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

is dense in X. Every such vector x is called hypercyclic for T. 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 C Φ on the space of entire functions on C; H(C) is said to be a composition operator if C Φ f(x) = f(Φ(x)) for some analytic map Φ : C → C. According to the Birkhoff result [2] the operator of composition with translation x 󳨃→ x + a, a ̸ = 0, and T a :


Introduction
Let  be a Fréchet linear space.An operator  :  →  is called hypercyclic if there is a vector  ∈  whose orbit under , Orb (, ) = {, ,  2 , . ..} , 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  Φ on the space of entire functions on C  ; (C  ) is said to be a composition operator if  Φ () = (Φ()) for some analytic map Φ : C  → C  .According to the Birkhoff result [2] the operator of composition with translation   →  + ,  ̸ = 0, and   : ()  → (+) is hypercyclic in the space of entire functions (C) on the complex plane C. Godefroy and Shapiro in [6] generalized this result for the translation operator on (C  ), 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 C.More detailed, they considered Hilbert spaces of entire functions of one complex variable for appropriated sequence of positive numbers and showed that if   / −1 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.

Journal of Function 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  0 ,  0 of , a sequence (  ) of positive integers, and a sequence of mappings (possibly nonlinear, possibly not continuous)    :  0 →  so that For background on analytic functions on Banach spaces we refer the reader to [12,13].

Symmetric Fock Spaces and Analytic Functions
Let  be a complex separable Hilbert space with an orthonormal basis (  ) ∞ =1 endowed with the scalar product ( | )  and the norm ‖‖  = ( | ) 1/2  , ,  ∈ .Clearly, for every  ∈ N 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 ‖ ⋅ ‖ ⊗  ℎ  on the vector space ⊗   such that the corresponding completion ⊗  ℎ  is a Hilbert space.More exactly, the scalar product on ⊗  ℎ  is defined by the equality for all   ,   ∈ ,  = 1, . . ., .Let [] denote a multi-index ( 1 , . . .,   ) ∈ N  .Since the system forms an orthonormal basis in ⊗  ℎ , every such vector  ∈ ⊗  ℎ  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 ⊗  ℎ ; that is, We denote by ⊗    the -fold symmetric algebraic tensor product of space .Every element from ⊗    can be defined by formula where form an orthogonal basis in the closure By Hermitian duality of a Hilbert space  we can define the relation Note that the classical symmetric Fock space F is the Hilbert direct sum of ⊗  ,ℎ ,  = 0, 1, . .., where ⊗ 0 ,ℎ  = C.This space is predual to a space of analytic functions on the unit ball of  [14].
We say that a Hilbert space F  with an arbitrary Hilbert norm ‖ ⋅ ‖  is (generalized) symmetric Fock space over a given Hilbert space  if vectors 1,  ()  [] = Thus F  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 F  .
Let us denote by H  the Hilbert space of analytic function   = ⟨(⋅) | ⟩  that is Hermitian duality to F  .We will use the same symbol ⟨⋅ | ⋅⟩  for the scalar product in H  .
We recall definition of reproducing kernel.
is a reproducing kernel for H  .
Since  generates the reproducing kernel of H  we say that  is a reproducing function of H  .Example 6.For an arbitrary positive integer  set where () = ( 1 , . . .,   ),  ∈ Z + .Thus We denote by  := { ∈  : ‖‖  < 1} unit ball on .It is easy to see that  () is an analytic map from the unit ball  ⊂  to F  () for every  and If  = 1 and  = C  , then this space is called Drury-Arveson Hardy space [17].As well the space coincides with Besov-Sobolev space  1/2 2 of analytic functions on open unit ball in C  .Note that H  () coincides with the classical Hardy space on the unit ball if (and only if) dim  = .

Hilbert Spaces of Entire Functions
In this section we consider the case when H  = F *  consists with entire functions on .Proposition 7. Suppose that there exists a constant  > 0 and a sequence of positive numbers (  ),   → 0 as  → ∞, such that where  ()  [] = ‖ and  is hence an entire mapping.
where  ∈ .Denote by  2 () the corresponding space H  .It is easy to see that  2 () consists of bounded-type entire functions on  and       ⊗() [] The reproducing kernel of this space is is analytic on the ball  ⊂  (where 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 () = 1/(1 − )  and the reproducing function  in Example 9 is generated by () =   .Corollary 11.Let () = ∑ ∞ =0     ,   > 0, be an entire function of one complex variable such that  +1 /  decreases to zero as n increases to ∞.Then  () is a reproducing function of a Hilbert space H  () of entire functions on the Hilbert space .
Let () = ∑ ∞ =0     be an entire function of one complex variable.We are interested to know the following: Under which conditions does  ∘ () = (()) belong to 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 () = ∑ ∞ =0     be an entire function of one complex variable.Then  ∘ () = (()) belongs to H  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

Differentiation and Translation Operators on H 𝜂
Let us consider a differentiation operator   : H  → H  : where  = (  ) ∈ ℓ 2 ,  ∈ .  is well defined on an appropriated dense subspace in We will make use of the following two lemmas (cf.[7]).
Proof.Let {   } ∈ be a maximal linearly independent subset of B, where  is a set of indexes.Fix  ∈  * , and assume that there exist nonzero constants   1 , . . .,    ∈ C so that Let  ∈  be arbitrary.Applying the differentiation operator   →   (⋅) in (39), it follows that Since {   } ∈ is linearly independent and   1 , . . .,    are nonzero, by (39) and (40) we have Proof.It is sufficient to establish that   ∈  for all  ∈  and  ≥ 1.To test this assertion we use the method of mathematical induction.When  = 1 the statement is obvious.Suppose the claim is true for  ≤  − 1.We prove this for  = .Since  ∈ , then for each 0 <  < 1 we have −1  ()  [] / √  (   ) [] } is bounded, where coefficients  ()   [] are defined in (13) Proof.The functions ) √ ()  [] = √ ()  []  ()   [] form an orthonormal basis for H  .Define the 1th-derivative of E ()  [] : where ( For a given  ∈  let an operator   : H  → H  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 H  , and where the series on the right converges in the norm operator topology.
Proof.It is well known, and not difficult to show, that   = ∑ ∞ =0 (1/!)   holds for the full space H  of entire functions, in the sense that when each term of the series on the right is applied to a function  ∈ H  , 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   = ∑ ∞ =0 (1/!)   converges in operator norm to a bounded operator on H  , and this bounded operator must be   .