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Kravtsiv, Victoriia; Vasylyshyn, Taras; Zagorodnyuk, Andriy

doi:https://doi.org/10.1155/2017/4947925

Journal of Function Spaces

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Abstract Introduction Materials and Methods Results and Discussion Conclusions Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 4947925 | https://doi.org/10.1155/2017/4947925

On Algebraic Basis of the Algebra of Symmetric Polynomials on

Victoriia Kravtsiv,¹Taras Vasylyshyn,¹and Andriy Zagorodnyuk¹

Academic Editor: Alberto Fiorenza

Received28 Apr 2017

Accepted28 Sept 2017

Published30 Oct 2017

Abstract

We consider polynomials on spaces of -summing sequences of -dimensional complex vectors, which are symmetric with respect to permutations of elements of the sequences, and describe algebraic bases of algebras of continuous symmetric polynomials on

1. Introduction

Algebras of polynomials and analytic functions on a Banach space which are invariant (symmetric) with respect to a group of linear operators acting on were studied by a number of authors [1–10] (see also a survey [11]). If has a symmetric structure, then it is natural to consider the case when is a group of operators which preserve this structure. In particular, if is a rearrangement-invariant sequence space, then is used to be the group of permutations of positive integers. In [8] Nemirovskii and Semenov described algebraic bases of algebras of continuous symmetric polynomials on real spaces , where Their results were generalized by González et al. [7] to real separable rearrangement-invariant sequence spaces.

Algebraic basis plays a crucial role in the problem of description of spectra of algebras generated by polynomials [1–4]. For example, each complex homomorphism on the algebra of symmetric polynomials on is completely defined by its values on the basis elements.

Note that an algebra of symmetric functions essentially depends on a representation of a given group on In particular, in [12–14] the group of permutations of positive integers was considered which acts on the complex space permutating “blocks” of coordinates. Polynomials which are invariant with respect to the action are called block-symmetric. It is natural to consider such polynomials as symmetric polynomials on

In this work we get an explicit description of algebraic bases of algebras of symmetric polynomials on , where

2. Materials and Methods

We denote by the set of all positive integers and by the set of all nonnegative integers.

A mapping , where is a complex Banach space, is called an -homogeneous polynomial if there exists an -linear form such that is the restriction to the diagonal of , that is, for every By [15, Corollary ], -homogeneous polynomial is continuous if and only if its norm is finite. Definition of -homogeneous polynomial implies the inequality for every A mapping , where and is a -homogeneous polynomial for every , is called a polynomial of degree at most

Let and Let us denote the vector space of all sequences where for , such that the series is convergent. The space with norm is a Banach space.

Definition 1. A function is called symmetric if for every and for every bijection , where

Let us denote the algebra of all symmetric continuous polynomials on

3. Results and Discussion

3.1. Power Sum Symmetric Polynomials on

For a multi-index let For every such that , where is a ceiling of , let us define a mapping by Also we set Note that is a symmetric -homogeneous polynomial. Polynomials are generalizations of so-called power sum symmetric polynomials on finite-dimensional spaces (see, e.g., [16, page 23] or [17, page 297]).

Proposition 2. For and for every such that , polynomial on is continuous and

Proof. Let such that Note that Since for every and , it follows that for every Note that Therefore, Since , it follows that for every and Therefore, Thus, Therefore, Hence, is bounded and, consequently, it is continuous.

For , let be the space of all sequences , where and Note that is isomorphic to Let Note that is a dense subspace in Also note that is well-defined on for every

For arbitrary , we set For , let Note thatAlso note that for every , such that ,

For every and , we setAlso we set For , letwhere for

Let us define a partial order on by the following way. For we set if and only if there exists such that for every We write , if and

Proposition 3. For such that and for arbitrary where, by the definition, product of an empty set of multipliers is equal to In particular,

Proof. By (12) and (14), By the definition of , Therefore,Let Then there exists such that for every For such that , we have that too. Consequently, for such we have , and, by (13),Therefore, by (18), In the case we have Let Then we have two cases. Case 1. There exists such that Then therefore, Case 2. There exists such that Then It is known thatfor every and Therefore, and, consequently,

Let us prove the following auxiliary proposition.

Proposition 4. A function , where and , is strictly decreasing.

Proof. Let us prove that for every Note that Therefore,Since and for every , it follows that

Corollary 5. For every and for every

For an arbitrary nonempty finite set let us define a mapping , where is the cardinality of , bywhere is an -dimensional vector of values of on , indexed by We endow the space with norm , where .

Theorem 6. Let be a finite nonempty subset of such that for every Then(i)there exists such that for every there exists such that ;(ii)there exists a constant such that if , then for every

Proof. (i) Let For every , let us define and by the following way. For minimal elements of the partially ordered set , let and , where is defined by (14) and For , which are not minimal elements of , we define and inductively byWe set . Note that , where For , by (12), Since is a -homogeneous polynomial,By Proposition 3, is not equal to zero only for such that Therefore, By Proposition 3, , and therefore, by (33), Hence, Taking into account (30), we have Hence,
(ii) Let be such that For let Note that for minimal elements we have
LetLetand for every let whereAlso we set
Note that for every Let us prove that for every We proceed by induction on In the case , we have , and therefore, Since , it follows that If , then (42) is proved. Let and Suppose that inequality (42) holds for every such that Let us prove (42) for such that By (31) and (37),By (30), Since is a -homogeneous polynomial on the space and , Therefore, taking into account , we have Therefore,By Proposition 4,Note that if , then Therefore,Since ,By the induction hypothesis, if , where , then Therefore,Since , it follows that for every , and therefore,Since , by (41),By (47)–(53),By (43) and (54), Hence, inequality (42) holds for every
By (11) and by Proposition 4, By (42),Set We have that if By Corollary 5, for every

Corollary 7. Let such that for every Then there exists such that for every polynomials are algebraically independent on

Proof. By Theorem 6, there exists such that for every there exists such thatfor every Let us show that are algebraically independent on for every Let be a polynomial such that for every Set Taking into account (57), we have for arbitrary , that is, Hence, are algebraically independent.

3.2. Algebraic Basis of the Algebra

Theorem 8. Every -homogeneous polynomial , where is an arbitrary positive integer, can be represented as an algebraic combination of polynomials , where such that

Proof. We proceed by induction on In the case for , we have where Suppose the statement holds for and prove it for Let and Then can be represented as a sum of terms where such that , and is an -homogeneous polynomial. Note that , and therefore, by the induction hypothesis, can be represented as an algebraic combination of , where such that Note that Therefore, can be represented as an algebraic combination of and Since and are symmetric, it follows that together with term where and , the sum must contain terms where Therefore, can be represented as a sum of terms Since , where , it follows that is an algebraic combination of polynomials , where such that

Theorem 9. Let be a symmetric -homogeneous polynomial. Let There exists a polynomial such that , where the mapping is defined by (28).

Proof. By Corollary 7, there exists such that for every polynomials , where , are algebraically independent. Therefore, the representation, given by Theorem 8 for the restriction of to , is unique. Thus, for every there exists a unique polynomial such that for every Since , it follows that is the restriction of to By Theorem 6, , and therefore, Let Then for every

Theorem 10. Polynomials , where , form an algebraic basis of the algebra

Proof. Let us prove that every symmetric continuous polynomial on can be uniquely represented as an algebraic combination of polynomials . It suffices to prove the statement only for homogeneous polynomials. Let be a symmetric continuous -homogeneous polynomial. By Theorem 9, the restriction of to can be uniquely represented as an algebraic combination of polynomials , where such that Since is dense in and polynomials are well-defined and continuous on , it follows that given representation can be extended to

3.3. Algebraic Basis of the Algebra

Let In this section, we describe an algebraic basis of the algebra

Let us prove a complex analog of [8, Lemma ].

Lemma 11. Let and be an orthogonal projection: Let and for every open set a set is unbounded. If polynomial is bounded on , then does not depend on

Proof. Suppose that depends on Then where and Note that on , and therefore, there exists point such that Since is open and is continuous, there exists such that and , where is an open ball with center and radius in the space Note that, for ,where and for Note that for the polynomial there exists such that if , then , that is, Therefore, if , thenSince is unbounded, there exists a sequence such that as Taking into account (66) and (67), we have as , which contradicts the boundedness of on

For , let and

Lemma 12. For if and , then

Proof. Since , there exists such that , and Therefore, if for some , then too. It means that On the other hand, Therefore, ; that is, for , we have that if and only if Therefore, for every we have that Since , there exists such that Therefore,

For and letBy Theorem 6, for there exists such that contains the open unit ball of the space with norm , where

Proposition 13. For , let be a polynomial on If is bounded on , then does not depend on such that and

Proof. Let such that and Let and be an orthogonal projection, defined by Let us show that, for every ball with center and radius such that , a set is unbounded. Since , there exists such that For , we set , where is defined by (14). Choose such that where is a Riemann zeta-function. Let Let us show that By (11), Therefore, By the triangle inequality, Since , it follows that Hence,
Note that for arbitrary such that , by (12), Let us show that For such that , by Proposition 3, , and therefore, by (78), Let be such that If , then , since If and , then by Lemma 12. Hence, in both cases. By (78), Since , it follows that Since , it follows that Taking into account and , we have that Since and are integer numbers and , it follows that , and therefore, Hence,Since , it follows that , and therefore,
By Proposition 3, , and therefore, by (78), as Hence, is unbounded. By Lemma 11, does not depend on

Theorem 14. Let be an -homogeneous polynomial. If , then Otherwise, there exists a unique polynomial such that , where and is defined by

Proof. Let be the restriction of to Note that is a continuous symmetric -homogeneous polynomial. By Theorem 9, there exists a unique polynomial , where such that Since is continuous, is bounded on , defined by (71). Therefore, is bounded on
Let us prove that does not depend on arguments such that by induction on By Proposition 13, for we have that does not depend on arguments such that and Suppose that the statement holds for , where , that is, does not depend on arguments such that and Then the restriction of to , by Proposition 13, does not depend on such that and Hence, does not depend on such that
Since polynomials , where , are well-defined and continuous on and is dense in , it follows that Note that in the case we have and, therefore,

Corollary 15. Polynomials , where , form an algebraic basis of the algebra

4. Conclusions

Power sum symmetric polynomials on are algebraically independent and form an algebraic basis of the algebra of all continuous symmetric polynomials on

Results of this work generalize results of works [7, 8, 14].

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

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Copyright © 2017 Victoriia Kravtsiv et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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