Hypercyclic convolution operators on entire functions of Hilbert-Schmidt holomorphy type

A theorem due to G. Godefroy and J. Shapiro states that every continuous convolution operator, that is not just multiplication by a scalar (non-trivial), is hypercyclic on the space of entire functions in n variables endowed with the compact-open topology. We study the space of entire functions of Hilbert-Schmidt type H H (E) on a Hilbert space E. We characterize its continuous convolution operators and prove the following: Every continuous non-trivial convolution operator is hypercyclic on H H (E).


Introduction
A cyclic (hypercyclic) vector for an operator T : : is a vector x such that the closed linear hull (closed hull) of the orbit O(T, x) -{x, Tx, ...} under the operator is the entire space. An operator T is cyclic (hypercyclic) whenever there exists a cyclic (hypercyclic) vector. Recall that an invariant subset for an operator T : X -X is a subset S C X such that T S C S. Thus every orbit constitutes an invariant set and the invariant sets are called trivial. Note that the closed linear hull of an orbit under a continuous operator is the smallest closed invariant subspace that contains the vector under consideration. Consequently, a continuous operator lacks non-trivial invariant closed subspaces (subsets) if and only if every non-zero vector is cyclic (hypercyclic).
The theory of cyclic and hypercyclic operators is a natural part of the study of invariant subspaces and the approximation theory. An overview of the theory is exposed in [7]. The most natural problems are maybe (1): given an operator T : X --~ X, is it hypercyclic and (2): given a space X, does it admit a hypercyclic operator T : X -~ X. For example, it is known that no linear operator on a finite dimensional space is hypercyclic but every separable infinite-dimensional Frechet space carries a hypercyclic operator (see [7] for more on this).
Godefroy and Shapiro show in [6] that every continuous non-trivial convolution operator is hypercyclic on the (Fréchet-) space of entire functions in n-variables (a convolution operator is an operator that commutes with all translations and it is called trivial when it is given by x H ax for soine scalar a). It is known that the continuous convolution operators are the operators of the form 03C6(D), p(D) f ~ 03A303B1~Nn 03C603B1D03B1 f where 03C6 = 03A303B1~Nn 03C603B1y03B1 is an entire exponential type function in n variables. Thus, in particular, every operator of translation is hypercyclic and the one variable version of this particular result was obtained by Birkhoff already in the twenties [2]. Before Godefroy and Shapiro obtained their general result, MacLanc [11] ha.d established the hypcrcyclicity of differentiation D on the one variable entire functions. Hypercyclic properties of exponential type differential operators on spaces of holomorphic functions with infinite dimensional domains, have 1991 Mathematics Subject Classification: Primary: 46G20, 47A15, Sccondary: 46A32, 46E25 also been studied (see for example [1]). In this note we prove the analogue of Godefroy and Shapiro's result for entire functions of Hilbert-Schmidt type HH(E) on a (separable) Hilbert space E (Theorem 3.1). HH(E) is a separable Fréchet space and is built up of homogenous Hilbert-Schmidt polynomials. A similar, but different, type of holomorphy is studied in [4]. In fact, we prove that every continuous non-trivial convolution operator has a dense set of hypercyclic vectors but that there is a certain dense subspace for which every such type of hypercyclic vector must be outside. This result is interesting in view of a result of the following type: There exists a continuous linear operator on ~1 for which every non-zero vector is hypercyclic (due to Read [14] and it is not known whether we can replace .~1 with an infinite-dimensional separable Hilbert space (see [7] page 359)).
For our purpose we make use of the following well-known theorem due to Gethner, Godefroy, Shapiro, Kitai ([5], [6], [10]). The theorem is based on the Baire Category Theorem and gives a criterion, known as the Hypercyclicity Criterion, for an operator to be hypercyclic. Theorem 1.1 (Hypercyclicity Criterion) Let X be a separable Fréchet space and let T : X -~ X be a continuous linear operator. Assume that T satisfies the following (hypercyclicity) criterion (HC): there are dense subsets Z, Y C X and a map S : Y -~ Y such that 1.
Then T is hypercyclic. , We emphasize that the subsets Z, Y and the operator S in the hypothesis need not to be linear. Moreover, it is not necessary that the map S is continuous. It is known that (HC) is not a necessary condition for an operator to be hypercyclic. We shall say that an operator T (on an arbitrary locally convex Hausdorff space X) satisfies the Strong Hypercyclicity Criterion (SHC) when it satisfies the condition (HC) in such a way that the set Z can be chosen as an invariant set for T.

Hilbert-Schmidt entire functions and convolution operators
In this section we introduce the space of entire functions of Hilbert-Schmidt type and characterize its continuous convolution operators.
If X is a complex vector space, we denote by ) the complex valued Gateaux holomorphic functions on X. . If f E we denote by the n:th directional derivative at x along y. Let E be a separable complex Hilbert space (we shall tacitly assume everywhere below that all vector spaces are complex and that all Hilbert spaces are separable). We denote by PF(nE) C HG(E) the space of n-homogenous polynomials on E of finite type. That is.
is the subspace of the n-homogenous polynomials on E, spanned by the elements ( ~, y) n, y E E, where (., .) denotes the inner product on E. We endow PF(nE) with the inner product defined by ((.,y)n, (.,z)n)n ' n!(z, y)n (More precisely, by the assumption on E we can identify the symmetric tensors @n,sE with PF(nE) and (., .)n is the inner product is induced from the inner product space in this way). The n-homogenous Hilbert-Schmidt polynomials, denoted by PH (nE), is the completion Of w.r.t. the inner product (~, ~)n. . ~ye use the symbol ,~~ for the corresponding norm. In view of our purposes, it is convenient to note that Let (e j) be an orthonormal basis in E. For a given multi-index a E N~ ~ ~~k=1N, let Here suppa D {~ : 0} and H = ~ aj. The elements ea, |03B1| = n, form an orthogonal basis for PH(nE) and ~e03B1~2n = a! = 03B11!...
(this follows from Lemma 1 in (4J). Thus PH(nE) can be identified with the space of all sequences (Pa) such that 03A3|03B1|=n |P03B1|203B1! oo and in this way we have that Let us note the following. The n-homogenous nuclear polynomials PN(nE) and the continuous polynomials PC(nE) can be put in duality by passing to the limit out of the inner product (., .)n on PF(nE). In this way we have that PC(nE) is the topological dual of PN(nE) (see Dineen [3] or Gupta [8] for further details). Recall that PN(nE) is the Banach space obtained from the completion of PF(nE) w.r.t. the nuclear norm. We have the following (continuous) injections (2. 3) The following lemma is crucial for our investigation and can, at this stage, only be found in a preprint [13j. Therefore we include here a proof. (2.5) It suffices to prove that the right hand side defines an element R in PH(n+mE), i.e. that 0 3 A 3 | 0 3 B 3 | = n + m | R 0 3 B 3 | 2 0 3 B 3 ! oo. Indeed, then both PQ and R define continuous polynomials and since they coincide on Ej M span{ el, ..., ej} for all j, we deduce that PQ = R.
We have that is the index set in the sum in (2.5) and denotes the number of elements in J03B3(m). We derive an estimate for by using arguments from the probability theory. Consider a bowl with |03B3| objects of # supp 03B3 different kinds and of 03B3j of sort j E supply respectively. Assume that we pick m objects from the bowl.  This estimate completes the proof.
Given r > 0 we denote by EXPr(E) the (Banach-) space of all p E H(E) such that for some AI > 0, ~03C6n~n n = 0,... equipped with the norm = supn The symbol EXPH (E) denotes the union Ur>0EXPr(E) equipped with the corresponding inductive locally convex topology. Thus EXPH (E) is given by all cp 2lH(E) such that l i m ( n !0 3 C 6 ñn ) 1 / n ~. E;ery cp E EXPH(E) defines an exponential type function, i.e. a Gateaux holomorphic function with |03C6|(y)| ~ Mer~yf or some > 0, and its power series converges in EXPH(E). A proof of the "finitedimensional" analogue of the following proposition can be found in [15] (see also [12] page 320). (2.10) Hence f = f ~ 03A3 n E HH(E). Further, we conclude that (a, f ) = F03BB,~ for all A E H'H(E) so F is weakly continuous.
We have proved that;: is an isomorphism which implies that :F is an isomorphism for the weak topologies T" E= H"H) and 03C3(EXPH, EXP'H). But we also proved that In view of our purposes, it is convenient to note the following. Let ey _ e(.,y) = 03A3(., y)n/n! E EXPH(E) C y E E. Then F is 'given by F03BB(y) = and cp(v) = ey, 03C6~, f(y) = f, ey) for all cp E EXPH(E) and f E HH(E).  [12]. However by virtue of Lemma 2.1 it is not difficult to prove that every homogenous convolution operator P(D), U ~ P E PH(nE)) is surjective). Thus a hypercyclic vector for a convolution operator must be outside the set H0 = U03C8~0 ker 03C8(D).
H0 is a dense subspace of Indeed, since ker 03C6(D) U ker 03C8(D) C H0 is a vector space. Further, assume that 0 ~ cp E H0. Since = Im 03C8, we have that H0 = ~03C8~0Im03C8. Clioose yo so that ~ 0 and let. y1 be a vector orthogonal to y0. VtG deduce that cp does not belong to Im 03C8 where 03C8 = (., y1)03C6. Thus H0 contains no non-zero vectors hence H0 is dense in HH(E). Thus ~ vanishes in a neighbourhood of the origin and hence ~ = 0. This is a contradiction which proves our claim for HV(E) and the assertion concerning HW(E) follows analogously. Next, let y E V be arbitrary. Then 03C6(D)ney = 03C6(y)ney for all n > 0. This shows that maps HVE) into HV(E) and that cp(D)n f ~ 0 for every f E HV(E) .
On HW(E) we define the operator S by Sey = ey/cp(y), yEW. . We conclude, in the same way as for T and that S maps HW(E) into HW(E) and that Sn f ~ 0 for every f E HW(E). Finally we note that TSey = 03C6(D)ey/03C6(y) = ey for yEW and thus T S f = f for all f E . This completes the proof. u