Hadamard Multipliers on Spaces of Holomorphic Functions

Several representation theorems of multipliers are derived: in terms of analytic functionals, and germs of holomorphic functions. The co-induced topology via the representation theorems is discussed. As an application of the representation theorems the density of non-invertible multipliers is proved. Moreover, Euler differential operators are distinguished among all multipliers.


Notations and Main Statements
Let Ω ⊂ C be a Runge open set. A linear continuous map M : H(Ω) → H(Ω) is called a Hadamard, or coefficient type multiplier if every monomial is an eigenvector of M. The corresponding sequence of eigenvalues (m n ) n∈N will be called the multiplier sequence.
The set of all multipliers on H(Ω), denoted by M(Ω), is a linear subspace of the space of all continuous operators L b (H(Ω)) on the Fréchet space H(Ω). Recall that the space L b (H(Ω)) is endowed with the strong topology τ b , i.e., the topology of uniform convergence on bounded subsets of H(Ω). We equip M (Ω) with the induced topology from L b (H(Ω)). Observe that M (Ω) is a closed subspace of L b (H(Ω)), thus complete.
Abel's Theorem shows that monomials form a Schauder basis of H(Ω) if and only if Ω = rD for some r > 0, or Ω = C. Thus, multipliers are not just diagonal operators. On the other hand, since Ω is a Runge open set, every multiplier acting on H(Ω) is unambiguously determined by its behaviour on monomials.

M. Trybu la IEOT
Example. Hadamard multiplication operators. (cf. [6,11,[16][17][18], as well as the survey article [20]) For any function f holomorphic around 0 let G f stand for the maximal starlike domain to which f is analytically continuable. Following Hadamard define the Hadamard multiplication on the space of germs of holomorphic functions around the origin: * : n a n z n * n b n z n := n a n b n z n .
Assume that U, U are arbitrary domains, 0 ∈ U ∩ U. An operator L g : H(U ) → H( U ) given by the formula L g (f ) = f * g, for some g ∈ H({0}) is called the Hadamard multiplication operator. If U ⊂ G f * g for every f ∈ H(U ), and U is Runge open, then L g ∈ M(U ). We shall prove later that every multiplier is in fact a Hadamard multiplication operator. However,thist requires to extend the Hadamard product to a larger class of sets than those which contain the origin.
Example. Euler differential operators, i.e., operators of the form ∞ n=0 a n θ n , where θf (z) := zf (z) a n ∈ C. The interested reader is invited to get acquainted with the references given in [9]. In the last section we shall provide a characterization of Euler type differential among M(Ω).
Example. Hardy averaging operator H : H(rD) → H(rD), r ∈ (0, ∞], is the operator that corresponds to the multiplier sequence 1 n+1 ∞ n=0 . It is not hard to check that The main aim of the paper is to find a representation of all multipliers on the space H(Ω). Our work is motived by the recent paper [9] where an analogous problem in the real analytic case has been considered. Similarly as Domański and Langenbruch we give the desciption of all Hadamard multipliers on the space H(Ω) via analytic functionals H(V (Ω)) on the space of holomorphic germs on the dilation set of Ω, i.e., In the one real variable case the core of the problem was hidden in two results: the so-called Köthe-Grothendieck Duality which represents an analytic functional as a holomorphic function on the complement of its support; and the fact which says that every analytic functional T ∈ H(S) , S ⊂ R open, has the minimal real carrier contained in S. In general it does not make sense to talk about the minimal carrier, due to the simple reason: intersection of two carriers might be empty. Nevertheless, on the asumption that we made about Ω, there is a simple way to omit this obstacle. Namely, intersection of two polynomially convex carriers is a carrier. Consequently, there is the . The inverse is defined as follows: for a given M ∈ M(Ω) and z −1 ∈ Ω * , the analytic functional (1)), does not depend on z and has a polynomially convex carrier contained in V (Ω); furthermore, T z has a unique extension T acting on H(V ) that solves the equation Φ(T ) = M.
The celebrated Hadamard's Theorem provides the answer to the question concerning the domain of existence of the Hadamard product of Taylor series around the origin (for the classical formulation cf. [4,Chapter 1.4], or [19,Chapter 6.3]; for the extended one see [18]). However, it says nothing about the possible holomorphic extension of f * g beyond its star product. On the other hand, we feel tempted to describe the set Ωg ∪ {0, ∞} , a finite union of closed curves such that γ separatesĈ\Ω f and z Ĉ \ 1 Ωg , and ind γ satisfies certain conditions (see [18,Theorem 2.4]). Define is bilinear continuous provided Ω 1 , Ω 2 are open sets satisfying Ω 1 * Ω 2 = ∅.
The preliminaries together with the Köthe-Grothendieck Duality lead to the following conclusion where f M is defined by the equation: is an algebra isomorphism between the multipliers on H(Ω) with composition of operators, and the space of holomorphic germs onĈ\V vanishing at ∞ with multiplication of Laurent series, i.e.,: n∈N a n z n+1 * n∈N b n z n+1 := n∈N a n b n z n+1 .
The second problem that we look at is related to the co-induced topology on H(V ) by Φ from M(Ω), i.e., the topology on H(V ) that makes Φ defined in Theorem 1.1 a topological isomorphism. Before we give the precise statement define where V K (Ω) := {ζ ∈ C : ζK ⊂ Ω}. Section 4 is entirely devoted to the proof of the fact that Φ : The proof of (1.4) relies on methods explored in [10,21].
The paper ends with two applications of the results from Sects. 3 and 4: a density property of noninvertible multipliers and a characterization of Euler differential operators.
Before we proceed to the main part of the paper we explain some notation which we use throughout the paper. Unless otherwise stated, the dual space to H(Ω) is equipped with the strong topology H(Ω) b , i.e., the topology of uniform convergence on bounded sets in H(Ω).
Recall that Ω ⊂ C is called a Runge open set ifĈ\Ω is connected.

Monomials are linearly dense in H(Ω) if and only if Ω is Runge.
For We assume that 0 ∈ N. For non-explained notions from Functional Analysis we refer to the book [15].

Dilation Set
For an open set Ω ⊂ C define V (Ω) will be called the dilation set of Ω. Clearly, it is always non-empty since 1 ∈ V (Ω).
The next proposition collects some basic facts about V (Ω). In contrast to the real case we add one condition which seems to be the most important.

Proposition 2.1. For an open set
Before we indicate the proposition we make two remarks: • 0 ∈ V if and only if 0 ∈ Ω; Consequently, • being a Runge set is invariant under dilation, i.e.,Ĉ\Ω has only one connected component if and only ifĈ\wΩ has only one connected component for every w ∈ C * .
Because wz = w z zn z n , and the factor w z zn is in Ω for n 1, we conclude that wz ∈ Ω. d-3): Take r > 0 so that D(r) ⊂ Ω, and α n :

d-4):
If Ω is a Runge open set, then by (2.1) setĈ\V is a union of connected sets with non-empty intersection. Hence,Ĉ\V is connected.
Then Ω has the required properties. Indeed, by d-2) Ω is open. So, it suffices to show that V (Ω) = V. Clearly, we may assume that Ω = C. First, let us observe that To get V ⊂ V (Ω), assume the contrary. Take v ∈ V, y ∈ Ω so that vy / ∈ Ω. Hence, vy = 1 z for some z ∈ V * . By d-1) zv ∈ V. From the observation we conclude that y / ∈ Ω; a contradiction. Using once more the observation, the condition d-1), and a · 1 a = 1 ∈ V, we obtain: a / ∈ V ⇒ a / ∈ V (Ω), i.e., the opposite inclusion.
(2) Fix θ ∈ [0, 2π) such that θ π / ∈ Q. As a candidate for a dilation set consider There is no Runge open set Ω with V 2 as its dilation set despite V 2 satisfies all the conclusions of Proposition 2.1. Indeed, assume that such Ω exists.
Vol. 88 (2017) Hadamard Multipliers on Spaces 255 Because (0, 1] ⊂ V 2 and Ω is Runge, some halfline l containg the origin is disjoint from Ω (otherwise for every ψ ∈ [0, 2π) there is a non-empty segment (0, r ψ e iψ ] contained in Ω, r ψ > 0; hence,Ĉ\Ω is a union of two disjoint closed non-empty sets: one that contains the origin, and the other containing infinity). But this contradicts the fact that e inθ : n ∈ N is dense in T. Indeed, because Ω is a non-empty open set, we may find a non-empty arc (3) Let H denote a halfplane. It is easy to check the following: Proof. We must show that for every choice of a neighborhood U of V we may find a Runge neighborhood W of V that is contained in U . First, let us assume that V = V. Our aim is to construct an ascending sequence of Runge open sets W n n∈N contained in U whose union covers V. If so, by Runge's Theorem, for W we may take the union (cf.
Directly from the construction we get that ∞ ∈ C\U + ∩ C\U − . So,

Topology on H(V ) and its Dual
By an analytic functional we mean any element T in the dual space H(C) . T is said to be carried by a compact K if for every neighborhood Ω of K there is a constant C Ω such that From the definition of the topology in H(C) it follows that every analytic functional has a carrier.
Observe that if K carries T and S is any polynomially convex set containing K, then T might actually be considered as an element of H(S) . Indeed, it is enough to take the extension of T to H(S). By the convexity the extension is unique.
For T ∈ H(C) let T z n := t n , n ∈ N. We call (t n ) n∈N the moment sequence of T.
Every T ∈ H(K) , where K ⊂ C compact, corresponds to a holomorphic function f T ∈ H 0 (Ĉ\K) defined by the equation The correspondence between f T and T is given by the so-called Köthe-Grothendieck Duality  Fix T ∈ H i (V ) . Let U be a Runge neighborhood of V. Since the composition is continuous we may apply the Köthe-Grothendieck Duality described above. We get a compact set K U, V (= the intersection of all polynomially convex carriers of T ) and f U, V ∈ H 0 (Ĉ\K U, V ) such that Take another Runge neighborhoodŨ of V and put it, as U, in the Köthe-Grothendieck Duality. Therefore, for z ∈Ĉ\(K U, V ∪ KŨ , V ) : Changing the roles of U,Ũ, we obtain that K U, V = KŨ , V . Consequently, T has a minimal polynomially convex carrier K T contained in V, and there is f T ∈ H 0 (Ĉ\K T ) such that (3.

3) holds for h ∈ H(V ). For the opposite inclusion, observe that (3.3) defines an analytic functional with carrier in
The linearity of the correspondence follows from the uniqueness.
While the topology on the dilation set V is clear, it is just the induced topology from C, we have two candidates for a topology on H(V ): • Define: n ∈ N * .
(3.5) Then the sequence K n n≥1 gives the exhaustion of V by polynomially convex compact sets, i.e., Hence, H p (V ) = proj n H(K n ). Moreover
Proof. Since polynomials are dense in every H(K n ), it suffices to use the result on a reduced projective spectrum (cf. [6, p. 57]). IEOT In general the inductive topology is finer than the productive one. However, we will show that they coincide if V is a dilation set. For that purpose recall:

Theorem 3.3. ([14, Theorem 1.1])
Let S ⊂ C n be a locally closed set (i.e., every point in S has a neighborhood U ⊂ C n such that U ∩ S is closed in U ). The following conditions are equivalent: We will show that the third condition in Theorem 3.3 holds for V.

a Runge open set. Then
Proof. To complete the demonstration let us note that: in algebraic sense. For a different proof see [14,Proposition 1.9]. where U n ⊂ C is a Runge neighborhood of K n and C n > 0, n ∈ N * (see (3.5)). Observe that n U n is a Runge open set (since ∞ / ∈ U n for every n, we getĈ\ n U n is connected), and obviously it contains V. Thus, B is a locally bounded family of holomorphic functions on n U n . Ad. 2: It follows from (1) and [3, Proposition 2.1].

Proofs of the Representation Theorems
Proof of Representation Theorem. Let T ∈ H(V ) . Take an arbitrary g ∈ H(Ω). By the openness of Ω the function g w is holomorphic near V provided w ∈ Ω. Therefore, Φ T (g)(w) is well defined.
Fix w 0 ∈ Ω. By the proof of Proposition 3.1 we can find a polynomially convex compact set K T ⊂ V and a holomorphic function f T ∈ H 0 (Ĉ\K T ) such that (3.3) holds. In particular, where γ ⊂Ω\K T is a suitably chosen union of curves andΩ ⊃ V is a Runge open set. Plainly, we might find open U w 0 in a such way that h w is defined on γ for w ∈ U. So, and Φ T (g) ∈ H(Ω). Φ is well-defined: We shall show that Φ(T ) is sequentially continuous. Because H(Ω) is a Frechet space it will imply the continuity of Φ(T ). Let g n → g in H(Ω) and w 0 ∈ Ω. Choose γ ⊂ ζ : w 0 ζ ∈ Ω as in the Köthe-Grothendieck Duality. There is r > 0 such that D(w 0 , 2r) · γ ⊂ Ω. Hence: The conclusion follows from the compactness of D(w 0 , r) · γ ⊂ Ω. Furthermore, So, z n is an eigenvector corresponding to the eigenvalue T (z n ). Observe that T w is an analytic functional whose moment sequence is the same as the multiplier sequence of M, say (m n ) n∈N . Because of the proof of Proposition 3.1, T w has the minimal polynomially convex carrier K Tw ⊂ wΩ and there is a holomorphic function f Tw ∈ H 0 (Ĉ\K Tw ) such that for a suitably chosen γ (wΩ is Runge and −γ means the negatively oriented curve). Taking in (4.1) h = z n for n ∈ N we see that Taking into account (3.2) and (3.3) we conclude that is a well defined holomorphic function onĈ\K T , where

M. Trybu la IEOT
Hence, every T w extends to an analytic functional T ∈ H(V ) given by the formula

Continuity of Φ and Φ −1
A natural question is to ask what is the coinduced topology on H(V ) by Φ, i.e., topolgy that makes Φ an isomorphism while on M(Ω) we have the strong topology t b . There is one natural candidate: τ k , is the union of all connected components of V K (Ω) which have a nonempty intersection with V (Ω), For the origin of the definition of τ k we refer to [21]. The main goal of this section is to demonstrate the following: Since Ω is a Runge open set, we have for every compact K ⊂ Ω (if K is a bounded component of C\K and zK ⊂ Ω for some z = 0, then zK ⊂ Ω as well-here we have used the fact thatĈ\Ω is connected). And, finally by the same cause Vol. 88 (2017) Hadamard Multipliers on Spaces 261 Without loss of generality we might assume that K ⊂ C * is compact. Moreover, since dilations do not change V (Ω) suppose 1 ∈ Ω and 1 ∈ K. Hence, Thus, Φ(T )(f ) = g. Φ is onto MC(Ω, K): Let T w be as in the proof of Theorem 1.1. Observe, that T w = T w on the set of polynomials for every w, w ∈ K −1 , i.e., they represent the same analytic functional T, and the minimal polynomially convex carrier of T is contained in every 1 z Ω, z ∈ K, so as well in V K (Ω). Φ, Φ −1 sends equicontinuous sets into equicontinuous sets: The first part is quite obvious because if the minimal polynomially convex carrier of T is contained in U V K (Ω) satisfying |T f| ≤ C f U for some C > 0, then Fix f ∈ H(C) and −1 ∈ K. We have Hence, the minimal polynomial convex carrier for T M is a compact set con- Continuity of Φ −1 : We shall recall the main result from [21]. Let X be a Banach space and d ∈ N * . We put where |α| := α 1 + . . . + α d . On F we consider for any positive null-sequence δ = δ n n∈N the continuous norm where δ (|α|) (|α|) := δ 0 δ 1 · . . . · δ |α| . Lemma 5.3. ( [21]) The norms δ n are a fundamental system of seminorms on F . Lemma 5.3 was the key observation that led Vogt in [21] to discover a canonical system of seminorms on H(K), where K ⊂ R compact. It turns out that after a small modification this proof works in our setting.
Since we are interested in the case d = 1, for simplicity we put δ (n) := δ First of all, let us observe that F is an imbedding spectrum of Banach spaces. Hence, in particular F is a webbed (DF)-space, and the family B n | n ∈ N , B n := {f ∈ F n : |f | n ≤ 1} is a fundamental system of bounded sets in F (note that B n is closed in F and apply [15,Lemma 25.16]). Next, becauseĈ\V is a compact set, the space H 0 (Ĉ\V ) is ultrabornological and Montel (because it is topologically isomorphic with H(L) for some compact L ⊂ C). Moreover, since H 0 (Ĉ\V ) is an inductive limit (even regular), continuity can be proved on the step spaces. Hence, let us fix an open set U ⊃Ĉ\V. It is enough to show that the graph of L| H(U ) is sequentially closed, according to [13,Theorem 13.3.4]. But, this easily follows from the Weierstrass's Theorem. Finally, we claim that L is actually an injective topological embedding. By Baernstein's Lemma this is so if L −1 is locally bounded. But, the preimage of B n by Weierstrass's Theorem is closed in the locally uniform topology since it consists all those f ∈ H 0 (Ĉ\V ) that are holomorphic near the set g D( 1 n ) ∪ z∈∂V D(z, 1 n ). So, the functions in L −1 (B n ) are uniformly bounded on ∂V + D( ) ∪ Ĉ \D( 1 ) for some > 0.
Fix an arbitrary strictly positive null-sequence δ. Let f T ∈ H 0 (Ĉ\V K (Ω)) be the representation of T ∈ H(V K (Ω)) b given by the Köthe-Grothendieck Duality.
We check easily that Since δ is a null sequence, for every compact S there is a positive number C such that δ (n) ≤ C z n S for all n ∈ N. Thus, the family z n δ (n+1) : n ∈ N is bounded in H(C).