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In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.
Let X be a complex Banach space and the algebra of all entire functions on This algebra is locally multiplicatively convex (locally m-convex) with respect to the topology of the uniform convergence on compact subsets of It is known (see, e.g., [1,2]) that if X has a Schauder basis, the spectrum of consists of the point evaluation functionals. Consequently, every homomorphism of can be represented as a composition operator with an analytic map. In applications we have to deal with some particular subalgebras of The subalgebra of comprising functions which are bounded on bounded subsets (functions of bounded type), , has been studied by many authors [3,4,5,6] and is always a proper subalgebra if X is infinite-dimensional (see , p. 157). Spectra and homomorphisms of can be explicitly described for some special cases of Banach spaces (e.g., if or X is the Tsirelson space [1,8,9]), while in the general case, the spectrum of may have a very complicated structure [3,6]. Thus, it is reasonable to consider some “smaller” subalgebras of analytic functions to obtain a complete and explicit description of their homomorphisms. There are a number of papers related to algebras, generated by linear functionals on X [4,10,11,12,13,14]. Another approach is related to the investigation of subalgebras of analytic functions that are invariant with respect to a group (or a semigroup) of operators on the underlying space. In [15,16], the authors considered algebras of symmetric analytic functions (with respect to the group of permutations of basis vectors) on These investigations were continued in [17,18] and in [19,20] for the case Various results in this direction for different subalgebras were obtained in [21,22,23,24,25,26,27,28,29,30,31,32].
In many cases, algebras of symmetric analytic functions with respect to a symmetric group are generated by a countable family of homogeneous polynomials. So, it makes sense to study algebras generated by a sequence of algebraically independent polynomials (countably generated algebras) in the general case. Such algebras were investigated in [33,34,35] and in  for the finite-dimensional case. In particular, spectra have been described for some special countably generated algebras of analytic functions on Banach spaces and including their homomorphisms, and some operations have been constructed on the spectra. In the presented paper, we continue these investigations for abstract countably generated algebras of analytic functions on Banach spaces.
Finitely generated algebras of polynomials and their spectra form a typical object in the classical invariant theory. For the countably generated case, it is important to work with algebras of analytic functions, and depending on the topology of the space of polynomials, we will have different algebras of analytic functions. The first question arising here is as follows: How many different countably generated algebras of analytic functions do we have? In other words, under which conditions can we guarantee that two given countably generated algebras of analytic functions are isomorphic? This question leads to another about the equivalence of algebraic bases and is connected to the theory on the automatic continuity of homomorphisms of Fréchet algebras. In this paper, we describe some conditions when two countably generated algebras are isomorphic and find many examples of non-isomorphic pairs of such algebras. Also, we construct a countably generated algebra, where all normalized algebraic bases of homogeneous polynomials are equivalent. In addition, we consider natural operators of derivations on countably generated algebras of analytic functions and find the conditions for which these operators are hypercyclic.
In Section 2, we introduce basic definitions and recall preliminary results on polynomials and analytic functions on Banach spaces. In Section 3, we consider homomorphisms on algebras of analytic functions on X that are invariant with respect to a semigroup of operators. Section 4 is devoted to the general case of algebras, generated by a sequence of homogeneous polynomials. We find some conditions when two such algebras are isomorphic and construct several examples of different countably generated algebras. In Section 5, we consider a special case of a countably generated algebra where all algebraic bases of homogeneous polynomials are equivalent. In Section 6, we investigate the hypercyclicity of derivations and operators constructed by derivations in countably generated algebras.
For general information on polynomials and analytic functions, we refer the readers to [7,37], and for information on hypercyclic operators, to [38,39]. Basic results of the Gelfand theory and its relationship with algebras of analytic functions on Banach spaces can be found in .
2. Definitions and Preliminary Results
Let X and Y be complex Banach spaces. A mapping is a continuous n-homogeneous polynomial if there exists a continuous n-linear map such that A map is said to be a continuous polynomial of degree n if where are k-homogeneous polynomials and For example, any continuous linear functional on X is a 1-homogeneous continuous polynomial with values in the set of complex numbers , and is an n-homogeneous continuous polynomial, while is a continuous polynomial of degree n, which is not homogeneous. The space of all continuous polynomials from X into Y is denoted by We will use notations for and for the space of continuous n-homogeneous complex-valued polynomials. The space is an algebra with pointwise multiplication. This algebra is locally m-convex with respect to the metrizable topology generated by the following countable family of norms
The completion of in this topology consists of all analytic functions on X which are bounded on every bounded subset of X (so-called functions of bounded type). Let us recall that a function is analytic (or holomorphic) if it is continuous and the restriction of f to every finite-dimensional subspace is an analytic function. Every analytic function on X can be represented by its Taylor’s series
We denote by the algebra of all analytic functions on is a proper subalgebra of whenever X is infinite-dimensional (, p. 157). Elements in are called analytic functions of unbounded type. The behavior of analytic functions of unbounded type was studied in [40,41].
A nonzero continuous complex valued homomorphism of a Fréchet algebra is a character of this algebra. The set of characters of a given algebra is called the spectrum of this algebra. The spectrum of is denoted by For every we denote by its restriction to the Banach space of n-homogeneous continuous polynomials Then, the radius function is defined as the infimum of such that is continuous with respect to the norm of uniform convergence in on the ball of radius r centered at the origin. According to ,
Conversely, if a sequence of functionals on spaces satisfies (1), then there exists a continuous linear functional on such that
where is the Taylor series expansion of . In , it is proved that Formula (1) is true for any subalgebra of For a given we denote by the Gelfand transform of that is,
A continuous linear operator acting on a separable Fréchet space E is called hypercyclic if there is a vector for which the orbit under T,
is dense in It is well known that any separable infinite-dimensional Fréchet space admits a hypercyclic operator. According to the Birkhoff result , for every complex number the operator is hypercyclic on the space A similar result for the space was obtained in . The hypercyclicity of translation operators for analytic functions on Banach spaces was considered in [44,45,46].
3. Subalgebras of Polynomials and Semigroups of Symmetry
Let be a subalgebra of Let be a set of affine continuous mappings such that for every and We will always suppose that contains the identity operator and is closed with respect to the operation of composition. Thus, is a unital semigroup which is called a semigroup of symmetry of , and polynomials that are invariant with respect to the action for any are called -symmetric. A subalgebra is said to be -complete if every -symmetric continuous polynomial belongs to Clearly, the set of all -symmetric continuous polynomials forms a -complete subalgebra in
Let us consider the natural partial order on the set of all semigroups of symmetry of with respect to the set theoretical inclusion. It is easy to see that if a polynomial is -symmetric and then P is -symmetric. Using standard arguments involving Zorn’s lemma, we have the following proposition.
For every subalgebra there exists a maximal semigroup of symmetry. The maximal semigroup is unique.
be a chain of semigroups of symmetry of Then, their union is a semigroup of symmetry of Thus, by Zorn’s lemma, the partially ordered set of all semigroups of symmetry of has a maximal element. Let and be two maximal elements. Clearly,
is a semigroup of symmetry of and since contains the identity operator it follows because for every From the maximality of we have that By the same reason, Hence, □
Let us consider two relations of equivalence on X associated with and a semigroup of symmetry of , respectively. For any pair , we say that if for every and if there are such that It is easy to check that implies but the converse statement is not true in general. We will use the following notations for the classes of equivalence
Let us denote by the closure of in In other words, is the minimal Fréchet subalgebra of containing Also, we denote by the minimal subalgebra of containing If is an analytic map, then the composition operator is a continuous homomorphism. If is of bounded type, then the restriction of to is a continuous homomorphism from to itself (see, e.g., ). The following theorem generalizes Theorem 2.2 in .
Let be a subalgebra of and a semigroup of symmetry of and Φ an analytic map from X to itself. If Φ is such that whenever and is -complete, then is a continuous homomorphism from to If, moreover, Φ is of bounded type, then the restriction of to is a continuous homomorphism from to itself.
The restriction of to is a continuous homomorphism from to By the definition of is -symmetric for every Let
be the Taylor series expansion of Since consists of affine maps, and
implies that is, all are -symmetric. But is -complete and so all are in Thus, For the case if is of bounded type, the proof is quite similar. □
For every , the functional is a continuous complex homomorphism (character) of and  and of any closed subalgebras of and as well. Note that with respect to the equivalence induced by if and only if act as functionals on In other words, the quotient set can be considered as a subset of the set of characters of (or ) with respect to the embedding Suppose that all characters of are of the form We do not know the answer to the following question: Is every homomorphism of the form for some analytic map of bounded type ?
Let us recall that a sequence of polynomials on X is algebraically dependent if there exists a number and a nonzero polynomial q of m complex variables such that
If a sequence of polynomials is not algebraically dependent then it is algebraically independent. The sequence is generating for if every polynomial in can be represented as a finite algebraic combination of polynomials in If is generating and algebraically independent, then it is called an algebraic basis of The algebra is countably generated if it has a generating sequence of polynomials. If is countably generated by a generating sequence , which is not a basis, then there are algebraic dependencies on that is, the family of nonzero polynomials of several but finite numbers of variables such that
Since the number of finite subsets of a countable set is countable, the set of algebraic dependencies is finite or countable.
The following proposition shows that any homomorphism of a countably generated algebra of polynomials can be defined on a solely algebraic basis.
Let be countably generated by a generating sequence and be a complex algebra. Then, any homomorphism is completely defined by its values on polynomials in That is, if and only if for every If are algebraic dependencies on then
Let be countably generated by a generating sequence and be a complex topological algebra. Then, any homomorphism is completely defined by its values on polynomials in
Since is dense in every continuous map on is completely defined by its values on and we can apply Proposition 2. □
Let us consider the following basic examples of algebras and the corresponding algebras of analytic functions.
Let Then, any semigroup of symmetry is trivial (consists of the identity operator). The set of continuous complex homomorphisms (the spectrum) of in the general case may be very complicated (see [3,5,6]) and due to the Aron–Berner extension [47,48] contains the second dual space If X has the approximation property, the spectrum of consists of point-evaluation functionals,
Let be the subalgebra of polynomials of finite type. Recall that P is a polynomial of finite type if it is a finite algebraic combination of continuous linear functionals. Every semigroup of symmetry, in this case, is also trivial. The closure of in is a so-called algebra of approximable analytic functions The spectrum of can be identified with up to the Aron–Berner extension. For some special cases like if X is finite-dimensional, , or X is the Tsirelson space, Note that is not countably generated if X is infinite-dimensional because to obtain all linear functionals, the generating sequence must contain a Hamel basis of but, as is well known, the Hamel basis of any infinite-dimensional Banach space is uncountable. However, if has a Schauder basis (i.e., topological basis) then the algebra of all algebraic combinations of functionals is a dense countably generated subalgebra of
Let and be the group of operators of the form
where σ passes over all bijections of to itself. Let be the algebra of all S-symmetric polynomials. In the literature, S-symmetric polynomials are referred as symmetric polynomials on It is well known [49,50] that is countably generated and polynomials
form an algebraic basis in By the definition of , it is S-complete but the group S is not maximal. For example, the following linear operator
does not belong to S but for every and According to , in if and only if for any and we have In other words, if we remove a finite number of polynomials then the algebra generated by the remaining polynomials will generate the same classes of equivalence on but it will obviously not be S-complete.
Spectra of algebras of symmetric analytic functions on were investigated in [15,16,17,18]. In particular, point evaluation functionals were constructed on for positive integers p, which are not point evaluation functionals.
Symmetric polynomials on and other Banach spaces are straightforward generalizations of classical symmetric polynomials of several variables. For classical symmetric polynomials and their applications, see . There are a number of other generalizations of symmetric polynomials for infinite dimensions.
Let be a semigroup of operators on generated by the following operators
-symmetric polynomials on are called subsymmetric. The algebra of subsymmetric polynomials on is denoted by It is known  that polynomials
form a linear basis in the space of N-homogeneous subsymmetric polynomials. From here it follows that is countably generated. However, we do not know how admits an algebraic basis (see [53,54] for some results in this direction). Also, nothing is known about the spectra of the corresponding algebras of analytic functions.
Let For any element X, we represent this as
(in the literature, the notation is also used), We consider the semigroup of symmetry generated by the following affine operators
where σ and ω are bijections of and -symmetric polynomials are called supersymmetric polynomials and it is known that the algebra of supersymmetric polynomials is -complete and admits the following algebraic basis
The properties of algebras generated by supersymmetric polynomials on Banach spaces, and structures of their spectra, were considered in [55,56]. Supersymmetric polynomials of several variables and related algebraic structures were studied in [57,58,59].
Let , and we denote by Ξ the group of measurable automorphisms of which preserve the Lebesgue measure. Ξ-symmetric polynomials are called symmetric polynomials on and the algebra of symmetric polynomials is denoted by If then is finitely generated and polynomials
form an algebraic basis in for . If then polynomials (2) are well defined and form an algebraic basis in . The algebra ) of symmetric analytic functions on its spectrum, and its analytic structures on the spectrum were studied in [19,20]. In particular, it was proved that the spectrum consists of the point evaluation functionals, homeomorphic to the space of exponential-type functions of a complex variable. These results can be generalized for spaces of essentially bounded integrable functions in the union of Lebesgue–Rohlin spaces .
If a semigroup has representations as semigroups of operators on Banach spaces then it naturally acts on the Cartesian product More precisely, if and is a representation of an element in then we can define a semigroup of operators on as a representation of in the following way:
Often, -symmetric polynomials on are called block-symmetric polynomials. Algebras of block-symmetric analytic functions were studied in [22,25,32,61,62,63] for the case when and are groups of permutations of basis vectors in , and in [20,31] for the case when and are groups of measurable, measure-preserving automorphisms of measure spaces
Let or Denote by the sequence of polynomials and by the algebra of polynomials on generated by The group of symmetry of is generated by linear operators
where is an n-power root of If then is not -complete. Indeed, every polynomial P on can be naturally defined on and is -invariant but does not belong to It is easy to check that for or is -complete.
4. Algebras Generated by Sequences of Polynomials
Let be a sequence of polynomials on a Banach space Throughout this section, we will assume that each is either a norm one n-homogeneous polynomial or is equal to zero, and nonzero polynomials in this sequence are algebraically independent. We consider the minimal unital algebra generated by polynomials in We denote by the closure of in and by the algebra of all entire functions
on X such that all Taylor polynomials are in The algebra is a topological algebra with respect to the topology of uniform convergence on compact subsets of Let be another algebraically independent sequence of n-homogeneous norm one (or zero) polynomials on a Banach space We consider the following question:
Under which conditions can the algebraic isomorphism
be extended to a topological isomorphism or to a topological isomorphism ?
Clearly, the sequence is an algebraic basis in the algebra , and if is another algebraic basis in then is isomorphic to However, in this case, is not necessarily continuous. The following example shows that may be discontinuous even if for some numbers with
Then, is the algebra of symmetric analytic functions of bounded type on and forms an algebraic basis in the algebra of all symmetric polynomials (see Example 3). In , it is proved that the mapping can be extended to a continuous homomorphism of if and only if a is a positive integer.
However, there are interesting and surprising examples where the isomorphism is continuous.
(see Example 6), and (see Example 8). In , it is proved that the mapping can be extended to a continuous isomorphism from to
Note that the existence of an isomorphism between and does not imply the existence of an isomorphism between and
In , it is proved that can be extended to a continuous isomorphism between and where as in Example 8. But, is not isomorphic to because contains a nontrivial function of unbounded type while .
We say that an algebraic basis in is equivalent to if is an isomorphism of to itself.
Since the sequence of polynomials forms an algebraic basis in and is a dense subalgebra in any continuous complex homomorphism on is completely defined by its values on the basic polynomials Thus, the spectrum of can be identified with the set
In particular, for point-evaluation functionals for we can write
Often, instead of it is convenient to use the following multi-valued map from X to defined by
where is the multi-valued n-root function. In , it is observed that maps the ball into the ball and if z is in the range of then
Let us suppose that that is, is a surjection from X to Then,
If , then the mapping can be extended to an algebraic isomorphism J from to
Since for every there exists such that If
and then So, g is well defined on Clearly, g is G-analytic (that is, the restriction of g to any finite-dimension subspace is analytic). If it is not continuous, then a homogeneous polynomial is discontinuous. However, any is an algebraic combination of and is continuous. Thus, and By the same reason, Hence, J is a bijection. The linearity and multiplicativity of J can be easily checked. □
Let Suppose that and Then, is a topological isomorphism from to
By Proposition 3, is an algebraic isomorphism. It is known (see, e.g., [65,66]) that there is a unique Fréchet topology on a commutative semi-simple algebra. Thus, any algebraic isomorphism between commutative semi-simple Fréchet algebras is topological, because if we have a discontinuous algebraic isomorphism, then preimages of open sets with respect to this isomorphism give us a different Fréchet topology. □
By Theorem 2 (c.f. ), if then So, we have the following corollary:
If then is isomorphic to
A subset is -bounding if every function in is bounded on
A Fréchet algebra is functionally continuous if every complex homomorphism on is continuous.
The statement that every Fréchet algebra must be functionally continuous is the famous Michael Conjecture, open since 1952 . There are many results (see, e.g., ) about classes of Fréchet algebra which are functionally continuous (e.g., algebras of analytic functions of several variables). On the other hand, it is known (, p. 240) that if X is an infinite dimensional Banach space with a topological basis, then is a so-called test algebra. That is, if every complex homomorphism on is continuous, then every Fréchet algebra is functionally continuous. In , it is proved that the countably generated algebra is a test algebra.
Suppose that and for every bounded subset there is an -bounding subset such that Then, the restriction of J to is an injective algebra homomorphism from to with a dense range. If is functionally continuous, then is continuous.
Let and Let W be a bounded subset in Then
because is bounding. Thus, g is of the bounded type. Hence, is an algebraic homomorphism from to The range of is dense because it contains all polynomials in It is known that every homomorphism from a functionally continuous Fréchet algebra to a semi-simple Fréchet algebra is continuous . Since is a semi-simple Fréchet algebra, the homomorphism is continuous if is functionally continuous. □
Let Suppose that for every bounded subset , there is an -bounding subset such that and vice versa, for every bounded subset there is an -bounding subset such that Then, is a topological isomorphism from to
By Theorem 3, both and are well defined and continuous, but Thus, is the isomorphism. □
The following theorem gives us a simple condition that is isomorphic to
Then, is a topological isomorphism from to
As in Proposition 3, for every , there exists such that If
and then So, the formal series
is well defined on We write that
If is well defined on and surjective, then it is bijective and so continuous. If is discontinuous, then it is not defined (as a homomorphism from to ), and so there is such is not of the bounded type.
Thus, the radius of boundedness of the formal series
is equal to Hence, there is a bounded sequence such that and the set
is unbounded for every subsequence of On the other hand, , and for the reason of compactness, the set of characters contains a cluster point Let As mentioned above,
is well defined. But,
and the right-hand series diverges because by the definition of there is a subsequence of such that
for every polynomial , which is a contradiction. □
In , it is proved that for every , there exists a point and such that is not defined. In other words, is an unbounded complex homomorphism of From this fact, it follows that
In , it is shown that the mapping from the algebra of supersymmetric analytic functions of the bounded type on (see Example 5) to the algebra of symmetric analytic functions of the bounded type on is a continuous homomorphism with a dense range. The inverse map is defined on a dense subspace and is unbounded. Thus, the spectrum of is a subset of in the sense that the restriction of any character in to the range of under is given as an element in On the other hand, from Theorem 4, we have that
Let be the backward shift operator on or Then, for a given algebra the operator is an isomorphism, where the sequence of polynomials is defined by Indeed, it is easy to check that and since is bounded, we can apply Corollary 4.
Let be the algebra of symmetric analytic functions of the bounded type on and
For every , there is a character such that in the following way. Let We set
Since the sequence is bounded and g is of the bounded type, the limit formally exists via a free ultrafilter. But, evidently, Thus, the limit does not depend on an ultrafilter and for every From here, it follows that maps onto Note that are not point evaluation functionals, in general. On the other hand, due to [17,18], we know that there is a family of “exceptional” characters on , which are not point-evaluation functionals, such that and for For the exceptional character , the point is such that Indeed,
If is a vector in such that only a finite number of coordinates and is not equal to zero. Then, there is no character such that for all (as in Example 15). The map is not projected.
If such a vector y and a character exist, then only for a finite number of But, this is impossible according to . If is projected, then it is bijective, and Thus,
is a character for every However, this is is not so, as we show above. □
5. The Equivalence of Algebraic Bases
We need the following technical result, which probably is known.
Let be a polynomial of n complex variables:
Then, for every such that for every and for every
We proceed by induction on In the case of , the polynomial p has the form
By the Cauchy estimate, for every and
Since it follows that
This completes the proof for the case.
Assume that the statement of the lemma holds for every Let be such that for every For fixed , the function is a polynomial of one complex variable. Note that
Since the statement of the lemma holds for polynomials of one complex variable, it follows that for every ,
The following theorem shows that for a special case of all normalized bases of homogeneous polynomials are equivalent.
Let be an algebraic basis in and and (i.e., is normalized). Then,
Since is an algebraic basis in the basis can be represented via in the following way
where for every and Let us first consider the case when and Then, from (8), it follows that each may depend only on Taking into account that and Lemma 1, we have that Thus, for every
where is the number of partitions of the integer number It is well known that
Conversely, suppose that Let us show that there exists such that Taking the inverse map of (8) for and , we have that
for some constants Using Lemma 1 and the same arguments as above, we obtain that Thus, , and so any algebraic basis in is equivalent to The general case can be obtained using Corollary 3. □
6. Derivations in Countable Generated Algebras
A linear operator D from algebra A to itself is a derivation if it satisfies the Leibniz rule
It is well known that any derivation on a semi-simple Fréchet commutative algebra with identity is continuous . To define a derivation D on the algebra of polynomials , it is enough to define operator D on the algebraic basis and extend it by the Leibniz rule and the linearity to the whole space If the operator D is continuous on then it can be extended to a derivation on , which we will denote by the same symbol
Let Z be a locally convex space endowed with topology generated on the basis of semi-norms A linear operator is continuous if and only if for every there exists such that for some constant C and every (see, e.g.,  (pp. 126–128)). In other words, if D is a derivation on then it is continuous if and only if for every number there is a number such that D is bounded as a linear operator between normed spaces and That is, there is a constant such that
Let be a function of exponential type on For a given derivation D on we consider the following operator :
Clearly, if then is an algebraic homomorphism. If D is continuous, then is so. Indeed,
for some and so this operator is continuous.
We denote by the closed subalgebra in generated by polynomials
Suppose that is such that is isomorphic to for every If D is a derivation on such that , then the operator
is hypercyclic, where γ is a nonconstant function of exponential type.
Let be the isomorphism from to given by Then, is an invariant subspace of and is a derivation on It is known that any derivation on is of the form
for some [70,71,72]. Taking into account that , we have
It is well known that this operator satisfies the hypercyclicity criterion for the entire sequence on Moreover, for every nonconstant function of exponential type the operator satisfies the hypercyclicity criterion for the entire sequence on . Thus, the restriction of to any satisfies the hypercyclicity criterion for the entire sequence. By , Lemma 3.2, the operator is hypercyclic on □
Let X be a complex Banach space. Let and be a set of polynomials on which have the following properties:
For every the mapping is a continuous -homogeneous polynomial, where ;
The set of polynomials is algebraically independent;
There exists a constant such that for every vector , there exists an element such that and for every where
Let A be a closed subalgebra of the Fréchet algebra such that for every function , each term of the Taylor series of this function is an algebraic combination of elements of the set Then, A and are isomorphic.
In particular, if each finite subsequence satisfies the conditions of Theorem 8 for every then is an isomorphism from to
A polynomial mapping
is said to be proper if for every compact set is a compact set in
It is known (see, e.g., , Chapter 15) that a proper map is surjective and open. In , it is proved that if polynomials are homogeneous, then is proper if and only if
The following proposition gives the conditions for which Theorem 7 is applicable.
Suppose that X contains an n-dimensional complex subspace such that the restriction of the polynomial map to is proper, then is an isomorphism from to
In the definition of the sequence , we already assumed that the homogeneous polynomials are algebraically independent. So, it is enough to check item (3) in Theorem 8. Since is proper, it is surjective and the preimage of the closed unit ball in is a bounded set in a ball of radius in Let and First, we consider the case when Then, is in the ball of radius R centered at the origin, and so if If then , because Thus, to check item (3) it is enough to show that there is a constant such that for every we have Let us suppose, on the contrary, that for every there exists such that We write where and Then,
Since we can find such for every there exists a sequence with such that as Let be a limit point of the sequence. Then, and But, this contradicts the property of proper maps in , mentioned above, suggesting that □
From Proposition 5, it follows that the algebras , as in Example 3, , as in Example 10, and , as in Example 8, and algebras of supersymmetric analytic functions of bounded type on satisfy the conditions of Theorem 8.
Let or and Then, for the algebra , we cannot apply Proposition 5 because the polynomial mapping
is not proper.
For the algebra , the derivation defined on the algebraic basis by is well defined and continuous only if (c.f. ). Here, and is the Kronecker symbol. The operator can be computed by
In other words, the homomorphism is defined on the algebraic basis by
For the same reason, if is continuous, then the homomorphism defined by
is a continuous homomorphism of such that From , it follows that is hypercyclic if
Considering various examples of countably generated algebras of analytic functions on Banach spaces, we can see that these algebras may have quite different structures. For example, all algebraic bases of homogeneous polynomials on the algebra of symmetric analytic functions on are equivalent, while the set of equivalent bases of the algebra of symmetric analytic functions of the bounded type on is very restricted. However, we find some general conditions when the algebras are isomorphic, and the corresponding algebraic bases are equivalent. In addition, we considered derivations in countably generated algebras of analytic functions of the bounded type, and find the conditions of the hypercyclicity of operators, related to the derivations.
The obtained results pave the way for further studies of algebraic and analytic structures on spectra of countably generated algebras of analytic functions on Banach spaces. In particular, for many reasons, it is interesting to define an analytic manifold structure on the spectrum of a given algebra of analytic functions so that the set of Gelfand transforms of the characters coincides with a natural algebra of analytic functions on this manifold. Also, it is interesting to consider derivatives of the countably generated algebras from the point of view of Lie Algebras Theory. These and other problems will be considered in further investigations.
Conceptualization, A.Z.; investigation, Z.N. and S.V.; writing—original draft preparation, Z.N. and S.V.; writing—review and editing, A.Z.; project administration, A.Z. All authors have read and agreed to the published version of the manuscript.
This research was supported by the National Research Foundation of Ukraine, 2020.02/0025.
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
Conflicts of Interest
The authors declare no conflict of interest.
Mujica, J. Ideals of Holomorphic Functions on Infinite Dimensional Spaces. In Finite or Infinite Dimensional Complex Analysis; Kajiwara, J., Li, Z., Shon, K.H., Eds.; CRC Press: Boca Raton, FL, USA, 2000; pp. 337–343. [Google Scholar]
Schottenloher, M. Spectrum and envelope of holomorphy for infinite dimensional Riemann domains. Math. Ann.1983, 263, 213–219. [Google Scholar] [CrossRef]
Aron, R.M.; Cole, B.J.; Gamelin, T.W. Spectra of algebras of analytic functions on a Banach space. J. Reine Angew. Math.1991, 415, 51–93. [Google Scholar]
Vasylyshyn, T.V. Symmetric polynomials on the Cartesian power of Lp on the semi-axis. Mat. Stud.2018, 50, 93–104. [Google Scholar] [CrossRef]
Vasylyshyn, T. Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power p∈[1,+∞) functions. Carpathian Math. Publ.2021, 13, 340–351. [Google Scholar] [CrossRef]
Vasylyshyn, T.V. The algebra of symmetric polynomials on (L∞)n. Mat. Stud.2019, 52, 71–85. [Google Scholar] [CrossRef]
Vasylyshyn, T. Symmetric analytic functions on the Cartesian power of the complex Banach space of Lebesgue measurable essentially bounded functions on [0,1]. J. Math. Anal. Appl.2022, 509, 125977. [Google Scholar] [CrossRef]
Zagorodnyuk, A.; Kravtsiv, V.V. Spectra of algebras of block-symmetric analytic functions of bounded type. Mat. Stud.2022, 58, 69–81. [Google Scholar] [CrossRef]
Halushchak, S. Spectra of Some Algebras of Entire Functions of Bounded Type, Generated by a Sequence of Polynomials. Carpathian Math. Publ.2019, 11, 311–320. [Google Scholar] [CrossRef]
Halushchak, S.I. Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces. Mat. Stud.2021, 56, 106–112. [Google Scholar] [CrossRef]
Vasylyshyn, S. Spectra of Algebras of Analytic Functions, Generated by Sequences of Polynomials on Banach Spaces, and Operations on Spectra. Carpathian Math. Publ.2023, 15, 104–119. [Google Scholar] [CrossRef]
Aron, R.M.; Falcó, J.; García, D.; Maestre, M. Algebras of symmetric holomorphic functions of several complex variables. Rev. Mat. Complut.2018, 31, 651–672. [Google Scholar] [CrossRef]
Mujica, J. Complex Analysis in Banach Spaces; North-Holland: Amsterdam, The Netherlands; New York, NY, USA; Oxford, UK, 1986. [Google Scholar]
Bayart, F.; Matheron, E. Dynamics of Linear Operators; Cambridge University Press: New York, NY, USA, 2009. [Google Scholar]
Grosse-Erdmann, K.G.; Peris Manguillot, A. Linear Chaos; Springer: London, UK, 2011. [Google Scholar] [CrossRef]
Ansemil, J.M.; Aron, R.M.; Ponte, S. Behavior of entire functions on balls in a Banach space. Indag. Math.2009, 20, 483–489. [Google Scholar] [CrossRef]
Ansemil, J.M.; Aron, R.M.; Ponte, S. Representation of spaces of entire functions on Banach spaces. Publ. Res. Inst. Math. Sci.2009, 45, 383–391. [Google Scholar] [CrossRef]
Birkhoff, G.D. Démonstration d’un théorème élémentaire sur les fonctions entières. C. R. Acad. Sci. Paris1929, 189, 473–475. [Google Scholar]
Godefroy, G.; Shapiro, J.H. Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal.1991, 98, 229–269. [Google Scholar] [CrossRef]
Chernega, I.; Holubchak, O.; Novosad, Z.; Zagorodnyuk, A. Continuity and hypercyclicity of composition operators on algebras of symmetric analytic functions on Banach spaces. Eur. J. Math.2020, 6, 153–163. [Google Scholar] [CrossRef]
Novosad, Z.; Zagorodnyuk, A. Polynomial automorphisms and hypercyclic operators on spaces of analytic functions. Arch. Math.2007, 89, 157–166. [Google Scholar] [CrossRef]
Aron, R.M.; Berner, P.D. A Hahn-Banach extension theorem for analytic mappings. Bull. Soc. Math. Fr.1978, 106, 3–24. [Google Scholar] [CrossRef]
Davie, A.M.; Gamelin, T.W. A theorem on polynomial-star approximation. Proc. Am. Math. Soc.1989, 106, 351–356. [Google Scholar] [CrossRef]
González, M.; Gonzalo, R.; Jaramillo, J.A. Symmetric polynomials on rearrangement-invariant function spaces. J. Lond. Math. Soc.1999, 59, 681–697. [Google Scholar] [CrossRef]
Nemirovskii, A.; Semenov, S. On polynomial approximation of functions on Hilbert space. Mat. USSR-Sb.1973, 21, 255–277. [Google Scholar] [CrossRef]
Macdonald, I.G. Symmetric Functions and Orthogonal Polynomials; University Lecture Serie 12; AMS: Providence, RI, USA, 1997. [Google Scholar]
Gonzalo, R. Multilinear forms, subsymmetric polynomials, and spreading models on Banach spaces. J. Math. Anal. Appl.1996, 202, 379–397. [Google Scholar] [CrossRef]
D’Alessandro, S.; Hájek, P.; Johanis, M. Erratum to: “Polynomial algebras on classical Banach spaces”. Isr. J. Math.2015, 207, 1003–1012. [Google Scholar] [CrossRef]
Hájek, P. Polynomial algebras on classical Banach Spaces. Isr. J. Math.1998, 106, 209–220. [Google Scholar] [CrossRef]
Chernega, I.V. A semiring in the spectrum of the algebra of symmetric analytic functions in the space ℓ1. J. Math. Sci.2016, 212, 38–45. [Google Scholar] [CrossRef]
Jawad, F.; Zagorodnyuk, A. Supersymmetric polynomials on the space of absolutely convergent series. Symmetry2019, 11, 1111. [Google Scholar] [CrossRef]
Olshanski, G.; Regev, A.; Vershik, A.; Ivanov, V. Frobenius-Schur Functions. In Studies in Memory of Issai Schur. Progress in Mathematics; Joseph, A., Melnikov, A., Rentschler, R., Eds.; Birkhauser: Boston, MA, USA, 2003; Volume 210, pp. 251–299. [Google Scholar]
Sergeev, A.N. On rings of supersymmetric polynomials. J. Algebra2019, 517, 336–364. [Google Scholar] [CrossRef]
Stembridge, J.R. A Characterization of Supersymmetric Polynomials. J. Algebra1985, 95, 439–444. [Google Scholar] [CrossRef]
Vasylyshyn, T.; Zhyhallo, K. Entire Symmetric Functions on the Space of Essentially Bounded Integrable Functions on the Union of Lebesgue-Rohlin Spaces. Axioms2022, 11, 460. [Google Scholar] [CrossRef]
Kravtsiv, V. Algebraic basis of the algebra of block-symmetric polynomials on ℓ1⊕ℓ∞. Carpathian Math. Publ.2019, 11, 89–95. [Google Scholar] [CrossRef]
Kravtsiv, V.V. Analogues of the Newton formulas for the block-symmetric polynomials. Carpathian Math. Publ.2020, 12, 17–22. [Google Scholar] [CrossRef]
Kravtsiv, V. Zeros of block-symmetric polynomials on Banach spaces. Mat. Stud.2020, 53, 206–211. [Google Scholar] [CrossRef]
Zagorodnyuk, A.; Hihliuk, A. Entire analytic functions of unbounded type on Banach spaces and their lineability. Axioms2021, 10, 150. [Google Scholar] [CrossRef]
Carpenter, R.L. Uniqueness of topology for commutative semisimple F-algebras. Proc. Am. Math. Soc.1971, 29, 113–117. [Google Scholar] [CrossRef]
Narici, L.; Beckenstein, E. Topological Vector Spaces; CRC Press: Boca Raton, FL, USA, 2010. [Google Scholar] [CrossRef]
Becker, J. A note on derivations of algebras of analytic functions. J. Reine Angew. Math.1978, 297, 211–213. [Google Scholar]
Brummelhuis, R.G.M.; de Paepe, P.J. Derivations on algebras of holomorphic functions. Indag. Math.1989, 92, 237–242. [Google Scholar] [CrossRef]
Nandakumar, N.R. An application of Nienhuys-Thiemann’s theorem to ring derivations on H(G). Indag. Math.1988, 50, 199–203. [Google Scholar] [CrossRef]
Rudin, W. Function Theory in the Unit Ball of n; Reprint of the 1980 Edition. Classics in Mathematics; Springer: Berlin, Germany, 2008. [Google Scholar]
Rudin, W. Proper holomorphic maps and finite reflection groups. Indiana Univ. Math. J.1982, 31, 701–720. [Google Scholar] [CrossRef]
Chernega, I.; Fushtei, V.; Zagorodnyuk, A. Power Operations and Differentiations Associated With Supersymmetric Polynomials on a Banach Space. Carpathian Math. Publ.2020, 12, 360–367. [Google Scholar] [CrossRef]
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