Hypercyclic Behavior of Translation Operators on Spaces of Analytic Functions on Hilbert Spaces

Abstract

We consider special Hilbert spaces of analytic functions of many infinite variables and examine composition operators on these spaces. In particular, we prove that under some conditions a translation operator is bounded and hypercyclic.

1. Introduction

Let be a Fréchet linear space. An operator is called hypercyclic if there is a vector whose orbit under , is dense in . Every such vector is called hypercyclic for . It is well known that a hypercyclic operator can exist only in separable infinite-dimensional spaces (see [1]).

As for first results related to hypercyclic operators there are classical works of Birkhoff [2] and MacLane [3] showing that the operators of translation and differentiation, acting on the space of entire functions of one complex variable, are hypercyclic. There are many results related to hypercyclic operators on spaces of analytic functions on finite and infinite-dimensional spaces (see, e.g., [1, 4, 5]). Motivated by these results, we examine the hypercyclic behavior of composition operators on Hilbert spaces of entire functions of many infinite variables.

Let us recall that an operator on the space of entire functions on ; is said to be a composition operator if for some analytic map . According to the Birkhoff result [2] the operator of composition with translation , , and is hypercyclic in the space of entire functions on the complex plane . Godefroy and Shapiro in [6] generalized this result for the translation operator on , endowed with the topology of uniform convergence on compact subsets. Aron and Bès in [7] proved that the operator of composition with translation is hypercyclic in the space of weakly continuous analytic functions on all bounded subsets of a separable Banach space which are bounded on bounded subsets. Hypercyclic composition operators on spaces of analytic functions of finite and infinite many variables were studied also in [8]. In [9] Chan and Shapiro show that is hypercyclic in various Hilbert spaces of entire functions on . More detailed, they considered Hilbert spaces of entire functions of one complex variable with norms for appropriated sequence of positive numbers and showed that if is monotonically decreasing, then is hypercyclic.

The purpose of this paper is to prove a generalization of the Chan and Shapiro’s result for Hilbert spaces of entire functions of infinite many variables. In order to do it we consider in Section 1 a general construction of analytic functions on a Hilbert space which is related to generalized Fock space. In Section 2 we study special cases of Hilbert spaces of entire functions on a separable Hilbert space. In Section 3 we establish some conditions under which the translation operator is bounded and hypercyclic on these special spaces.

There is a general sufficient condition for hypercyclicity. This condition is inspired in the so-called Hypercyclicity Criterion given by Kitai [10] in her unpublished Ph.D. thesis and rediscovered by Gethner and Shapiro [11]. We use the general form of this Criterion as given in [7]. It may be stated as follows.

Theorem 1 (Hypercyclicity Criterion). Let be separable complete linear metric space and let be linear continuous operator. Suppose there exist , of , a sequence of positive integers, and a sequence of mappings (possibly nonlinear, possibly not continuous) so that (i) for pointwise on ,(ii) for pointwise on ,(iii) on ( is identity operator);
then is hypercyclic.

For background on analytic functions on Banach spaces we refer the reader to [12, 13].

2. Symmetric Fock Spaces and Analytic Functions

Let be a complex separable Hilbert space with an orthonormal basis endowed with the scalar product and the norm , . Clearly, for every the th tensor power is defined to be complex linear span of elements It is well-known that it is possible to define a norm on the vector space such that the corresponding completion is a Hilbert space. More exactly, the scalar product on is defined by the equality for all , . Let denote a multi-index . Since the system forms an orthonormal basis in , every such vector can be represented by the Fourier series expansion and we put It is clear that the above norm, generated by the scalar product, is a cross-norm on ; that is,

We denote by the -fold symmetric algebraic tensor product of space . Every element from can be defined by formula where and is the group of permutations on the set .

We will use the following notations for any . We denote by an arbitrary multi-index , , and . The vectors form an orthogonal basis in the closure of in and By Hermitian duality of a Hilbert space we can define the relation

Note that the classical symmetric Fock space is the Hilbert direct sum of , , where . This space is predual to a space of analytic functions on the unit ball of [14].

We say that a Hilbert space with an arbitrary Hilbert norm is (generalized) symmetric Fock space over a given Hilbert space if vectors , , (, , ) form an orthogonal basis in . Thus can be represented by the Hilbert direct sum of symmetric tensor powers:

Evidently, the norm is completely defined by its value on the basis vectors. Hence, setting by arbitrary positive numbers, we can get various symmetric Fock spaces over . Let be the scalar product in .

Put and . Let us consider a power series for any .

Theorem 2. Suppose that there are constants and such that for all multi-indexes , , and inequalities hold. Then there exists an open subset , such that (i)the series (13) is convergent for every and is an analytic map from into ,(ii)for every the map is an analytic function on ,(iii)the function is an -homogeneous polynomial and

We can find the proof in [15, Proposition 4.22].

Let us denote by the Hilbert space of analytic function that is Hermitian duality to . We will use the same symbol for the scalar product in .

For any vector denote such that . In particular, . Also by , we mean a vector from such that .

We recall definition of reproducing kernel.

Definition 3. Let be an abstract set and let be an Hilbert space of complex valued functions on with the scalar product . A function defined on is called reproducing kernel of closed subspace if (i)for any fixed , the kernel belongs to as a function of ;(ii)for any and for any Hilbert space is called space with reproducing kernel or functional Hilbert space.

Let be a function on such that for every and

Theorem 4. The function is reproducing kernel for .

We may see the proof in [16, p. 21].

Proposition 5. A map defined by is a reproducing kernel for .

The proof immediately follows from Theorem 4 for .

Since generates the reproducing kernel of we say that is a reproducing function of .

Example 6. For an arbitrary positive integer set where , . Thus We denote by unit ball on . It is easy to see that is an analytic map from the unit ball to for every and

If and , then this space is called Drury-Arveson Hardy space [17]. As well the space coincides with Besov-Sobolev space of analytic functions on open unit ball in . Note that coincides with the classical Hardy space on the unit ball if (and only if) .

Note that various Hilbert spaces of analytic functions of infinite many variables are studied in [18–22].

3. Hilbert Spaces of Entire Functions

In this section we consider the case when consists with entire functions on .

Proposition 7. Suppose that there exists a constant and a sequence of positive numbers , as , such that where and is an orthogonal basis in . Then is a Hilbert space of entire functions of bounded type that is bounded on bounded subsets on .

The proof is in [15, Proposition 4.25].

The next proposition gives another test for to be a space of entire functions.

Proposition 8. Suppose that decreases to zero as increases to infinity. Then consists with bounded-type entire functions, where

Proof. By the ratio test the power series is absolutely convergent for every . Thus, by the Cauchy-Hadamard formula, and is hence an entire mapping.

Example 9. Let where . Denote by the corresponding space . It is easy to see that consists of bounded-type entire functions on and The reproducing kernel of this space is and for every function from there exists such that for any vector . According to [22] is an infinite tensor product of where is the Lebesgue measure on .

Let be the open unit disk in . Denote by , , the set of all analytic functions on if and on if , such that , . Let be a stand for both and . Evidently, is an open convex subset of the Fréchet space of all analytic functions on .

Proposition 10. For a given and a Hilbert space the function is analytic on the ball (where if ) to and for every .

The proof is in [15, Proposition 4.28].

We say that is generated by . Note that the reproducing function in Example 6 is generated by and the reproducing function in Example 9 is generated by .

Corollary 11. Let , , be an entire function of one complex variable such that decreases to zero as n increases to . Then is a reproducing function of a Hilbert space of entire functions on the Hilbert space .

Let be an entire function of one complex variable. We are interested to know the following: Under which conditions does belong to for a given ? Let . It is easy to see that if where and then . So

If is generated by an analytic function in the means of (30), then (33) can be rewritten by So we have proved the following proposition.

Proposition 12. Let be an entire function of one complex variable. Then belongs to for a given if and only if And if is generated by an analytic function in the means of (30), then the condition may be written by In the case when we can write respectively.

4. Differentiation and Translation Operators on

Let us consider a differentiation operator : where , .

is well defined on an appropriated dense subspace in containing linear functionals. It is clear that is defined on functions , , if .

We will make use of the following two lemmas (cf. [7]).

Lemma 13. , is a linearly independent subset of .

Proof. Let be a maximal linearly independent subset of , where is a set of indexes. Fix , and assume that there exist nonzero constants so that Let be arbitrary. Applying the differentiation operator in (39), it follows that Since is linearly independent and are nonzero, by (39) and (40) we have Hence the set such that is maximal linearly independent subset which coincides with and so .

Lemma 14. Let be a nonempty open subset of a ball in with radius and center in . Suppose that for every . Then is dense in .

Proof. It is sufficient to establish that for all and . To test this assertion we use the method of mathematical induction. When the statement is obvious.
Suppose the claim is true for . We prove this for . Since , then for each we have So given , Thus, for in and . So the claim holds.

We will be interested in the operators of differentiation and translation on the Hilbert space .

Theorem 15. The operator is bounded on if and only if the set is bounded, where coefficients are defined in (13), , , , , and .

Proof. The functions form an orthonormal basis for . Define the 1th-derivative of : where .
We denote , , .
Let . If is bounded, then for each positive integer So .
Conversely, if , then for every holomorphic polynomial , we have Since the polynomials are dense in , it follows immediately that is bounded on , with norm .

For a given let an operator be defined as where is the th-Fréchet derivative of at the point towards .

Corollary 16. Suppose the set is bounded. Then each translation operator is bounded on , and where the series on the right converges in the norm operator topology.

Proof. It is well known, and not difficult to show, that holds for the full space of entire functions, in the sense that when each term of the series on the right is applied to a function , the result converges uniformly on bounded subsets of to the function (see, e.g., [7]).
Once we know this, it only remains to note that since is bounded (Theorem 15), the series on the right side of converges in operator norm to a bounded operator on , and this bounded operator must be .

It is relevant to remark that this result can be represented in the form whenever is bounded on .

Theorem 17. Let be a separable Hilbert space, , , , and the norm is defined on such that the differentiation operator is continuous, and satisfies condition of Proposition 12 for and every linear functional , for some . That is, , . Then the operator is hypercyclic.

Proof. Let be fixed element from . Consider the function defined by where . It is clear that is continuous and nonconstant function. So the sets, where , are both open and nonempty. Hence, according to Lemma 14, are both dense subspaces of . Next, notice that if , given , By (53), Also, by Lemma 13 there exists a linear map determined by which by (53) and (56) satisfies By Theorem 1 (Hypercyclicity Criterion), is hypercyclic.

Note that the translation operator is not hypercyclic in in Example 9, because it is discontinuous (see [23]).

Now we consider an example of a special Hilbert space of analytic functions satisfying conditions of Theorem 17.

Example 18. Let us denote by , where a Hilbert space which consists of bounded-type entire functions on and The reproducing kernel of this space is Since the set is bounded, the translation operator is bounded and since for , satisfies conditions of Proposition 12 and so is hypercyclic on .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.