

$$P_{{\alpha }_1,\dots ,{\alpha }_m}(\check{x})=P_{{\alpha }_m,\dots ,{\alpha }_1}(x),x\in c_{00}.$$

(8)

For a given polynomial $P\in {\mathcal{P}}_{\mathfrak{S}}(c_{00})$ we denote by $\check{P}={(P)}^{\vee }$ as the polynomial in ${\mathcal{P}}_{\mathfrak{S}}(c_{00})$ such that $\check{P}(x)=P(\check{x})$ for every $x\in c_{00}.$ Clearly, the map $P\mapsto \check{P}$ is linear and multiplicative. For example, the multiplicativity follows from the computations

$${(PQ)}^{\vee }(x)=(PQ)(\check{x})=P(\check{x})Q(\check{x})=\check{P}(x)\check{Q}(x),x\in c_{00}.$$

From (8) we have that ${\check{P}}_{{\alpha }_1,\dots ,{\alpha }_m}=P_{{\alpha }_m,\dots ,{\alpha }_1}.$ Thus the map $P\mapsto \check{P}$ is a homomorphism from ${\mathcal{P}}_{\mathfrak{S}}(c_{00})$ to itself. Moreover, this map is an involution because ${(\check{P})}^{\vee }=P$ for every $P\in {\mathcal{P}}_{\mathfrak{S}}(c_{00}).$

Proposition 4.

For every pair $k,m\in \mathbb{N}$ the polynomial

$$Q(x)=P_{k,m}(\check{x}⊲x),x\in c_{00}$$

is symmetric and

$$Q(x)=2P_k(x)P_m(x)-P_{k+m}(x).$$

Proof. 

From (7) we have

$$P_{k,m}(\check{x}⊲x)=P_{k,m}(\check{x})+P_k(x)P_m(x)+P_{k,m}(x)=P_{k,m}(x)+P_{m,k}(x)+P_k(x)P_m(x).$$

From Proposition 3 it follows that $Q(x)$ is symmetric and from (4) we have $Q(x)=2P_k(x)P_m(x)-P_{k+m}(x).$ □

Note that for every $x\in c_{00}$ we have

$$P_{k,m}(x•x)=4P_{k,m}(x)+P_{k+m}(x).$$

Let us introduce two types of equivalences on $c_{00}.$ We say that x and y in $c_{00}$ are equivalent with respect to the actions of $\mathfrak{S},$ notation $x\simeq y,$ if there are ${\tau }_1$ and ${\tau }_2$ in $\mathfrak{S}$ such that ${\tau }_1(x)={\tau }_2(y).$ Similarly, we say that x and y are equivalent with respect to the semigroup S of all permutations (that is, injective mappings) of $\mathbb{N}$ acting on $c_{00}$ as $\sigma (x)=(x_{\sigma (1)},x_{\sigma (2)},\dots ),$ notation $x\sim y,$ if there are ${\sigma }_1$ and ${\sigma }_2$ in S such that ${\sigma }_1(x)={\sigma }_2(y).$ We denote by ${\mathcal{M}}_0^{\mathfrak{S}}$ the quotient set $c_{00}/\simeq ,$ and by ${\mathcal{M}}_0^S$ the quotient set $c_{00}/\sim .$ It is known that ${\mathcal{M}}_0^S$ can be identified with the set of all finite multisets (that is, unordered tuples with possible repetitions) of nonzero complex numbers. Moreover, $x\sim y$ if and only if $P(x)=P(y)$ for every symmetric polynomial $P.$ Since $P_n$ form an algebraic basis in ${\mathcal{P}}_s(c_{00}),$ it is enough to check the equality $P_n(x)=P_n(y)$ for every $n.$ Finally, if we know that both x and y have only m nonzero coordinates each, then it is enough to check $P_n(x)=P_n(y)$ for $1\le n\le m.$ We denote by $[x]\in {\mathcal{M}}_0^S$ the equivalence class containing x and by $[[x]]$ the equivalence class in ${\mathcal{M}}_0^{\mathfrak{S}}$ containing $x.$ Clearly $x\simeq y$ if and only if after removing all zero coordinates in x and y we have that $(x_{i_1},\dots ,x_{i_m})$ and $(y_{j_1},\dots ,y_{j_m})$ are equal as vectors.

Note that the operation “⊲” can be lifted to ${\mathcal{M}}_0^{\mathfrak{S}}$ by

$$[[x]]⊲[[y]]=[[x⊲y]],$$

while “•” cannot. Indeed, let $x^{\prime }=(0,x_1,x_2,\dots ).$ Then $x\simeq x^{\prime }$, but

$$x^{\prime }•y=(0,y_1,x_1,y_2,\dots )\not\simeq (x_1,y_1,x_2,y_2,\dots )=x•y.$$

However, the unary operation $x\mapsto x•x$ can be lifted to the equivalence classes by $[[x]]•[[x]]=[[x•x]].$ The restriction of “⊲” to the subalgebra of symmetric polynomials coincides with “•”, and “•” can be lifted to ${\mathcal{M}}_0^S.$

Theorem 2.

The equivalence $x\simeq y$ holds for some vectors $x,y\in c_{00}$ if and only if $P(x)=P(y)$ for every subsymmetric polynomial $P.$ Moreover, it is enough to check the equality of subsymmetric polynomials of the form $P_k$ and $P_{k,2k},$ $k\in \mathbb{N}.$ More explicitly, if x and y have no more than m nonzero coordinates, then we can check the equalities $P_k(x)=P_k(y)$ for all $k\le m$ and $P_{k,2k}(x)=P_{k,2k}(y)$ for $k\le {\displaystyle \frac{m(m-1)}{2}}.$

Proof. 

Clearly, if $x\simeq y,$ then $P(x)=P(y)$ for every subsymmetric polynomial $P,$ by the definition of subsymmetric polynomials. Suppose that $P(x)=P(y)$ for every subsymmetric polynomial $P.$ In particular, $P(x)=P(y)$ for every symmetric polynomial $P.$ It is well-known that in this case the multiset $\{x_1,\dots ,x_m\}$ of non-zero coordinates of x coincides with the multiset of non-zero coordinates of $y,$ $\{y_1,\dots ,y_m\}.$ To make sure that $x\simeq y$ we need check that after removing zero coordinates, $(x_1,\dots ,x_m)=(y_1,\dots ,y_m)$ as vectors. Consider the following multisets

$$M_{1,2}(x)=\{x_i{x}_j^2:i<j\}\mathrm{and}{M}_{1,2}(y)=\{y_i{y}_j^2:i<j\}.$$

We claim that if $(x_1,\dots ,x_m)\ne (y_1,\dots ,y_m),$ then $M_{1,2}(x)\ne M_{1,2}(y).$ Indeed, if $(x_1,\dots ,x_m)\ne (y_1,\dots ,y_m),$ then there are $x_r\ne x_s$ such that $r<s$ and a bijection $\sigma :\{1,\dots ,m\}\to \{1,\dots ,m\}$ such that $(y_1,\dots ,y_m)=(x_{\sigma (1)},\dots ,x_{\sigma (m)})$ and $\sigma (r)>\sigma (s).$ Then the symmetric difference $M_{1,2}(x)▵M_{1,2}(y)$ is nonempty, because it contains, at least, elements $x_r{x}_s^2$ and $x_s{x}_r^2.$ Here $M_{1,2}(x)▵M_{1,2}(y)=(M_{1,2}(x)\setminus M_{1,2}(y))\cup (M_{1,2}(y)\setminus M_{1,2}(x)),$ and the set-theoretical substraction “∖” takes into account the multiplicity of each element. Hence, we have two different multisets $M_{1,2}(x)$ and $M_{1,2}(y)$ of length $N={\displaystyle \frac{m(m-1)}{2}}.$ As we mentioned above, there exists a symmetric polynomial P of N variables such that

$$P(x_1{x}_2^2,x_1{x}_3^2,\dots ,x_2{x}_3^2,\dots ,x_{N-1}{x}_N^2)\ne P(y_1{y}_2^2,y_1{y}_3^2,\dots ,y_2{y}_3^2,\dots ,y_{N-1}{x}_N^2).$$

Equivalently, there is $1\le k\le N$ such that

$$\sum _{i<j}{\left(x_i{x}_j^2\right)}^k\ne \sum _{i<j}{\left(y_i{y}_j^2\right)}^k,$$

that is, $P_{k,2k}(x)\ne P_{k,2k}(y).$ This is a contradiction. Thus $(x_1,\dots ,x_m)=(y_1,\dots ,y_m)$ and so $x\simeq y.$ □

Let us denote by ${\mathcal{P}}_{\mathfrak{S}}^{(n)}(c_{00})$ the subalgebra of ${\mathcal{P}}_{\mathfrak{S}}(c_{00}),$ generated by polynomials $P_{{\alpha }_1,\dots ,{\alpha }_k}$ for $0\le k\le n.$ Clearly, ${\mathcal{P}}_{\mathfrak{S}}^{(1)}(c_{00})={\mathcal{P}}_s(c_{00})$ is the algebra of symmetric polynomials and it is not equal to ${\mathcal{P}}_{\mathfrak{S}}^{(2)}(c_{00}).$ Let $(\Theta (P))(x)=P(\check{x}⊲x).$ From Proposition 4 it follows that the restriction of $\Theta$ to ${\mathcal{P}}_{\mathfrak{S}}^{(2)}(c_{00})$ is a homomorphism from ${\mathcal{P}}_{\mathfrak{S}}^{(2)}(c_{00})$ to ${\mathcal{P}}_s(c_{00}).$

The following example shows that ${\mathcal{P}}_{\mathfrak{S}}^{(2)}(c_{00})$ is a proper subalgebra in ${\mathcal{P}}_{\mathfrak{S}}^{(3)}(c_{00}).$

Example 3.

Let $k\ne n$ be positive integers. We claim that $P_{k,n,k}\notin {\mathcal{P}}_{\mathfrak{S}}^{(2)}(c_{00}).$ Indeed,

$$\begin{array}{cc}\hfill \Theta (P_{k,n,k})(x)& =P_{k,n,k}(\check{x}⊲x)=P_{k,n,k}(\check{x})+P_k(\check{x})P_{n,k}(x)+P_{k,n}(\check{x})P_k(x)+P_{k,n,k}(x)\hfill \\ \hfill & =2P_{k,n,k}(x)+2P_k(x)P_{n,k}(x).\hfill \end{array}$$

Polynomial $2P_{k,n,k}(x)+2P_k(x)P_{n,k}(x)$ is not symmetric because, for example,

$$2P_{k,n,k}(1,2,0,\dots )+2P_k(1,2,0,\dots )P_{n,k}(1,2,0,\dots )=10·2^k$$

and

$$2P_{k,n,k}(2,1,0,\dots )+2P_k(2,1,0,\dots )P_{n,k}(2,1,0,\dots )=10·2^n.$$

But, if $P_{k,n,k}\in {\mathcal{P}}_{\mathfrak{S}}^{(2)}(c_{00}),$ then by Proposition 4, $\Theta (P_{k,n,k})$ must be symmetric.

The following theorem delivers some nontrivial sequences of algebraically independent subsymmetric polynomials.

Theorem 3.

Let $(R_n),$ $n\in \mathbb{N}$ be a sequence of subsymmetric polynomials. If there exists an algebra homomorphism $\Phi :{\mathcal{P}}_{\mathfrak{S}}(c_{00})\to {\mathcal{P}}_S(c_{00})$ such that the sequence $(\Phi (R_n)),$ $n\in \mathbb{N}$ is algebraically independent, then polynomials $R_1,\dots ,R_n,\dots$ are algebraically independent as well.

Proof. 

If $(R_n),$ $n\in \mathbb{N}$ are not algebraically independent, then for some $r\in \mathbb{N}$ there is a nontrivial polynomial of several variables $q(t_1,\dots ,t_r)$ such that

$$q(R_1,\dots ,R_r)\equiv 0.$$

Since $\Phi$ is a homomorphism,

$$\Phi \left(q(R_1,\dots ,R_r)\right)=q\left(\Phi (R_1),\dots ,\Phi (R_r)\right)=\Phi (0)=0.$$

But it contradicts the algebraic independence of polynomials $\Phi (R_n),$ $n\in \mathbb{N}.$ □

Let $Z_1=P_1,$ $Z_2=P_2$ and for $n>2,$

$$Z_n(x)=\sum _{\begin{array}{c}k+m=n,\\ 0<k<m\end{array}}{P}_{k,m}(x),x\in c_{00}.$$

Corollary 1. 

(i) 

Polynomials $Z_n,$ $n\in \mathbb{N}$ are algebraically independent in ${\mathcal{P}}_{\mathfrak{S}}^{(2)}(c_{00}).$

(ii) 

Polynomials $P_{1,m},$ $m\in {\mathbb{Z}}_{+}$ are algebraically independent in ${\mathcal{P}}_{\mathfrak{S}}^{(2)}(c_{00}).$ Here $P_{1,0}=P_1.$

Proof. 

(i). Let $\Phi =\Theta .$ According to Proposition 4,

$$\begin{array}{cc}\Theta (Z_1)\hfill & =2P_1,\hfill \\ \Theta (Z_2)\hfill & =2P_2,\hfill \\ \Theta (Z_3)\hfill & =2P_1{P}_2-P_3,\hfill \\ \hfill & \cdots \hfill \\ \Theta (Z_n)\hfill & =2P_1{P}_{n-1}+\cdots +2P_N{P}_{n-N}-N{P}_n,\hfill \\ \hfill & \cdots ,\hfill \end{array}$$

(9)

where N is the number of pairs $(k,m)$ such that $0<k<m$ and $k+m=n.$ It is easy to check that $N=\lfloor {\displaystyle \frac{n-1}{2}}\rfloor .$ For every fixed $n,$ relations (9) give a polynomial automorphism

$$(P_1,P_2,\dots ,P_n)\mapsto \left(\Theta (Z_1),\Theta (Z_2),\dots ,\Theta (Z_n)\right).$$

Since $P_1,P_2,\dots ,P_n$ are algebraically independent for every $n,$ the same is true for $\Theta (Z_1),\Theta (Z_2),\dots ,\Theta (Z_n).$ By Theorem 3, polynomials $Z_n,$ $n\in \mathbb{N}$ are algebraically independent.

The item (ii) can be proved by the same way taking into account that

$$\begin{array}{cc}\Theta (P_1)\hfill & =2P_1,\hfill \\ \Theta (P_{1,1})\hfill & =2{(P_1)}^2-P_2,\hfill \\ \Theta (P_{1,2})\hfill & =2P_1{P}_2-P_3,\hfill \\ & \cdots \hfill \\ \Theta (P_{1,n})\hfill & =2P_1{P}_n-P_{n+1},\hfill \\ & \cdots .\hfill \end{array}$$

□

Note that both sequences $(Z_n)$ and $(P_{1,m})$ of algebraically independent subsymmetric polynomials in Corollary 1 do not contain symmetric polynomials, excepting $P_1$ and $P_2.$ If we set $R_n=P_n$ if n is odd and $R_n=Z_n$ if n is even, we can show using Theorem 3 for $\Phi =\Theta$ that polynomials $R_n$ are algebraically independent. But we do not know the following: Is there an algebraically independent sequence of subsymmetric polynomials containing polynomials $P_n$ for all $n\in \mathbb{N}$ and an infinite subsequence of nonsymmetric polynomials?

4. Extension of Subsymmetric Polynomials

4.1. Extension to Banach Spaces ${\ell }_p$

Throughout the rest of the paper we consider spaces ${\ell }_p$ only for $1\le p<\infty .$ The linear space $c_{00}$ is dense in the Banach space ${\ell }_p$ for every $1\le p<\infty .$ Thus, if a subsymmetric polynomial P on $c_{00}$ is continuous in the topology of ${\ell }_p,$ we can extend it to a continuous subsymmetric polynomial on ${\ell }_p.$ As we observed the standard subsymmetric polynomials $P_{{\alpha }_1,\dots ,{\alpha }_n}$ are well-defined and continuous on ${\ell }_p$ if and only if each ${\alpha }_j\ge \lceil p\rceil$ [19]. In particular, each subsymmetric polynomial on $c_{00}$ can be extended to a subsymmetric polynomial on ${\ell }_1$ so that for every $x\in {\ell }_1,$

$$P_{{\alpha }_1,\dots ,{\alpha }_n}(x)=\sum _{i_1<\cdots <i_n}{x}_{i_1}^{{\alpha }_1}\cdots x_{i_n}^{{\alpha }_n}.$$

We denote by ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ the algebra of all continuous subsymmetric polynomials on ${\ell }_p.$

For every $x\in {\ell }_p$ the mapping

$${\delta }_x:P\mapsto P(x)$$

is called a point evaluation complex homomorphism of ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$ As in the case of $c_{00},$ we define the equivalence on ${\ell }_p$ so that $x\simeq y$ if there are operators ${\tau }_1$ and ${\tau }_2$ in $\mathfrak{S}$ such that ${\tau }_1(x)={\tau }_2(y).$ Similarly, $x\sim y$ if there are operators ${\sigma }_1$ and ${\sigma }_2$ in S such that ${\sigma }_1(x)={\sigma }_2(y).$ The quotient set ${\mathcal{M}}_p^S={\ell }_p/\sim$ can be associated with the set of infinite multisets $[x]$ consisting of nonzero complex numbers $x_i$ such that ${\sum }_{i=1}^{\infty }{|x_i|}^p<\infty .$ The following proposition was proved in [4].

Proposition 5. 

Let $x,$ $y\in {\ell }_p.$ Then $x\sim y$ in ${\ell }_p$ if and only if there exists a natural number $m\ge \lceil p\rceil$ such that $P_n(x)=P_n(y)$ for every $n\ge m.$ In this case, of course, $P(x)=P(y)$ for every symmetric polynomial $P.$

The following result extends Theorem 2 to ${\ell }_p.$

Theorem 4. 

Let $x,$ $y\in {\ell }_p.$ Then $x\simeq y$ if there exists a natural number $m\ge \lceil p\rceil$ such that $P_n(x)=P_n(y)$ and $P_{n,2n}(x)=P_{n,2n}(y)$ for every $n\ge m.$ In this case, $P(x)=P(y)$ for every subsymmetric polynomial $P.$

Proof. 

If $x\simeq y,$ then it is trivially true that $P(x)=P(y)$ for all $P\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$ Conversely, from Proposition 5 it follows that $x\sim y,$ and removing zero coordinates, we may assume $x_i$ are elements of the multiset $[x]$ and $y_j$ are elements of the multiset $[y],$ $i,$ $j\in \mathbb{N}.$ As in Theorem 2, we consider the following multisets

$$M_{1,2}(x)=\{x_i{x}_j^2:i<j\}\mathrm{and}{M}_{1,2}(y)=\{y_i{y}_j^2:i<j\}.$$

We claim that if $(x_1,x_2,\dots )\ne (y_1,y_2,\dots )$ (as vectors) then $M_{1,2}(x)\ne M_{1,2}(y)$ (as multisets). Indeed, suppose that there are $x_r\ne x_s$ such that $r<s$ and a bijection $\sigma :\mathbb{N}\to \mathbb{N}$ such that $(y_1,y_2,\dots )=(x_{\sigma (1)},x_{\sigma (2)},\dots )$ and $\sigma (r)>\sigma (s).$ Then, as in the proof of Theorem 2, the symmetric difference $M_{1,2}(x)▵M_{1,2}(y)$ is nonempty, because it contains elements $x_r{x}_s^2$ and $x_s{x}_r^2.$ Applying Proposition 5 to multisets $u=M_{1,2}(x)$ and $v=M_{1,2}(y)$ we have that for every $m\ge \lceil p\rceil$ there is $n\ge m$ such that $P_n(u)\ne P_n(v).$ But

$$P_n(u)=\sum _{i<j}{\left(x_i{x}_j^2\right)}^n=\sum _{i<j}{x}_i^n{x}_j^{2n}=P_{n,2n}(x),$$

and so $P_{n,2n}(x)\ne P_{n,2n}(y).$ This is a contradiction. Thus, $M_{1,2}(x)=M_{1,2}(y),$ that is, $x\simeq y.$ In particular, $P(x)=P(y)$ for every subsymmetric polynomial $P.$ □

From Theorem 4 it follows that $x\simeq y$ if and only if ${\delta }_x={\delta }_y$ on ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$

It is easy to see that if $x\in {\ell }_p$ has infinitely many nonzero coordinates, then the element $\check{x}$ does not exist in ${\ell }_p.$ But we can define a complex homomorphism ${\delta }_{\check{x}}$ by

$${\delta }_{\check{x}}(P)=\underset{k\to \infty }{lim}P(x_k,x_{k-1},\dots ,x_1,0,0,\dots ),P\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$$

Similarly, if x and y are in ${\ell }_p,$ the $x⊲y$ may be not defined as a vector in ${\ell }_p,$ but we can define

$${\delta }_{x⊲y}(P)=\underset{k\to \infty }{lim}P(x_1,x_2,\dots ,x_k,y_1,y_2,\dots ),P\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$$

Moreover, we can define

$${\delta }_{\check{x}⊲y}(P)=\underset{k\to \infty }{lim}P(x_k,x_{k-1},\dots ,x_1,y_1,y_2,\dots ),P\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$$

Proposition 6. 

Complex homomorphisms ${\delta }_{\check{x}},$ ${\delta }_{x⊲y},$ and ${\delta }_{\check{x}⊲y}$ are well-defined on ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ for every x and y in ${\ell }_p.$ Moreover,

$${\delta }_{\check{x}}(P_{{\alpha }_1,\dots ,{\alpha }_n})=P_{{\alpha }_n,\dots ,{\alpha }_1}(x),$$

and

$${\delta }_{x⊲y}=\sum _{j=0}^{n+1}{P}_{{\alpha }_1,\dots ,{\alpha }_j}(x)P_{{\alpha }_{j+1},\dots ,{\alpha }_n}(y)$$

for every standard polynomial $P_{{\alpha }_1,\dots ,{\alpha }_n}\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$

Proof. 

Let $x^{(k)}=(x_1,x_2,\dots ,x_k,0,0,\dots ).$ For every fixed k and a standard polynomial $P_{{\alpha }_1,\dots ,{\alpha }_n},$

$$P_{{\alpha }_1,\dots ,{\alpha }_n}\left({(x^{(k)})}^{\vee }\right)=P_{{\alpha }_n,\dots ,{\alpha }_1}\left(x^{(k)}\right).$$

Since $x^{(k)}\to x$ as $k\to \infty$, from the continuity of $P_{{\alpha }_n,\dots ,{\alpha }_1}$ it follows

$${\delta }_{\check{x}}(P_{{\alpha }_1,\dots ,{\alpha }_n})=\underset{k\to \infty }{lim}{P}_{{\alpha }_n,\dots ,{\alpha }_1}\left(x^{(k)}\right)=P_{{\alpha }_n,\dots ,{\alpha }_1}(x).$$

Since every polynomial in ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ is a finite linear combination of standard polynomials, ${\delta }_{\check{x}}$ is well-defined on ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$

Similarly, using equality (7) and the density of $c_{00}$ in ${\ell }_p$ we can prove the rest of the proposition concerning ${\delta }_{x⊲y},$ and ${\delta }_{\check{x}⊲y}.$ □

For every $P\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ we define $\check{P}(x)={\delta }_{\check{x}}(P).$ Clearly, if $x\in c_{00},$ then ${\delta }_{\check{x}}(P)=P(\check{x}).$

Corollary 2. 

The mapping $P\mapsto \check{P}$ is an involution of the algebra ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ and $\parallel P\parallel =\parallel \check{P}\parallel .$

Proof. 

We already have

$${\check{P}}_{{\alpha }_1,\dots ,{\alpha }_n}(x)={\delta }_{\check{x}}(P_{{\alpha }_1,\dots ,{\alpha }_n})=P_{{\alpha }_n,\dots ,{\alpha }_1}(x).$$

Thus, the homomorphism $P\mapsto \check{P}$ is well-defined for every $P\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$ Also, ${(\check{P})}^{\vee }=P,$ that is, P is an involution. Since $c_{00}$ is dense in ${\ell }_p$ and $\parallel x\parallel =\parallel \check{x}\parallel$ in ${\ell }_p$ for every $x\in c_{00}\subset {\ell }_p,$

$$\parallel \check{P}\parallel =\{sup|\check{P}(x)|:x\in c_{00}\subset {\ell }_p,\parallel x\parallel \le 1\}=\{sup|\check{P}(\check{x})|:x\in c_{00}\subset {\ell }_p,\parallel x\parallel \le 1\}=\parallel P\parallel .$$

□

4.2. Extension to ${\ell }_p(\mathfrak{A})$ for a Well-Ordered Set $\mathfrak{A}$

Let us recall that a set $\mathfrak{A}$ is well-ordered if it is linearly ordered and every nonempty subset of $\mathfrak{A}$ has a minimal element. Throughout this subsection we assume that $\mathfrak{A}$ is an infinite well-ordered set with respect to an order $⪯.$ The Banach space ${\ell }_p(\mathfrak{A})$ is the space of functions

$$x:\mathfrak{A}\to \mathbb{C},x={(x_{\gamma })}_{\gamma \in \mathfrak{A}}$$

such that

$$\parallel x\parallel :={\left(\sum _{\gamma \in \mathfrak{A}}{|x_{\gamma }|}^p\right)}^{1/p}<\infty .$$

Clearly, for any $x\in {\ell }_p(\mathfrak{A})$ only a countable number of coordinates $x_{\gamma }$ is not equal to zero.

Let

$$\iota :\mathbb{N}\to \mathfrak{A}$$

be an embedding of the set of natural numbers with the natural order ≤ to $\mathfrak{A}$ such that if $n_1\le n_2$ then $\iota (n_1)⪯\iota (n_2).$ Thus, ${\ell }_p$ can be considered as a subspace of ${\ell }_p(\mathfrak{A})$ consisting of vectors

$$\left(x_{\iota (1)},x_{\iota (2)},\dots ,x_{\iota (n)},\dots \right).$$

Let $P_{{\alpha }_1,\dots ,{\alpha }_n}$ be a standard polynomial. We can write

$$P_{{\alpha }_1,\dots ,{\alpha }_n}(x_{\iota (1)},x_{\iota (2)},\dots )=\sum _{i_1<\cdots <i_n}{x}_{\iota (i_1)}^{{\alpha }_1}\cdots x_{\iota (i_n)}^{{\alpha }_n}.$$

Let us denote by $P_{{\alpha }_1,\dots ,{\alpha }_n}^{\mathfrak{A}}$ the following polynomial on ${\ell }_p(\mathfrak{A}),$

$$P_{{\alpha }_1,\dots ,{\alpha }_n}^{\mathfrak{A}}(x)=\sum _{{\gamma }_1\prec \cdots \prec {\gamma }_n}{x}_{{\gamma }_1}^{{\alpha }_1}\cdots x_{{\gamma }_n}^{{\alpha }_n}.$$

(10)

Clearly, $P_{{\alpha }_1,\dots ,{\alpha }_n}^{\mathfrak{A}}$ is an extension of $P_{{\alpha }_1,\dots ,{\alpha }_n}$ and this extension does not depend on the embedding $\iota .$ Since every subsymmetric polynomial is a linear combination of standard polynomials, we have the following proposition.

Proposition 7. 

Every subsymmetric polynomial P on ${\ell }_p$ can be extended to a polynomial $P^{\mathfrak{A}}$ on ${\ell }_p(\mathfrak{A}).$ The operator of extension $\mathcal{J}:P\mapsto P^{\mathfrak{A}}$ is linear and$P_{{\alpha }_1,\dots ,{\alpha }_n}^{\mathfrak{A}}$is defined by (10).

Let us denote by ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p(\mathfrak{A}))$ the range of ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ under the mapping $\mathcal{J}.$ We can see that ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p(\mathfrak{A}))$ is an algebra and $\mathcal{J}$ is an algebra isomorphism. In particular, for every point $u\in {\ell }_p(\mathfrak{A}),$ the functional $P\mapsto P^{\mathfrak{A}}(u)$ is a complex homomorphism.

Example 4. 

As we mentioned above, there is no point z in ${\ell }_p$ such that $P(z)={\delta }_{x⊲y}(P)$ if $y\ne 0$ and x has infinitely many nonzero coordinates. However, such a point exists in ${\ell }_p(\mathfrak{A})$ for some set of indexes $\mathfrak{A}.$ Indeed, let $\mathfrak{A}={\mathbb{N}}_1\cup {\mathbb{N}}_2,$ where ${\mathbb{N}}_1$ and ${\mathbb{N}}_2$ are copies on natural numbers with the usual order, and if $i\in {\mathbb{N}}_1$ and $j\in {\mathbb{N}}_2,$ then $i\prec j.$ For given x and y in ${\ell }_p$ we set

$$u=(x_1,\dots ,x_i,\dots )+(y_1,\dots ,y_j,\dots )\in {\ell }_p({\mathbb{N}}_1\cup {\mathbb{N}}_2),i\in {\mathbb{N}}_1,j\in {\mathbb{N}}_2.$$

Then ${\delta }_{x⊲y}(P)=P(u).$ Note that ${\parallel u\parallel }^p={\parallel x\parallel }^p+{\parallel y\parallel }^p.$

Proposition 8. 

For every $P\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p),$ $\parallel P\parallel =\parallel P^{\mathfrak{A}}\parallel .$

Proof. 

Since ${\ell }_p$ is isometrically embedded into ${\ell }_p(\mathfrak{A}),$ $\parallel P\parallel \le \parallel P^{\mathfrak{A}}\parallel .$ For every $\epsilon >0$ there is a vector $u=(u_{{\gamma }_1},\dots ,u_{{\gamma }_n},\dots )$ in ${\ell }_p(\mathfrak{A}),$ $\parallel u\parallel =1$ such that $|P^{\mathfrak{A}(u)}|\ge \parallel P^{\mathfrak{A}}\parallel -\epsilon .$ Let $x\in {\ell }_p$ be such that $x_n=u_{{\gamma }_n},$ $n\in \mathbb{N}.$ Then

$${\parallel x\parallel }^p=\sum _{n=1}^{\infty }|u_{{\gamma }_n}{|}^p={\parallel u\parallel }^p=1.$$

Also, by the definition of the extension operator $\mathcal{J},$ $P(x)=P^{\mathfrak{A}}(u)$ and so $|P(x)|\ge \parallel P^{\mathfrak{A}}\parallel -\epsilon .$ Since such a vector exists for every $\epsilon >0,$ $\parallel P\parallel \ge \parallel P^{\mathfrak{A}}\parallel .$ Thus, $\parallel P\parallel =\parallel P^{\mathfrak{A}}\parallel .$ □

5. Algebras of Subsymmetric Analytic Functions

5.1. General Properties of the Algebra of Subsymmetric Analytic Functions of Bounded Type

Let $B_p^r$ be the ball in ${\ell }_p$ of radius $r>0$ and centered at the origin. Denote by $H_{u\mathfrak{S}}(B_p^r)$ the completion of ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ with respect to the norm

$${\parallel P\parallel }_r=\underset{{\parallel x\parallel }_{{\ell }_p}\le r}{sup}|P(x)|.$$

$H_{u\mathfrak{S}}(B_p^r)$ is a Banach algebra of uniformly continuous subsymmetric analytic functions on $B_p^r.$ It is a closed subalgebra of the algebra of all uniformly continuous analytic functions on $B_p^r,$ $H_u(B_p^r).$ Also, we denote by $H_{b\mathfrak{S}}({\ell }_p)$ the projective limit of algebras $H_{u\mathfrak{S}}(B_p^r)$ as $r\to \infty .$ In other words, $H_{b\mathfrak{S}}({\ell }_p)$ is a locally convex Fréchet algebra consisting of subsymmetric entire functions of bounded type (i.e., bounded on each bounded subset), and

$$H_{b\mathfrak{S}}({\ell }_p)=\bigcap _{r>0}{H}_{u\mathfrak{S}}(B_p^r).$$

Any function $f\in H_{b\mathfrak{S}}({\ell }_p)$ can be expressed using the Taylor series expansion

$$f(x)=\sum _{n=0}^{\infty }{f}_n(x),x\in {\ell }_p,$$

where $f_n$ are n-homogeneous subsymmetric polynomials. The Taylor series converges to f uniformly on every bounded subset. Moreover, a formal series ${\sum }_{n=0}^{\infty }{f}_n(x)$ of n-homogeneous polynomials converges to an entire function of bounded type if and only if the radius of uniform converges at zero,

$${\varrho }_0(f)={\left(\underset{n\to \infty }{lim\; sup}{\parallel f_n\parallel }^{1/n}\right)}^{-1}=\infty .$$

Note that $H_{b\mathfrak{S}}({\ell }_p)$ is a closed subalgebra of the algebra $H_b({\ell }_p)$ of all entire functions of bounded type and $H_{b\mathfrak{S}}({\ell }_p)$ contains the closed subalgebra $H_{bs}({\ell }_p)$ of all symmetric entire functions of bounded type.

Let $\mathfrak{A}$ be an infinite well-ordered set. For every function $f\in H_{b\mathfrak{S}}({\ell }_p)$ we assign

$$f^{\mathfrak{A}}=\sum _{n=0}^{\infty }{f}_n^{\mathfrak{A}},$$

and denote by $H_{b\mathfrak{S}}({\ell }_p(\mathfrak{A}))$ the set $\{f^{\mathfrak{A}}:f\in H_{b\mathfrak{S}}({\ell }_p)\}.$

Proposition 9. 

For every $f\in H_{b\mathfrak{S}}({\ell }_p),$ $f^{\mathfrak{A}}$ is an entire function of bounded type on ${\ell }_p(\mathfrak{A})$ and the mapping $f\mapsto f^{\mathfrak{A}}$ is a topological isomorphism of algebras such that

$${\parallel f\parallel }_r={\parallel f^{\mathfrak{A}}\parallel }_r,r>0.$$

Proof. 

From Proposition 8 it follows that $\parallel f_n^{\mathfrak{A}}{\parallel }_r={\parallel f_n\parallel }_r$ for every $r>0.$ Thus ${\varrho }_0(f^{\mathfrak{A}})=\varrho (f)=\infty$ and so $f^{\mathfrak{A}}$ is an entire function of bounded type. Clearly, ${(fg)}^{\mathfrak{A}}=f^{\mathfrak{A}}{g}^{\mathfrak{A}}$ and ${(f+g)}^{\mathfrak{A}}=f^{\mathfrak{A}}+g^{\mathfrak{A}}.$ Also, for every $r>0,$

$$\parallel f^{\mathfrak{A}}{\parallel }_r=\underset{m\to \infty }{lim}\parallel \sum _{n=0}^m{f}_n^{\mathfrak{A}}{\parallel }_r=\underset{m\to \infty }{lim}\parallel \sum _{n=0}^m{f}_n{\parallel }_r={\parallel f\parallel }_r.$$

□

5.2. Spectrum of $H_{b\mathfrak{S}}({\ell }_p)$

Let us denote by $M_{b\mathfrak{S}}({\ell }_p)$ the spectrum (i.e., the set of continuous complex homomorphisms (characters)) of $H_{b\mathfrak{S}}({\ell }_p).$ The set $M_{b\mathfrak{S}}({\ell }_p)$ is the inductive limit of the spectra of $H_{u\mathfrak{S}}(B_p^r),$ $M_{u\mathfrak{S}}(B_p^r)$ as $r\to \infty ,$ in particular,

$$M_{b\mathfrak{S}}({\ell }_p)=\bigcup _{r>0}{M}_{u\mathfrak{S}}(B_p^r)$$

and each $M_{u\mathfrak{S}}(B_p^r)$ is a compact subset of $M_{b\mathfrak{S}}({\ell }_p)$ (cf. [20]). Thus, every character $\phi \in M_{b\mathfrak{S}}({\ell }_p)$ belongs to $M_{u\mathfrak{S}}(B_p^r)$ for some $r>0.$ The infimum of $r>0$ such that $\phi \in M_{u\mathfrak{S}}(B_p^r)$ is called the radius function of $\phi$ and denoted by $R(\phi ).$ Thus, for every character $\phi ,$ the radius function is well-defined and finite. Let ${\phi }_n$ be the restriction of $\phi$ to the normed space ${\mathcal{P}}_{\mathfrak{S}}{(}^n{\ell }_p)$ of subsymmetric n-homogeneous polynomials and

$$\parallel {\phi }_n\parallel =\{sup|{\phi }_n{(P)|:\parallel P\parallel }_1\le 1\}.$$

It is known that

$$R(\phi )=\underset{n\to \infty }{lim\; sup}{\parallel {\phi }_n\parallel }^{1/n}.$$

(11)

Formula (11) was proved in [20] for characters on the algebra $H_b(X)$ for any Banach space X and in [10] for characters on any subalgebra of $H_b(X).$ From (11) it follows that if $\phi$ is a linear multiplicative functional on ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ such that ${lim\; sup}_{n\to \infty }{\parallel {\phi }_n\parallel }^{1/n}=r_0<\infty ,$ then it is bounded in $H_{u\mathfrak{S}}(B_p^r)$ for every $0<r<r_0.$ Thus, $\phi$ can be extended to a continuous complex homomorphism of $H_{u\mathfrak{S}}(B_p^r).$ In particular, the extension of $\phi$ is in $M_{b\mathfrak{S}}({\ell }_p).$ This can be formulated as following.

Proposition 10. 

A linear multiplicative functional φ on ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p)$ can be extended to a character in $M_{b\mathfrak{S}}({\ell }_p)$ if and only if

$$R(\phi )=\underset{n\to \infty }{lim\; sup}{\parallel {\phi }_n\parallel }^{1/n}<\infty .$$

Theorem 5. 

The following linear multiplicative functionals are continuous on $H_{b\mathfrak{S}}({\ell }_p).$

(i) 

${\delta }_x$ for every $x\in {\ell }_p.$ Moreover, $R({\delta }_x)=\parallel x\parallel .$

(ii) 

${\delta }_u$ for every $u\in {\ell }_p(\mathfrak{A})$ for every well-ordered set $\mathfrak{A}.$ Moreover, $R({\delta }_u)=\parallel u\parallel .$

(iii) 

${\delta }_{x⊲y}$ for all $x,$ $y\in {\ell }_p.$ Moreover, $R({\delta }_{x⊲y})=\sqrt[p]{{\parallel x\parallel }^p+{\parallel y\parallel }^p}.$

(iv) 

${\delta }_{\check{x}}$ for every $x\in {\ell }_p.$ Moreover, $R({\delta }_{\check{x}})=\parallel x\parallel .$

Proof. 

According to Proposition 10, it is enough to compute radius functions for the complex homomorphisms.

(i) From the definition of radius function we have $R({\delta }_x)\le \parallel x\parallel .$ On the other hand, in [7] it is proved that in the algebra of entire symmetric function of bounded type $R({\delta }_x)=\parallel x\parallel .$ Since $H_{bs}({\ell }_p)\subset H_{b\mathfrak{S}}({\ell }_p)$ we have $R({\delta }_x)=\parallel x\parallel$ in $H_{b\mathfrak{S}}({\ell }_p).$

(ii) From Propositions 8 and 9 we have that

$${\delta }_u:f\mapsto f^{\mathfrak{A}}(u),f\in H_{b\mathfrak{S}}({\ell }_p)$$

is a complex homomorphism and $R({\delta }_u)=\parallel u\parallel .$

(iii) As we observed in Example 4, there is a well-ordered set $\mathfrak{A}$ and $u\in \mathfrak{A}$ such that ${\delta }_{x⊲y}={\delta }_u,$ and ${\parallel u\parallel }^p={\parallel x\parallel }^p+{\parallel y\parallel }^p.$ Thus,

$$R({\delta }_{x⊲y})=R({\delta }_u)=\parallel u\parallel =\sqrt[p]{{\parallel x\parallel }^p+{\parallel y\parallel }^p}.$$

(iv) Let ${({\delta }_{\check{x}})}_n$ be the restriction of ${\delta }_{\check{x}}$ to the subspace ${\mathcal{P}}_{\mathfrak{S}}{(}^n{\ell }_p)$ of n-homogeneous subsymmetric polynomials. Then

$${({\delta }_{\check{x}})}_n(P)={({\delta }_x)}_n(\check{P})$$

and so $\parallel {({\delta }_{\check{x}})}_n\parallel =\parallel {({\delta }_x)}_n\parallel .$ Taking into account (11), $R({\delta }_{\check{x}})=R({\delta }_x)=\parallel x\parallel .$ □

5.3. A Convolution on the Spectrum

For a Banach space X and a vector $x\in X$, the translation operator $T_x$ on $H_b(X)$ is defined by

$$(T_{x}f)(y)=f(x+y),f\in H_b(X).$$

Using the translation operator in [20] (see also [22]) a convolution operation $\phi \ast \theta$ was constructed for linear functionals $\phi$ and $\theta$ in $H_b^{\ast }(X)$ by

$$(\phi \ast \theta )(f)=\phi \left(\theta (T_{x}f)\right)=\phi (g),\mathrm{where}g(x)=\theta (T_x(f))$$

and proved that $\phi \ast \theta \in H_b^{\ast }(X).$ Here $H_b^{\ast }(X)$ is the space of all continuous linear functionals on $H_b(X).$ Moreover, if $\phi$ and $\theta$ are characters of $H_b(X),$ then $\phi \ast \theta$ is so (Theorem 6.9. in [20]), and by Lemma 6.2. in [20], $R(\phi \ast \theta )\le R(\phi )+R(\theta ).$ Note that the subalgebra $H_{b\mathfrak{S}}({\ell }_p)\subset H_b({\ell }_p)$ is not invariant with respect to $T_x.$ Thus, $T_x$ cannot to be used for a construction of a convolution on $H_{b\mathfrak{S}}({\ell }_p).$ In [6], they proposed a symmetric translation operator $({\mathcal{T}}_{x}f)(y)=f(x•y)$ on the algebra of symmetric entire functions of bounded type. In this section we consider a convolution on $H_{b\mathfrak{S}}({\ell }_p)$ based on the following “subsymmetric translation”, $({\mathfrak{T}}_{x}f)=f(x⊲y).$

Theorem 6. 

For every $\theta \in H_{b\mathfrak{S}}^{\ast }({\ell }_p)$ and $f\in H_{b\mathfrak{S}}({\ell }_p)$ the function $g(x)=\theta ({\mathfrak{T}}_{x}f)$ is in $H_{b\mathfrak{S}}({\ell }_p).$ If $\theta \in M_{b\mathfrak{S}}({\ell }_p),$ then the map $f\mapsto g$ is a continuous homomorphism of $H_{b\mathfrak{S}}({\ell }_p)$ and for every $\phi \in M_{b\mathfrak{S}}({\ell }_p),$

$$(\phi ⊲\theta )(f):=\phi \left(\theta ({\mathfrak{T}}_{x}f)\right)=\phi (g)$$

(12)

belongs to $M_{b\mathfrak{S}}({\ell }_p).$

Proof. 

Let $\mathfrak{A}={\mathbb{N}}_1\cup {\mathbb{N}}_2$ be the well-ordered set as in Example 4. The space ${\ell }_p(\mathfrak{A})$ has a Schauder basis $\{e_i^1\}\cup \{e_j^2\},$ $i\in {\mathbb{N}}_1,$ $j\in {\mathbb{N}}_2,$

$$e_i^1=(\underset{i\in {\mathbb{N}}_1}{\underbrace{0,\dots ,0,1}},0,\dots ),e_i^2=(\underset{j\in {\mathbb{N}}_2}{\underbrace{0,\dots ,0,1}},0,\dots ).$$

We denote by ${\theta }^{\mathfrak{A}}$ the extension of the linear functional $\theta$ to a linear functional on $H_{b\mathfrak{S}}({\ell }_p(\mathfrak{A}))$ defined as

$${\theta }^{\mathfrak{A}}(P_{{\alpha }_1,\dots ,{\alpha }_m}^{\mathfrak{A}})=\theta (P_{{\alpha }_1,\dots ,{\alpha }_m}).$$

Since every homogeneous subsymmetric polynomial is a finite linear combination of standard polynomials,

$${\theta }^{\mathfrak{A}}(f^{\mathfrak{A}})=\underset{m\to \infty }{lim}\left(\sum _{n=1}^m{\theta }^{\mathfrak{A}}(f_n^{\mathfrak{A}})\right)=\underset{m\to \infty }{lim}\left(\sum _{n=1}^m\theta (f_n)\right)=\theta (f).$$

On the other hand, as we observed in Example 4,

$$({\mathfrak{T}}_{x}f)(y)={\delta }_{x⊲y}(f)=f^{\mathfrak{A}}(\tilde{x}+\tilde{y})=(T_{\tilde{x}}f)(\tilde{y}),$$

where

$$\tilde{x}=\sum _{i\in {\mathbb{N}}_1}{x}_i{e}_i^1,\mathrm{and}\tilde{y}=\sum _{j\in {\mathbb{N}}_2}{y}_j{e}_j^2.$$

Note that Theorem 6.1. in [20] asserts that for fixed $\phi \in H_b^{\ast }(X)$ and $f\in H_b(X)$ the function $x\mapsto \phi (T_x(f))$ belongs to $H_b(X).$ Thus, by Theorem 6.1. in [20], the function $\tilde{x}\mapsto {\theta }^{\mathfrak{A}}\left(T_{\tilde{x}}{f}^{\mathfrak{A}}\right)$ is an entire function of bounded type on ${\ell }_p(\mathfrak{A}).$ Since $\parallel \tilde{x}\parallel =\parallel x\parallel$ and $f\mapsto f^{\mathfrak{A}}$ is a topological isomorphism of algebras, the function $g(x)={\theta }^{\mathfrak{A}}\left(T_{\tilde{x}}{f}^{\mathfrak{A}}\right)$ is an entire function of bounded type on ${\ell }_p.$ Clearly, for every operator ${\mathcal{C}}_j$ as in (1), $g({\mathcal{C}}_j(x))=g(x).$ Hence, $g\in H_{b\mathfrak{S}}({\ell }_p).$ So, for every linear continuous functional $\phi ,$ the functional

$$(\phi ⊲\theta )(f)=\phi (g)=\phi \left(\theta ({\mathfrak{T}}_{x}f)\right)$$

is well-defined. On the other hand,

$$(\phi ⊲\theta )(f)=({\phi }^{\mathfrak{A}}\ast {\theta }^{\mathfrak{A}})(f^{\mathfrak{A}}).$$

By 6.2. Lemma in [20],

$$R({\phi }^{\mathfrak{A}}\ast {\theta }^{\mathfrak{A}})\le R({\phi }^{\mathfrak{A}})+R({\theta }^{\mathfrak{A}})$$

and so ${\phi }^{\mathfrak{A}}\ast {\theta }^{\mathfrak{A}}$ is continuous. Thus, $\phi ⊲\theta$ is continuous as well. Moreover, if both $\phi$ and $\theta$ are characters, then ${\phi }^{\mathfrak{A}}$ and ${\theta }^{\mathfrak{A}}$ are characters too and, by Theorem 6.9. in [20], ${\phi }^{\mathfrak{A}}\ast {\theta }^{\mathfrak{A}}$ is a character. Thus, $\phi ⊲\theta$ is a character. Finally, since for every $\phi \in M_{b\mathfrak{S}}({\ell }_p)$ the function $f\mapsto \phi (g)=\phi \left(\theta ({\mathfrak{T}}_{x}f)\right)$ is a complex homomorphism of $H_{b\mathfrak{S}}({\ell }_p)$ and $H_{b\mathfrak{S}}({\ell }_p)$ is a semisimple commutative algebra, the map $f\mapsto g$ is a continuous homomorphism of $H_{b\mathfrak{S}}({\ell }_p)$ to itself. □

From the definition of $\phi ⊲\theta$ (Equation (12)) it follows that ${\delta }_{x⊲y}={\delta }_x⊲{\delta }_y.$

Corollary 3. 

The linear multiplicative functional ${\delta }_{\check{x}⊲x}={\delta }_{\check{x}}⊲{\delta }_x$ is continuous for every $x\in {\ell }_p.$

Proposition 11. 

Let $\phi ,$ $\theta \in H_{b\mathfrak{S}}^{\ast }({\ell }_p).$ Then

$$(\phi ⊲\theta )(P_{{\alpha }_1,\dots ,{\alpha }_m})=\sum _{j=0}^{m+1}\phi (P_{{\alpha }_1,\dots ,{\alpha }_j})\theta (P_{{\alpha }_{j+1},\dots ,{\alpha }_m}).$$

Proof. 

From the density of $c_{00}$ in ${\ell }_p$ it follows that formula (7) is true for polynomials in ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$ Thus, it is enough to apply $\phi ⊲\theta$ to $P_{{\alpha }_1,\dots ,{\alpha }_m}$ by the definition and taking into account that $({\mathfrak{T}}_x{P}_{{\alpha }_1,\dots ,{\alpha }_m})(y)=P_{{\alpha }_1,\dots ,{\alpha }_m}(x⊲y)$ can be represented by (7). □

5.4. “Exceptional” Characters of $H_{b\mathfrak{S}}({\ell }_p)$

The complex homomorphism ${\delta }_{x⊲y}$ is not a point evaluation functional on $H_{b\mathfrak{S}}({\ell }_p)$ but it is the evaluation at a point in ${\ell }_p(\mathfrak{A}),$ and the character ${\delta }_{\check{x}}$ cannot be represented as an evaluation at any point of ${\ell }_p(\mathfrak{A}).$ However, the restriction of ${\delta }_{\check{x}}$ to the subalgebra $H_{bs}({\ell }_p)$ coincides with ${\delta }_x.$ It is known (see e.g., [6]) that for a natural number $p>1$, there exists a family of characters of $H_{bs}({\ell }_p)$ (so called exceptional characters) that are not point evaluation functionals. Let us show that the construction of such functionals works for subsymmetric analytic functions as well.

Let us recall the definition of an ultrafilter. A filter on a nonempty subset D is a family $\mathfrak{F}$ of subsets of D with the following properties:

(1)

$D\in \mathfrak{F}$ and $⌀\notin \mathfrak{F};$

(2)

If $A,$ $B\in \mathfrak{F},$ then $A\cap B\in \mathfrak{F};$

(3)

If $A\in F$ and $A\subseteq B\subseteq D,$ then $B\in \mathfrak{F}.$

An ultrafilter on D is a filter on D which is not properly contained in any other filter on $D.$ An ultrafilter is free if there is no point $x\in D$ such that $x\in A$ for every $A\in \mathfrak{F}.$ It is well-known that for every free ultrafilter $\mathcal{U}$ on the set of natural numbers and for every bounded sequence $(x_n),$ there exists a unique limit

$$w=\underset{\mathcal{U}}{lim}{x}_n,$$

which we understand so that for every $\epsilon >0$ there is $A\subset \mathfrak{F}$ such that $|w-x_n|<\epsilon$ if $n\in A.$ For details on ultrafilters we refer the reader to [37].

For a given complex number $\lambda$ and $p\in \mathbb{N}$, we consider a sequence

$$v_n(\lambda )=\left(\underset{n}{\underbrace{{\displaystyle \frac{\lambda }{\sqrt[p]{n}}},\dots ,{\displaystyle \frac{\lambda }{\sqrt[p]{n}}}}},0,\dots \right).$$

Since $\parallel v_n(\lambda )\parallel =|\lambda |,$ $R\left({\delta }_{v_n(\lambda )}\right)=|\lambda |.$ The sequence ${\delta }_{v_n(\lambda )}$ has a cluster point (not unique) in the spectrum $M_b({\ell }_p)$ of the algebra of all entire functions of bounded type, $H_b({\ell }_p).$ More precisely, for any free ultrafilter $\mathcal{U}$ on $\mathbb{N}$ there exists a character ${\psi }_{\lambda }^{\mathcal{U}}\in M_b({\ell }_p)$ defined by

$${\psi }_{\lambda }^{\mathcal{U}}(f)=\underset{\mathcal{U}}{lim}{\delta }_{v_n(\lambda )}(f)=\underset{\mathcal{U}}{lim}f(v_n(\lambda )).$$

However, the restriction of any such cluster point to the subalgebra $H_{bs}({\ell }_p)$ defines a unique character ${\psi }_{\lambda }$ (does not depend on $\mathcal{U}$) such that ${\psi }_{\lambda }(P_p)={\lambda }^p$ and ${\psi }_{\lambda }(P_k)=0$ for $k>p.$ It is known [4,6] that ${\psi }_{\lambda }$ is not a point evaluation functional on $H_{bs}({\ell }_p).$ Let us consider the action of this character on $H_{b\mathfrak{S}}({\ell }_p).$ From simple combinatorial arguments we observe that for $n\ge m,$

$$P_{{\alpha }_1,\dots ,{\alpha }_m}(\underset{n}{\underbrace{1,\dots ,1}},0,\dots )=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)={\displaystyle \frac{n!}{m!(n-m)!}}.$$

Thus, in ${\ell }_p,$ $p\in \mathbb{N}$ for $n\ge m$ we have

$$\begin{array}{cc}\hfill P_{{\alpha }_1,\dots ,{\alpha }_m}(v_n(\lambda ))& =P_{{\alpha }_1,\dots ,{\alpha }_m}\left(\underset{n}{\underbrace{{\displaystyle \frac{\lambda }{\sqrt[p]{n}}},\dots ,{\displaystyle \frac{\lambda }{\sqrt[p]{n}}}}},0,\dots \right)\hfill \\ \hfill & ={\lambda }^{{\alpha }_1+\cdots +{\alpha }_m}{\displaystyle \frac{n(n-1)\cdots (n-m+1)}{m!n^{\frac{{\alpha }_1+\cdots +{\alpha }_m}{p}}}}.\hfill \end{array}$$

(13)

The numerator of the right-hand part of (13) is $O(n^m).$ So, to get a nonzero limit as $n\to \infty ,$ we need to have ${\alpha }_1+\cdots +{\alpha }_m\le mp.$ Since each ${\alpha }_j\ge p,$ this is possible only if ${\alpha }_1=\cdots ={\alpha }_m=p.$ In this case, ${\alpha }_1+\cdots +{\alpha }_m=mp$ and so

$$P_{\underset{m}{\underbrace{p,\dots ,p}}}(v_n(\lambda ))={\lambda }^{mp}{\displaystyle \frac{n(n-1)\cdots (n-m+1)}{m!n^m}}⟶{\displaystyle \frac{{\lambda }^{mp}}{m!}}\mathrm{as}n\to \infty ,$$

and the limit does not depend of any ultrafilter. Thus, we have the following theorem.

Theorem 7. 

There is a unique restriction ${\psi }_{\lambda }$ of ${\psi }_{\lambda }^{\mathcal{U}}$ to the subalgebra $H_{b\mathfrak{S}}({\ell }_p)\subset H_{bs}({\ell }_p),$ and ${\psi }_{\lambda }$ is defined on the standard polynomials by

$${\psi }_{\lambda }(P_{{\alpha }_1,\dots ,{\alpha }_m})=\left\{\begin{array}{cc}{\displaystyle \frac{{\lambda }^{mp}}{m!}}\hfill & if{\alpha }_1=\cdots ={\alpha }_m=p,\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$$

Example 5. 

Let us compute $({\psi }_{\lambda }⊲{\delta }_x)(P_{p,k})$ for some $k>p,$ $P\in {\mathcal{P}}_{\mathfrak{S}}({\ell }_p),$ and $p\in \mathbb{N}.$ Using Proposition 11 and Theorem 7 we have

$$({\psi }_{\lambda }⊲{\delta }_x)(P_{p,k})={\delta }_x(P_{p,k})+{\psi }_{\lambda }(P_p){\delta }_x(P_k)+{\psi }_{\lambda }(P_{p,k})=P_{p,k}(x)+{\lambda }^p{P}_k(x).$$

Note that $({\psi }_{\lambda }⊲{\delta }_x)(P_{k,p})=P_{k,p}(x)$ because, for $k>p,$ ${\psi }_{\lambda }(P_k)=0.$ Thus, if $A_{p,k}=P_{p,k}-P_{k,p},$ $k>p,$ then

$$({\psi }_{\lambda }⊲{\delta }_x)(A_{p,k})=A_{p,k}(x)+{\lambda }^p{P}_k(x).$$

Proposition 12. 

Let $p\in \mathbb{N}.$ Polynomials $A_{p,k}=P_{p,k}-P_{k,p},$ $k>p$ are algebraically independent in ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$

Proof. 

Let $\lambda \ne 0.$ Then

$$({\psi }_{\lambda }⊲{\delta }_{\check{x}⊲x})(A_{p,k})={\delta }_{\check{x}⊲x}(A_{p,k})+{\lambda }^p{\delta }_{\check{x}⊲x}(P_k)={\lambda }^{p}2P_k(x),$$

because ${\delta }_{\check{x}⊲x}(A_{p,k})=0.$ Since the map $P\mapsto {\delta }_{\check{x}⊲x}(P)$ is a homomorphism and polynomials $\{{\lambda }^{p}2P_k\},$ $k>p$ are algebraically independent, we can apply Theorem 3 and get that polynomials $\{A_{p,k}\}$ are algebraically independent as polynomials on $c_{00}.$ So they are algebraically independent as polynomials on ${\ell }_p$ as well. □

5.5. Subsymmetric Polynomials on Abstract Hilbert Spaces

Let $\mathcal{H}$ be an abstract separable Hilbert space with the inner product $\langle ·|·\rangle$ and an orthonormal basis $(e_n).$ Then $\mathcal{H}$ is isomorphic to ${\ell }_2$ with respect to the isomorphism J define on the basis vectors by

$$J:e_n\mapsto (\underset{n}{\underbrace{0,\dots ,0,1}},0,\dots )$$

and so each subsymmetric polynomial P on ${\ell }_2$ is well-defined on $\mathcal{H}$ as $P(h):=P(J(h)).$ However, some specific structures of $\mathcal{H}$ can provide additional properties of subsymmetric polynomials.

Example 6. 

Let $\mathcal{H}$ be the Hardy space $H^2(\mathbb{D})$ of analytic functions

$$h(z)=\sum _{k=0}^{\infty }{h}_k{z}^k$$

on the open unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ such that

$${\parallel h\parallel }^2=\langle h|h\rangle =\underset{|z|=1}{\int }h(z)\overline{h(z)}dz=\sum _{k=0}^{\infty }{|h_k|}^2<\infty .$$

Thus, polynomial $e_k(z)=z^k,$ $k=0,1,2,\dots$ form an orthonormal basis in $H^2(\mathbb{D})$ and so each standard subsymmetric polynomial on $H^2(\mathbb{D})$ is of the form

$$P_{{\alpha }_1,\dots ,{\alpha }_n}(h)=\sum _{i_1<\cdots <i_n}{h}_{i_1}^{{\alpha }_1}\cdots h_{i_n}^{{\alpha }_n},i_m\ge 2,$$

where coefficients $h_k$ can be computed as

$$h_k=\underset{|z|=1}{\int }h(z)\overline{z^k}dz.$$

The operators

$${\mathcal{C}}_j:(h_0,\dots ,h_n,\dots )\mapsto (h_0,\dots ,h_{j-1},0,h_j,h_{j+1}\dots ),j=0,1,2,\dots$$

can by represented as operators of multiplication by the independent variable z:

$${\mathcal{C}}_j(h)={\mathcal{C}}_j\left(\sum _{k=0}^{\infty }{h}_k{z}^k\right)=\sum _{k=0}^{j-1}{h}_k{z}^k+z\sum _{k=j}^{\infty }{h}_k{z}^k.$$

In particular, ${\mathcal{C}}_0(h)=zh$ is the classical shift operator on $H^2(\mathbb{D})$ (see [38] for details). Note that in [38] invariant subspaces on $H^2(\mathbb{D})$ with respect to the shift ${\mathcal{C}}_0$ were investigated.

The operators ${\mathcal{C}}_j$ naturally act on the space $H_b\left(H^2(\mathbb{D})\right)$ of analytic functions of bounded type on $H^2(\mathbb{D})$ as

$${\widehat{\mathcal{C}}}_j(f)(h)=f({\mathcal{C}}_j(h)).$$

Since ${\widehat{\mathcal{C}}}_j$ is the operator of composition with the continuous linear operator ${\mathcal{C}}_j,$ it is a continuous algebra homomorphism from $H_b\left(H^2(\mathbb{D})\right)$ to itself (see e.g., [20]).

Proposition 13. 

The space $H_{b\mathfrak{S}}\left(H^2(\mathbb{D})\right)$ of subsymmetric analytic functions of bounded type on $H^2(\mathbb{D})$ is a maximal subspace of $H_b\left(H^2(\mathbb{D})\right)$ consisting of invariant elements for all operators ${\widehat{\mathcal{C}}}_j.$

Proof. 

According to the definition of subsymmetric functions, $f\in H_b\left(H^2(\mathbb{D})\right)$ is subsymmetric if and only if it is invariant about every operator ${\widehat{\mathcal{C}}}_j.$ □

Note that not every ${\widehat{\mathcal{C}}}_0$-invariant polynomial is subsymmetric.

Example 7. 

Let

$$P(h)=\sum _{k=0}^{\infty }{h}_k{h}_{k+1}.$$

Then P is ${\widehat{\mathcal{C}}}_0$-invariant but not subsymmetric. Indeed,

$${\widehat{\mathcal{C}}}_0(P)(h)=P({\mathcal{C}}_0(h))=P(0,h_0,h_1,h_2,\dots )=0h_0+h_0{h}_1+h_1{h}_2+\cdots =P(h),$$

while

$${\widehat{\mathcal{C}}}_1(P)(h)=P({\mathcal{C}}_1(h))=P(h_0,0,h_1,h_2,\dots )=0h_0+0h_1+h_1{h}_2+\cdots \ne P(h).$$

Let us consider another example of the Hilbert space $\mathcal{H}.$

Example 8. 

Let $\mathcal{H}$ be the real Hilbert space with the orthogonal basis consisting of Chebyshev polynomials of the first kind $T_n,$ $n\in \mathbb{N},$ and inner product

$$\langle u|v\rangle =\underset{-1}{\overset{1}{\int }}u(t)v(t){\displaystyle \frac{dt}{1-t^2}}.$$

Let us recall that Chebyshev polynomials of the first kind can be determined by the equality

$$T_n(cos\vartheta )=cosn\vartheta ,\vartheta \in \mathbb{R}.$$

(14)

The Hilbert space $\mathcal{H}$ can be considered as the weighted space $L_2[-1,1],$ namely, $\mathcal{H}=L_2\left([-1,1],{\displaystyle \frac{1}{1-t^2}}\right).$ The sequence of polynomials

$${\displaystyle \frac{1}{\pi }},{\displaystyle \frac{2}{\pi }}{T}_1,{\displaystyle \frac{2}{\pi }}{T}_2,\dots$$

form an orthonormal basis in $L_2\left([-1,1],{\displaystyle \frac{1}{1-t^2}}\right).$ More information about Chebyshev polynomials can be found in [39]. Thus, standard subsymmetric polynomials on $L_2\left([-1,1],{\displaystyle \frac{1}{1-t^2}}\right)$ are of the form

$$P_{{\alpha }_1,\dots ,{\alpha }_n}(g)=\sum _{i_1<\cdots <i_n}{h}_{i_1}^{{\alpha }_1}\cdots g_{i_n}^{{\alpha }_n},i_m\ge 2,$$

where $g\in L_2\left([-1,1],{\displaystyle \frac{1}{1-t^2}}\right),$

$$g_0={\displaystyle \frac{1}{\pi }}\underset{-1}{\overset{1}{\int }}g(t){\displaystyle \frac{dt}{1-t^2}},and{g}_n={\displaystyle \frac{2}{\pi }}\underset{-1}{\overset{1}{\int }}{T}_n(t)g(t){\displaystyle \frac{dt}{1-t^2}}.$$

From (14) it follows that for the composition $T_k\circ T_m$ of polynomials $T_k$ and $T_m$ we have $T_k\circ T_m(t)=T_k(T_m(t))=T_{km}(t),$ $k,$ $m\in \mathbb{N}.$ Hence, each polynomial $T_k$ generates a continuous linear operator $g\mapsto T_k\circ g$ on $L_2\left([-1,1],{\displaystyle \frac{1}{1-t^2}}\right)$ so that

$$T_k\circ g(t)={\displaystyle \frac{g_0}{\pi }}+{\displaystyle \frac{2}{\pi }}\sum _{n=1}^{\infty }{g}_n{T}_{km}.$$

In other words, the operator of composition with $T_k$ maps $(g_0,g_1,g_2,\dots )$ to

$$(g_0,\underset{k}{\underbrace{0,\dots ,0,g_1}},\underset{k}{\underbrace{0,\dots ,0,g_2}},\underset{k}{\underbrace{0,\dots ,0,g_3}},\dots ).$$

Clearly, if f is a subsymmetric function, then $f(T_k\circ g)=f(g).$ Thus, if we set ${\widehat{T}}_k(f)=f(T_k\circ g),$ then ${\widehat{T}}_k$ is a continuous operator on $H_b\left(L_2\left([-1,1],{\displaystyle \frac{1}{1-t^2}}\right)\right),$ and each function in $H_{b\mathfrak{S}}\left(L_2\left([-1,1],{\displaystyle \frac{1}{1-t^2}}\right)\right)$ is invariant with respect to every ${\widehat{T}}_k.$ Note that the space of $T_k$-invariant analytic functions of bounded type is larger than $H_{b\mathfrak{S}}\left(L_2\left([-1,1],{\displaystyle \frac{1}{1-t^2}}\right)\right).$ Indeed, the polynomial $Q(g)=g_0^2$ is invariant with respect to ${\widehat{T}}_k$ for every $k\in \mathbb{N}$ but it is not subsymmetric.

6. Zeros of Subsymmetric Polynomials and Subspaces of Symmetry

Let P be an n-homogeneous polynomial on a complex infinite-dimensional linear space. According to known result in [25], for every $x_0\in kerP$ there is an infinite-dimensional subspace in $kerP$ containing $x_0.$ Actually, the main idea of the proof of the result in [25] can be formulated as the following proposition.

Proposition 14.

Let X be an infinite dimensional complex linear space and P be an n-homogeneous $\mathbb{C}$-valued polynomial. For every $x_0\in kerP$ there exists an infinite linearly independent sequence $(x^{(k)})\subset X$ such that for every $m\in \mathbb{N},$

$$P(t_1{x}^{(1)}+t_2{x}^{(2)}+\cdots +t_m{x}^{(m)})=t_1^{n}P(x^{(1)})+t_2^{n}P(x^{(2)})+\cdots +t_m^{n}P(x^{(m)}),$$

(15)

and $x_0$ belongs to the linear span of $(x^{(k)}).$

The diagonal representation (15) suggests that a restriction of P to a subspace is symmetric with respect to an appropriate basis, i.e., $g_k=x^{(k)}/P(x^{(k)}).$ However, in the general case, the values $P(x_m)$ can be nonzero only for a finite number of vectors and so we cannot claim that any polynomial on X is symmetric with respect to a basis on an infinite-dimensional subspace. In this section, we consider zeros of subsymmetric polynomials and a linear subspace V for a given subsymmetric polynomial P such the restriction of P to V is a nontrivial symmetric polynomial in some basis.

For a given natural number N we define by ${\Xi }_N$ the following homomorphism of ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p),$

$${\Xi }_N(P)(x)=P(x⊲{(-1)}^{1/N}x),$$

where ${(-1)}^{1/N}$; we understand the principal value of the $N^{th}$ root of $-1.$

Proposition 15.

Let $P_{{\alpha }_1,\dots ,{\alpha }_n}$ be an M-homogeneous standard polynomial in ${\mathcal{P}}_{\mathfrak{S}}({\ell }_p).$ Then ${\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})$ is a homogeneous polynomial, $deg{\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})=M$ or ${\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})=0.$ If $M=N,$ then ${\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})\in {\mathcal{P}}_{\mathfrak{S}}^{(n-1)}({\ell }_p).$

Proof. 

According to (7) we have

$$\begin{array}{cc}\hfill {\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})(x)& =P_{{\alpha }_1,\dots ,{\alpha }_n}(x)\hfill \\ & +\sum _{k=1}^n{(-1)}^{{\displaystyle \frac{{\alpha }_{k+1}+\cdots +{\alpha }_n}{N}}}{P}_{{\alpha }_1,\dots ,{\alpha }_k}(x)P_{{\alpha }_{k+1},\dots ,{\alpha }_n}(x)+{(-1)}^{M/N}{P}_{{\alpha }_1,\dots ,{\alpha }_n}(x).\hfill \end{array}$$

Thus, $deg{\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})={\alpha }_1+\cdots +{\alpha }_n=M$ if ${\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})\ne 0.$ If $M=N,$ then

$${\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})(x)=\sum _{k=1}^n{(-1)}^{{\displaystyle \frac{{\alpha }_{k+1}+\cdots +{\alpha }_n}{N}}}{P}_{{\alpha }_1,\dots ,{\alpha }_k}(x)P_{{\alpha }_{k+1},\dots ,{\alpha }_n}(x).$$

Hence, for this case, ${\Xi }_N(P_{{\alpha }_1,\dots ,{\alpha }_n})$ is an algebraic combination of polynomials in ${\mathcal{P}}_{\mathfrak{S}}^{(n-1)}({\ell }_p).$ □

Note that if $P_{{\alpha }_1,\dots ,{\alpha }_n}=P_N,$ then ${\Xi }_N(P_N)=0.$

Corollary 4.

Let P be a homogeneous subsymmetric polynomial on ${\ell }_p.$ Then there is a finite sequence of natural numbers $N_1,N_2,\dots ,N_m$ such that

$${\Xi }_{N_m}\circ \cdots \circ {\Xi }_{N_1}(P)=0,$$

where “∘” is the composition of mappings.

Proof. 

Suppose that $P\in {\mathcal{P}}_{\mathfrak{S}}^{(n)}({\ell }_p)$ and $P\notin {\mathcal{P}}_{\mathfrak{S}}^{(n-1)}({\ell }_p)$ for some $n\in \mathbb{N}.$ Then P is a finite algebraic combination of standard polynomials $Q_1,Q_2,\dots$ in ${\mathcal{P}}_{\mathfrak{S}}^{(n)}({\ell }_p).$ Some of them, say $Q_{i_1},Q_{i_2}\dots Q_{i_s}$, are in ${\mathcal{P}}_{\mathfrak{S}}^{(n)}({\ell }_p)\setminus {\mathcal{P}}_{\mathfrak{S}}^{(n-1)}({\ell }_p).$ Let $N_1=deg{Q}_{i_1},$ $N_2=deg{Q}_{i_2}$ and so on. Then, by Proposition 15,

$${\Xi }_{N_s}\circ \cdots \circ {\Xi }_{N_1}(P)\in {\mathcal{P}}_{\mathfrak{S}}^{(n-1)}({\ell }_p),$$

and either $deg{\Xi }_{N_s}\circ \cdots \circ {\Xi }_{N_1}(P)=degP$ or $P=0.$ If $P\ne 0,$ then we can apply this process to ${\Xi }_{N_s}\circ \cdots \circ {\Xi }_{N_1}(P).$ Since, ${\Xi }_N(P_N)=0,$ after a finite number m of steps, we will have ${\Xi }_{N_m}\circ \cdots \circ {\Xi }_{N_1}(P)=0.$ □

Let us recall that a linearly independent sequence $e_n$ in a Banach space X is a Schauder basis of X (see [40] for detailed) if every $x\in X$ has a unique representation $x={\sum }_{n=1}^{\infty }{x}_n{e}_n$ for some numbers $x_n$ such that

$$\underset{m\to \infty }{lim}\parallel x-\sum _{n=1}^m{x}_n{e}_n\parallel =0.$$

Theorem 8. 

(i) 

For every subsymmetric homogeneous polynomial P on ${\ell }_p$ and a natural number d, there is a d-dimensional subspace $V_d$ in ${\ell }_p$ and a linear basis $(g_i)$ in $V_d$ such that P is symmetric on $V_d$ with respect to the basis.

(ii) 

For every subsymmetric homogeneous polynomial P on ${\ell }_p$ there exists a well-ordered set $\mathfrak{A}$ an infinite-dimensional subspace V in ${\ell }_p(\mathfrak{A}),$ and a Schauder basis $(g_i)$ in V such that $P^{\mathfrak{A}}$ is symmetric on V with respect to the basis.

Proof. 

(i) For given d and $N_1\in \mathbb{N}$ we consider the following subspace

$$V_d^{(1)}=\left\{(x_1,\dots ,x_d,{(-1)}^{1/N_1}{x}_1,\dots ,{(-1)}^{1/N_1}{x}_d,0,0,\dots )\right\}\subset {\ell }_p.$$

The space $V_d^{(1)}$ is a d-dimensional subspace in ${\ell }_p$ with basis

$$g_k^{(1)}=(\stackrel{d+k}{\overbrace{\underset{k}{\underbrace{0,\dots ,0,1}},0,\dots ,0,{(-1)}^{1/N_1}}},0,\dots ),k=1,\dots ,d.$$

If

$$y=\left(x_1,\dots ,x_d,{(-1)}^{1/N_1}{x}_1,\dots ,{(-1)}^{1/N_1}{x}_d,0,0,\dots \right),$$

then $P(y)={\Xi }_{N_1}(P)(x_1\dots ,x_d,0,\dots ).$

For another number $N_2,$ we can construct a space $V_d^{(2)}$ such that for every $z\in V_d^{(2)},$

$$P(z)={\Xi }_{N_2}\circ {\Xi }_{N_1}(P)(x_1\dots ,x_d,0,\dots ).$$

Indeed, let $V_d^{(2)}$ consist of the vectors of the form

$$\begin{array}{cc}\hfill z& =(x_1,\dots ,x_d,{(-1)}^{1/N_1}{x}_1,\dots ,{(-1)}^{1/N_1}{x}_d,{(-1)}^{1/N_2}{x}_1,\hfill \\ & \dots ,{(-1)}^{1/N_2}{x}_d,{(-1)}^{1/(N_1{N}_2)}{x}_1,\dots ,{(-1)}^{1/(N_1{N}_2)}{x}_d,0,\dots ).\hfill \end{array}$$

Then,

$${\Xi }_{N_2}\circ {\Xi }_{N_1}(P)(x_1,\dots ,x_d,0,\dots )={\Xi }_{N_2}(P)\left(x_1,\dots ,x_d,{(-1)}^{1/N_1}{x}_1,\dots ,{(-1)}^{1/N_1}{x}_d,0,\dots \right)=P(z).$$

Clearly, $V_d^{(2)}$ is a d-dimensional subspace and

$$g_k^{(2)}=\left(\stackrel{d+k}{\overbrace{\underset{k}{\underbrace{0,\dots ,0,1}},0,\dots ,0,{(-1)}^{1/N_1}}},\stackrel{d+k}{\overbrace{\underset{k}{\underbrace{0,\dots ,0,{(-1)}^{1/N_2}}},0,\dots ,0,{(-1)}^{1/(N_1{N}_2)}}}0,\dots \right),$$

$k=1,\dots ,d,$ is a basis. By continuing this process, we can construct for a given finite sequence $N_1,\dots ,N_m$ such that for every $1\le j\le m$ there exists a d-dimensional subspace $V_d^{(j)}$ with a basis $g_1^{(j)},\dots ,g_d^{(j)}$ such that

$$P\left(\sum _{i=1}^j{x}_i{g}_i^{(j)}\right)={\Xi }_{N_j}\circ \cdots \circ {\Xi }_{N_1}(P)(x_1,\dots ,x_d,0,\dots ).$$

From here, in particular, it follows that the restriction of P to $V_d^{(j)}$ is subsymmetric with respect to the basis $\left(g_i^{(j)}\right).$ Let us take the sequence $N_1,\dots ,N_m$ as in Corollary 4. Then for some $j\le m$ the polynomial ${\Xi }_{N_j}\circ \cdots \circ {\Xi }_{N_1}(P)$ is in ${\mathcal{P}}_{\mathfrak{S}}^{(1)}({\ell }_p)$ while ${\Xi }_{N_{j-1}}\circ \cdots \circ {\Xi }_{N_1}(P)$ is not. Thus, the space $V_d=V_d^{(j)}$ and its basis $(g_i)=\left(g_i^{(j)}\right)$ are such that the restriction of P to $V_d=V_d^{(j)}$ is symmetric with respect to $(g_i).$

(ii) As we observed above, the functional ${\delta }_{x⊲y}$ is not a point evaluation but it can be interpreted as a point evaluation functional on ${\mathcal{P}}_{\mathfrak{S}}^{\mathfrak{A}}({\ell }_p(\mathfrak{A}))$ for some appropriate $\mathfrak{A}.$ Let $(\mathfrak{A},⪯)$ be the following well-ordered set

$$\mathfrak{A}=\bigcup _{k=1}^{\infty }{\mathbb{N}}_k,$$

where each ${\mathbb{N}}_k$ is a copy of $\mathbb{N}$ with the natural order, that is, $n⪯m$ if and only if $n\le m,$ $n,$ $m\in {\mathbb{N}}_k.$ If $n\in {\mathbb{N}}_k$ and $m\in {\mathbb{N}}_j,$ $k\ne j,$ then $n\prec m$ if and only if $k<j.$ Let $(e_i^k)$ be the standard basis in ${\ell }_p({\mathbb{N}}_k),$ that is,

$$e_i^k=(\underset{i}{\underbrace{0,\dots ,0,1}},0\dots ),i\in {\mathbb{N}}_k.$$

Every vector $y\in {\ell }_p(\mathfrak{A})$ can be represented as

$$y=\sum _{k=1}^{\infty }\sum _{i=1}^{\infty }{y}_i^k{e}_i^k$$

for some complex numbers $y_i^k$ such that

$${\parallel y\parallel }^p=\sum _{k=1}^{\infty }\sum _{i=1}^{\infty }{|y_i^k|}^p<\infty .$$

For a given $N_1\in \mathbb{N}$ we consider the following subspace $V^{(1)}$ of ${\ell }_p(\mathfrak{A}),$

$$V^{(1)}=\left\{\sum _{i=1}^{\infty }{x}_i{e}_i^1+{(-1)}^{1/N_1}\sum _{i=1}^{\infty }{x}_i{e}_i^2\right\},$$

where $x=(x_1,x_2,\dots )$ goes over ${\ell }_p.$ We can see that

$${\Xi }_{N_1}(P)(x)=P^{\mathfrak{A}}\left(\sum _{i=1}^{\infty }{x}_i{e}_i^1+{(-1)}^{1/N}\sum _{i=1}^{\infty }{x}_i{e}_i^2\right).$$

Also, it is easy to check that vectors $g_i^{(1)}=e_i^1+{(-1)}^{1/N_1}{e}_i^2$ form a Schauder basis in $V^{(1)}.$ Suppose that for given natural numbers $N_1,\dots ,N_m$, we already constructed a subspace $V^{(m-1)}\subset {\ell }_p(\mathfrak{A})$ of the form

$$\left\{\sum _{i=1}^{\infty }{y}_i^1{e}_i^1+\sum _{i=1}^{\infty }{y}_i^2{e}_i^2+\cdots +\sum _{i=1}^{\infty }{y}_i^r{e}_i^r\right\}$$

for some finite number r and a Schauder basis $(g_i^{(m-1)})$ in $V^{(m-1)}$ such that

$${\Xi }_{N_{m-1}}\circ \cdots \circ {\Xi }_{N_1}(P)(x)=P\left(\sum _{i=1}^{\infty }{x}_i{g}_i^{(m-1)}\right).$$

Then the space $V^{(m)}$ is defined as

$$V^{(m)}=\left\{\sum _{i=1}^{\infty }{y}_i^1{e}_i^1+\sum _{i=1}^{\infty }{y}_i^2{e}_i^2+\cdots +\sum _{i=1}^{\infty }{y}_i^r{e}_i^r+{(-1)}^{1/N_m}\left(\sum _{i=1}^{\infty }{y}_i^1{e}_i^{r+1}+\sum _{i=1}^{\infty }{y}_i^2{e}_i^{r+2}+\cdots +\sum _{i=1}^{\infty }{y}_i^r{e}_i^{2r}\right)\right\},$$

and the basis $(g_i^{(m)})$ as

$$g_i^{(m)}=g_i^{(m-1)}+{(-1)}^{1/N_m}{g}_i^{(m-1)}.$$

Taking the sequence $N_1,\dots ,N_m$ as in Corollary 4, we have that for some $j\le m$ the polynomial ${\Xi }_{N_j}\circ \cdots \circ {\Xi }_{N_1}(P)$ is symmetric with respect to the basis $(g_i^{(j)}).$ Thus, the space $V_d=V_d^{(j)}.$ □

Let us show that there are some finitely dimensional subspaces such that the restriction of any subsymmetric polynomial of some special form to these subspaces are symmetric with respect to some basis.

Proposition 16.

For a number $d\in \mathbb{N}$ there is a d-dimensional subspace $V_d$ in ${\ell }_p$ such that every polynomial $P_{k,m}$ is nontrivial and symmetric on $V_d$ with respect to a basis.

Proof. 

Indeed, we know that $\Theta (P_{k,m})(x)={\delta }_{\check{x}⊲x}(P_{k,m})$ is a nontrivial symmetric polynomial. Thus, restriction of $P_{k,m}$ to the d-dimensional subspace spanned on the vectors

$$g_k=(\underset{k}{\underbrace{0,\dots ,0,1}},0\dots ,0,\underset{k}{\underbrace{1,0\dots ,0}}),1\le k\le d$$

is symmetric with respect to $(g_k).$ □

Proposition 17.

For a number $d\in \mathbb{N}$ there is a d-dimensional subspace $V_d$ in ${\ell }_p$ such that every polynomial of the form $A_{k,m}=P_{k,m}-P_{m,k},$ $0<k<m$ vanishes on $V_d.$

Proof. 

It is easy to check that $\Theta (A_{k,m})=0$ for every $A_{k,m}.$ Thus each $A_{k,m}$ vanishes on the subspace $V_d$ as in Proposition 16. □

Let us recall that a polynomial map

$$\mathcal{Q}:(z_1,\dots ,z_n)\mapsto (Q_1(z_1,\dots ,z_n),\dots ,Q_n(z_1,\dots ,z_n)),(z_1,\dots ,z_n)\in {\mathbb{C}}^n$$

is proper if for every compact subset $K\subset {\mathbb{C}}^n,$ the preimage ${(\mathcal{Q})}^{-1}(K)$ is compact in ${\mathbb{C}}^n.$ It is known (see, e.g., [41], Chapter 15) that a proper map is always surjective and open. Moreover, in [42], it is shown that if $Q_1,\dots ,Q_n$ are homogeneous polynomials, then $\mathcal{Q}$ is proper if and only if ${\mathcal{Q}}^{-1}(0)=\{0\}.$ Note that the surjectivity of a proper map implies that polynomials $Q_1,\dots ,Q_n$ are algebraically independent. Note that according to Proposition 12, polynomials $A_{1,k},$ $k>1$ on ${\ell }_1$ are algebraically independent but the restriction of $A_{1,k}$ to any M-dimensional subspace spanned on the basis vectors $e_i,$ $i=1,\dots ,M,$ $M\ge 2$ generates an improper map $(A_{1,2}(x_1,\dots ,x_M),\dots ,A_{1,M+1}(x_1,\dots ,x_M))$ from ${\mathbb{C}}^M$ to ${\mathbb{C}}^M,$ because each $A_{1,k}$ vanishes on the d-dimensional subspace $V_d,$ $d=[M/2]$ as in Proposition 17.

7. Applications

7.1. Polynomial Hash Function

An important question in cryptography is about the construction of a hash function h that is assigned to a given sequence of numbers $x=(x_1,\dots ,x_N)$ of some sequence $h(x)=w=(w_1,\dots ,w_M)$ so that h is injective (i.e., h is a collision-free hash function) and the inverse map $h^{-1}:w\mapsto x$ is difficult for computation.

Our approach allows us to work with finite sequences of nonzero numbers. From Theorem 2 it follows that we can take $M={\displaystyle \frac{N(N+1)}{2}}$, and the sequence $w=h(x)$ is defined as

$$\begin{array}{cc}\hfill & w_1=P_1(x),\hfill \\ \hfill & w_2=P_2(x),\hfill \\ \hfill & \dots \hfill \\ \hfill & w_N=P_N(x),\hfill \\ \hfill & w_{N+1}=P_{1,2}(x),\hfill \\ \hfill & w_{N+2}=P_{2,4}(x),\hfill \\ \hfill & \dots \hfill \\ \hfill & w_{N(N+1)/2}=P_{N(N-1)/2,N(N-1)}.\hfill \end{array}$$

(16)

Indeed, according to Theorem 2, $x\simeq y$ if and only if $P(x)=P(y)$ for every symmetric polynomial (that is, $P_i(x)=P_i(y),$ $i=1,\dots ,N$) and $P_{k,2k}(x)=P_{k,2k}(x),$ $k=1,N(N-1)/2.$ Thus, together we have $N+N(N-1)/2=N(N+1)/2=M$ polynomials and so the mapping $w=h(x)$ (16) is injective on the sequences of nonzero complex numbers. Note that if x contains a zero element, then it will be ignored by (16) and the value $h(x)$ will be the same as when we remove this element. To find the inverse mapping $h^{-1},$ we need to solve the system of high-degree polynomial Equation (16), which is a hard computational problem. More precisely, the problem of solving a polynomial system is $NP$-hard. In other words, the answer can be verified in polynomial time [43,44]. In [43], it is conjectured that the problem of solving a polynomial system over a finite field is exponentially complex.

The proposed construction has the advantage over general methods for polynomial systems that the mapping $x\mapsto h(x)$ is always injective. So h is a collision-free hash function, that is, every value of h has a unique preimage.

A hash function based on symmetric polynomials on ${\ell }_1$ was considered in [45]. In this case, we have a system of symmetric polynomials that can be reduced to a high-degree polynomial of a single variable and solved by approximation methods. The proposed hash function h is subsymmetric and cannot be reduced to a single polynomial.

Example 9. 

Let $x=(5,3,7).$ Then $N=3,$ and $h(x)=(w_1,\dots ,w_6),$ where

$$\begin{array}{cc}\hfill & w_1=P_1(x)=x_1+x_2+x_3=15,\hfill \\ \hfill & w_2=P_2(x)=x_1^2+x_2^2+x_3^2=83,\hfill \\ \hfill & w_3=P_3(x)=x_1^3+x_2^3+x_3^3=495,\hfill \\ \hfill & w_4=P_{1,2}(x)=x_1{x}_2^2+x_1{x}_3^2+x_2{x}_3^2=437,\hfill \\ \hfill & w_5=P_{2,4}(x)=x_1^2{x}_2^4+x_1^2{x}_3^4+x_2^2{x}_3^4=83659,\hfill \\ \hfill & w_6=P_{4,8}=x_1^4{x}_2^8+x_1^4{x}_3^8+x_2^4{x}_3^8=4074050131.\hfill \end{array}$$

Example 9 shows that evaluations of $P_{k,mk}$ rapidly grow, and so the hash function h may have big values that depend on the size of $x.$ This disadvantage can be reduced using multiplication in a finite field and other known methods.

Some different approaches to the construction of polynomial hash functions can be found in [46,47].

7.2. Spectra of Operators

Let E be a separable Hilbert space and $A:E\to E$ be a normal operator with a point spectrum such that the eigenvalues $x_i$ of A are nonzero and ${\sum }_{i=1}^{\infty }{|x_i|}^p<\infty .$ In other words, A is a p-nuclear (or p-trace) operator (see [48]). Let $(e_i)$ be the orthonormal basis of eigenvectors of $A.$ Such operators (with real eigenvalues) naturally appear in quantum physics as $A=e^{-\beta \mathcal{H}},$ where $\mathcal{H}$ is the Hamiltonian of a quantum system, and $\beta >0$ is a physical parameter (see for details e.g., [49,50,51]). It is well known that the operator A cannot be determined by its spectrum. Establishing additional conditions or some additional spectral data allowing us to determine $A,$ is an important problem of the inverse spectral theory (see e.g., [52] for some particular operators).

All eigenvalues $x_i$ of $A,$ counting with their multiplicities, form a multiset $[x]\in {\mathcal{M}}_p^S,$ where $x=(x_1,x_2,\dots )\in {\ell }_p.$ For every $k\ge \lceil p\rceil ,$

$$P_k(x)=\sum _{i=1}^{\infty }{x}_i^k=\mathrm{tr}{A}^k,$$

where $\mathrm{tr}$ is the trace of an operator. More information about the relationship between traces of p-nuclear operators and symmetric analytic functions on ${\ell }_p$ can be found in [53]. From Proposition 5 we have the following corollary.

Corollary 5. 

Let A and B be normal p-nuclear operators with point spectra and the same eigenvectors such that the eigenvalues $x_i$ of A and the eigenvalues $y_i$ of B are nonzero. Then $[x]=[y]$ if and only if there exists a natural number $m\ge \lceil p\rceil$ such that $\mathrm{tr}{A}^n=\mathrm{tr}{B}^n$ for every $n\ge m.$

Since we suppose that all eigenvalues $x_i$ of A are nonzero, the ordered multiset $[[x]]$ completely defines the operator $A.$

Let $E\wedge E$ be the skew-symmetric tensor product of $E.$ That is, $E\wedge E$ is the closed subspace of the Hilbert tensor product $E\otimes E$ spanned on the orthogonal vectors $e_i\wedge e_j={\displaystyle \frac{e_i\otimes e_j-e_j\otimes e_i}{2}}.$ In other words, the vectors $e_i\wedge e_j,$ $i<j$ form an orthogonal basis in $E\wedge E.$ For given operators A and B on E we denote by $A\wedge B$ the restriction of $A\otimes B$ to $E\wedge E.$ That is, $(A\wedge B)(e_i\wedge e_j)=A(e_i)\wedge B(e_j).$ Then, for any natural numbers k and $m,$

$$(A^k\wedge A^m)(e_i\wedge e_j)=x_i^k{x}_j^m,$$

and if A is p-nuclear, and $k\ge p$ and $m\ge p,$ then

$$\mathrm{tr}(A^k\wedge A^m)=\sum _{i<j}{x}_i^k{x}_j^m=P_{k,m}(x).$$

Thus, applying Theorem 4, we have the following result.

Theorem 9. 

Let A and B be normal p-nuclear operators with point spectra and the same eigenvectors such that the eigenvalues $x_i$ of A and the eigenvalues $y_i$ of B are nonzero. Then $A=B$ if and only if there exists a natural number $m\ge \lceil p\rceil$ such that $\mathrm{tr}{A}^n=\mathrm{tr}{B}^n$ and $\mathrm{tr}\left[{(A\wedge A^2)}^n\right]=\mathrm{tr}\left[{(B\wedge B^2)}^n\right]$ for every $n\ge m.$

Proof. 

As we mentioned above, it is enough to show that $[[x]]=[[y]].$ This equality follows from Theorem 4 because

$$\mathrm{tr}\left[{(A\wedge A^2)}^n\right]=\mathrm{tr}(A^n\wedge A^{2n})=P_{n,2n}(x),$$

and

$$\mathrm{tr}\left[{(B\wedge B^2)}^n\right]=\mathrm{tr}(B^n\wedge B^{2n})=P_{n,2n}(y).$$

□

8. Conclusions

In the paper, we establish algebraic properties of subsymmetric polynomials on $c_{00}$ and ${\ell }_p$ and describe some nontrivial sequences of algebraically independent subsymmetric polynomials. However, the problem of the existence of an algebraic basis in the algebra of subsymmetric polynomials of an infinite number of variables remains open. We proved that an ordered multiset $[[x]]$ is completely determined by evaluations $P_{n,2n}(x)$ and used this fact to propose some applications to cryptography and operator theory. We considered, probably, the most general case when we can extend subsymmetric polynomials to subsymmetric polynomials on a larger space such that the operator of extension is a continuous homomorphism of algebras. We applied these results to the algebra $H_{b\mathfrak{S}}({\ell }_p)$ of subsymmetric analytic functions of bounded type and constructed some nontrivial characters of this algebra. In the paper, we introduced a subsymmetric translation operator on $H_{b\mathfrak{S}}({\ell }_p)$ and extended it to a convolution on the set of characters of this algebra. Additionally, using a constructed homomorphism, we considered certain subspaces of sets of zeros of subsymmetric polynomials and subspaces where a subsymmetric polynomial is symmetric with respect to a basis.

We proposed some applications of the obtained results to cryptography and operator theory. In particular, we constructed a collision-free hash function based on subsymmetric polynomials.

Further investigations will focus on the problem of the existence of an algebraic basis in the algebra of all subsymmetric polynomials and on the description of the spectrum of $H_{b\mathfrak{S}}({\ell }_p)$ as it was done in [7] for the algebra $H_{bs}({\ell }_1).$ In addition, it would be interesting to consider an inner product on ${\mathcal{P}}_{\mathfrak{S}}(c_{00})$ and its corresponding Hilbert space—the completion of ${\mathcal{P}}_{\mathfrak{S}}(c_{00})$ with respect to the inner product norm. These kinds of Hilbert spaces in the case of symmetric polynomials were considered in [54].