Multipliers on Spaces of Holomorphic Functions

We consider multipliers on the space of holomorphic functions of one variable H(Ω),Ω⊂C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\Omega ),\,\Omega \subset \mathbb {C}$$\end{document} open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} and Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on H(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\Omega )$$\end{document}. We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is convex.


Introduction
By a (Hadamard type) multiplier on the space of holomorphic functions H ( ) we mean each linear continuous map M : H ( ) → H ( ) for which the monomials (ζ n ) n∈N are eigenvectors with a corresponding multiple sequence of eigenvalues (m n ) n∈N .Troughout the paper denotes a Runge open set in C. Recall that an open set is called Runge if every holomorphic function on can be approximated uniformly on every compact subset of by polynomials.It can be easily seen that if zero belongs to , then the map just multiplies the sequence of Taylor coefficients at zero of a function f by the multiplier sequence.Nevertheless multipliers are not in general diagonal operators since the monomials form a basis of H ( ) if and only if is a disc centered at the origin, bounded or unbounded.Please note that since polynomials are dense in H ( ) the multiplier sequence uniquely determines the multiplier.Let us mention also that the class of multipliers M( ) forms a closed subalgebra of the algebra of all linear continuous operators on H ( ) equipped with the topology of uniform convergence on bounded sets.
In the present paper we consider three main problems.First, we find a representation of all multipliers on the space H ( ) for arbitrary Runge open set in C via analytic functionals, that is, those analytic functionals T ∈ H ( ) which have a system of supports contained in z −1 for every nonzero z ∈ (see Theorem 3.1).This representation asserts only a linear isomorphism between M( ) and some subspace of H ( ) , hereinafter denoted by H r ( ) (for a precise definition see Section 3).There naturally arises the question which topology it induces on H r ( ) from M( ) ⊂ L(H ( )), where M( ) is equipped with the topology of uniform convergence on bounded sets inherited from the space of all linear continuous operators L(H ( )).This is the second problem we discuss (see Theorem 4.9).The third problem considered here is to characterize multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator (Theorem 5. 1

and 5.2).
There is an extensive literature on multipliers acting on spaces of holomorphic functions, on open planar sets containing zero, see, for instance, [2,3,5,[8][9][10].Multipliers are related to the celebrated Hadamard multiplication Theorem (see [1,4], and the survey article [11]).In this paper we take up the challenge of developing the theory of multipliers on spaces of holomorphic functions on Runge planar sets.
Notation and helpful facts are gathered in Sect. 2.

Notation
H ( ) -space of holomorphic functions on an open set with the topology of uniform convergence on compact sets -space of germs of holomorphic functions on S with the inductive topology

Duality
By an analytic functional we mean any element T in the dual space H (C) .Recall that T is said to be carried by a compact K if for every neighborhood U of K there is a constant C U so that Every analytic functional with a carrier K corresponds to a holomorphic function f T ∈ H 0 ( Ĉ \ K ) via the Köthe-Grothendieck Duality (see [6]): f T corresponding to T is given by the formula Conversely, for every f ∈ H 0 ( Ĉ \ K ) there exists an analytic functional T ∈ H (C) carried by K so that f T = f .T can be easily obtained from where γ ⊂ C \ U is a finite union of closed curves such that the index n(γ , z) = 1 for any z ∈ K , and n(γ , z) = 0 if z / ∈ U .

Representation M(Ä) Via Analytic Functionals
is an isomorphism, where T z denotes the extension of T to H (z −1 ) and f z (•) = f (z•).The multiplier sequence of (T ) is equal to the sequence of moments of the analytic functional T , that is, (T ζ n ) n∈N .The inverse is defined as follows: for a given M ∈ M( ) the analytic functional extends to H (z −1 ) for every z ∈ * and solves the equation (T ) = M, where δ 1 denotes the point evaluation at 1.

Proof
We realize the proof in several steps.
Step I.For every T ∈ H r ( ) , f ∈ H ( ), z ∈ * the formula describes a function holomorphic around z.Moreover, if 0 ∈ , then the function is holomorphic around 0.
Proof Fix z ∈ * .Let K z be a carrier of T z .Without loss of generality we may assume that 1 for some C z > 0 and all f ∈ H (C).Moreover, by (4): Hence, by continuity (4) and ( 5) hold for all f ∈ H ( ).Thus, for every f ∈ H ( ), z ∈ the function g ( f ,z) is well defined on zD(1, δ z ) and is given there by the formula By the Köthe-Grothendieck Duality there are ϕ (z) ∈ H 0 ( Ĉ\z −1 ) and γ z ⊂ z −1 \K z such that Taking smaller δ z , we may assume that which proves that g ( f ,z) has a complex derivative in zD(1, δ z ).
The remaining part is clear since ( T ζ n ) n has a geometric growth.
Step II For every T ∈ H r ( ) , z ∈ there is a neighborhood U z of z such that the operator Proof Let K z and δ z be as in Step I.
Case z = 0 : Put U z = zD(1, δ z ).We have In other words, M z (g n ) → M z (g) uniformly on U z .By the Weierstrass Theorem M z (g) is holomorphic on U z .We have just shown that M z has a sequentially closed graph, and so it is continuous.Case z = 0 : Choose 0 < r << 1 and R > 1 so that D(3r Consequently Step III For every T ∈ H r ( ) , f ∈ H ( ), z ∈ the formula defines a holomorphic function.
on the space of entire functions, and so on H ( ), either.Here The above argument shows that g ( f ) ∈ H ( * ).If 0 ∈ take any z ∈ (U 0 ) * .We have for every polynomial f and w ∈ U z ∩ U 0 .Recall that is a Runge open set, so (8) holds on H ( ) as well.
Step IV (T ) is a multiplier.Moreover, is injective.
Step II implies that there is N ∈ N so that Hence We have proved that (T ) is sequentially continuous.It remains to check how (T ) behaves on monomials.For this fix n ∈ N. Compute: Step V is surjective.
T is a composition of M and the evaluation at 1, so it is continuous.
for some compact set J z ⊂ and C > 0. Clearly, (9) remains true for every neighborhood of z −1 J z .Hence, T has a carrier contained in z −1 .Define Hence, M = M on polynomials what completes the proof.
1. Every analytic functional has the smallest convex carrier (with respect to inclusion).Hence, if is convex then H r ( ) = H (V ) .

There are Runge open sets for which we have a proper inclusion H (V )
H r ( ) .Consider for instance the function for every z ∈ * , and so, f corresponds to some M ∈ M( ) by Theorem 3.1.However, V = {0, 1} and f / ∈ H 0 ( Ĉ \ V ).

Topological Representation
In the present section we find the coinduced topology on H r ( ) via from M( ) ⊂ L(H ( )), that is, a topology τ on H r ( ) such that the map : H r ( ) , τ −→ M( ), τ b specified in Theorem 3.1, is a topological isomorphism, where M( ) is equipped with the topology of uniform convergence on bounded sets τ b inherited from the space of all linear continuous operators L(H ( )).
Recall that the topology on H ( ) is given by the system of seminorms | M admits all monomials as eigenvectors .
Proposition 4.1 M( , K ) = MC( , K ) as sets.The map : T → M T (as in Theorem 3.1) is a linear isomorphism between H r ( , K ) and M( , K ).Its inverse map is given as follows Clearly, T M on H (C) does not depend on z.From the continuity of M it follows that there exist a compact L ∈ K and a constant C > 0 such that for every polynomial f .This shows that the functional can be extended on , and so we get that T M ∈ H r ( , K ) .
Since is Runge both and are injective.Hence, we have injective maps: whose composition gives the identity.Thus, ι • is onto, and so M( , K ) = MC( , K ).
Let V ⊂ C is a nonempty polynomially convex open set.For any positive nullsequence δ = δ n n∈N , that is, δ n → 0 + , and f ∈ H ( Ĉ\V ) set where We claim that the function is a seminorm on H 0 ( , K ).Hereafter, f (z) ∈ H 0 ( Ĉ\z −1 ) denotes an extension of f on a neighborhood of Ĉ\z −1 .
Proof Fix f ∈ H 0 ( , K ).Define M T = (T ) with T = T f defined via (3) ( is as in Proposition 4.1).There are a compact K ⊂ and a constant With an abuse of notation we denote the extension by the same symbol.From the proof of Proposition 4.1, we have Hence: Derivatives can be estimated similarly Proposition 4.3 There extists a unique locally convex topology on H 0 ( , K ) for which the family of seminorms υ K ,δ | δ strictly positive null-sequence is a fundamental system of seminorms.
Proof It suffices to show that υ K ,δ ( f ) = 0 for every positive null-sequence δ and a nonzero germ f ∈ H 0 ( , K ).But Proof This is a direct consequence of [12].

Proposition 4.5 The countable inductive system
is an imbedding spectrum of Banach spaces.Consequently, H 0 ( , K ) is a DF-space.
In particular, H 0 ( , K ) has a web, is barrelled and ultra-bornological.
n and there is a constant C, depending on δ so that: The second inequality indicates that the inductive system is actually a strict imbedding spectrum, and so H 0 ( , K ) is an LB-space.
Finally, for T ∈ H r ( , K ) define ( f T is given by the Köthe-Grothendieck Duality).Equip H r ( , K ) with the locally convex topology given by the system of seminorms Proof Only the second part requires a comment.By virtue of the Montel Theorem we easily get that the set K ) .It remains to apply [7,Proposition 25.16].
We need the following result.Proposition 4.7 Supposing T n → T in (H r ( , K ) , τ ,K ), there exists a system of compacts {L z } such that z∈K * zL z , and Choose γ z as in the Köthe-Grothendieck Duality and r z ∈ (0, 1) so that and Assume 0 / ∈ .Then, the family {z D z } z∈K * forms an open covering of K .By the compactness: Consequently, it is enough to repeat the above argument for K \ D(min{r , r /R}).
At this place we would like to highlight two issues.If then as L z we may also take z −1 L. In this sense {T n }, T are uniformly carried by L. Further, for every there exists N 0 such that for every z.In particular, there exists N 0 so that sup 1≤ j≤m From this easily follows the claim.
Proposition 4.8 defined in Proposition 4.1 is a topological isomorphism between (H r ( , K ) , τ ,K ) and M( , K ).Moreover, and send equicontinuous sets into equicontinuous sets.Proof We will derive the statement in several steps.
Step I Image of equicontinuous sets under , Proof Assume there are a compact L ⊂ and a constant Then, by Proposition 4.1 we have Suppose now that we have a nonempty family In other words, Step II Continuity of Proof Bearing in mind the uniform boundedness principle for barrelled spaces, we would like to emphasize what we have shown in Step I is actually a local boundedness of .In the considered case, that is, in the case of the space H r ( , K ) which is ultra-bornological and barrelled, it means precisely the same as saying that is continuous.
Step III Continuity of Proof Fix T ∈ H r ( , K ) .Let f T ∈ H 0 ( , K ) denotes the representation given by the Köthe-Grothendieck Duality corresponding to T , whereas T z is an extension of T on H (z −1 ).Fix a positive null-sequence δ.We will compute a little where Since δ n → 0 + , for every compact S there is a constant The remaining part of |( f T ) (z) | δ we will evaluate separately sup n∈N, w∈∂(z −1 ) where It remains to show that B is bounded in H ( ).But this easily follows from the fact that sup η∈J , w∈∂(z −1 ), z∈K Hence, if J , the last estimate is bounded, and clearly independent of z.
The proof has been completed.As a direct consequence we shall demonstrate: Theorem 4.9 For every Runge open set ⊂ C the mapping: Proof It follows from Proposition 4.8 by going to the projective limit over K n , where Remark 4.10 Following Remark 3.2 for a compact K ⊂ we define Consequently, proj K (H r ( , K ) , τ ,K ) is coarser than the strong topology on H (V ) , that is, the topology of uniform convergence on bounded subsets of H (V ).

Representation M(Ä) Via Sequences
For z ∈ Ĉ and A ⊂ 2 Ĉ put H (z, A) , is an isomorphism between the algebra of multipliers on H ( ) with composition of operators and the space of germs of holomorphic functions at ∞ that extend holomorphically on every set of the form Ĉ \ z −1 , z ∈ * with multiplication of Laurent series at ∞, that is, The multiplier sequence of the given multiplier is equal the Laurent coefficients at ∞ of the corresponding germ.Moreover, the action of the multiplier M ψ ∈ M( ) associated with ψ ∈ H 0 (∞, O ) can be calculated explicitly as follows where γ is a finite union of curves chosen for ψ (z) , an extension of ψ on a neighborhood of Ĉ\z −1 , and f z .

Theorem 5.2 The map
: where is an extension of ψ to Ĉ \ z −1 , establishes an isomorphism between the algebra of multipliers on H ( ) with composition of operators and the space of germs of holomorphic functions at 0 that extend holomorphically on every set of the form Ĉ \ z −1 , z ∈ * with Hadamard multiplication of Taylor series at 0, that is, The multiplier sequence of the given multiplier is equal to the Taylor coefficients at 0 of the corresponding germ.Moreover, the action of the multiplier M ψ ∈ M( ) associated with ψ ∈ H (0, Ô ) can be calculated explicitly as follows where γ is a finite union of curves chosen for w → Hence Put −1 (ϕ) := (T ψ ) ( as in Theorem 3.1).We shall check whether (T ψ ) = ψ.The first part of the proof gives (T ψ ) ∈ H 0 (∞, O ).Hence, by the identity principle for holomorphic functions it is enough to know how (T ψ ) behaves near ∞.But its Laurent series is described by the multiplier sequence of (T ψ ) , that is,
n=0  . is an algebra homomorphism: Fix S, M ∈ M( ).Clearly, S • M ∈ M( ) with the multiplier sequence equals n m n n∈N if S(ζ n ) = s n ζ n , M(ζ n ) = m n ζ n , n ∈ N.