On Algebraic Basis of the Algebra of Symmetric Polynomials on
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
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 ). 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  Nemirovskii and Semenov described algebraic bases of algebras of continuous symmetric polynomials on real spaces , where Their results were generalized by González et al.  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