Toeplitz operators on the space of all holomorphic functions on finitely connected domains

We define and study Toeplitz operators on the Fréchet space of all holomorphic functions on finitely connected domains in the Riemann sphere. We completely characterize Fredholm, semi-Fredholm and invertible operators belonging to this class. As a result, we obtain a characterization of these classes of operators in the unit disk case. As a motivation we formulate and analyze the Riemann–Hilbert problem in the space of all holomorphic functions on the domains which we consider.


Introduction
We study Toeplitz operators on the space of all holomorphic functions on finitely connected domains in C or to put it in another way on finitely connected submanifolds of the Riemann sphere with boundary. We introduced and investigated the class of Toeplitz operators on the space of all real analytic functions on the real line in [18,[28][29][30]. The methods developed therein turned out to be general and universal enough to be applied to the other spaces of holomorphic functions. Therefore in [31] we developed a theory of Toeplitz operators on the space of all entire functions. The goal of this paper is to apply these methods to study Toeplitz operators on the space of all holomorphic functions on domains in C, the boundary of which consists of finitely many C ∞ smooth Jordan curves. Here we apply the techniques introduced in our previous papers-however some arguments are simplified in a rather interesting way. Importantly, there are some phenomena which are completely new and characteristic to the case which we study (degenerate symbols). The fact that the domain is not simply connected causes the arguments to be more involved, even if the main idea is that our previous papers. The style of the paper is rather concise. We strive however to make it self-contained.
We feel that the objects which we study, in particular Toeplitz operators on the space of all holomorphic functions in the unit disk H (D), are natural and important enough to justify thorough investigation. Also, Toeplitz operators on locally convex spaces of holomorphic functions appear in a natural way in some branches of mathematics such as approximation theory and number theory. They are related to Newton series, close cousins of Dirichlet series. We will not repeat these applications here and refer the reader to [28,31] and especially to [30,Introduction] for an explanation. The theory of continuous linear operators on locally convex spaces, including the spaces of analytic functions, is much less developed than the Banach space counterpart, not to mention the Hilbert one. Studying the classical operators such as Toeplitz operators seem just a natural idea to broaden our knowledge on this not so easy subject. Here we concentrate on the space of all holomorphic functions on finitely connected domains. It should be mentioned that such domains are a very important subject of study. We refer the reader to the beautiful book by Bell [6]. The study therein is however restricted to Banach spaces of holomorphic functions on such domains such as the Hardy spaces and the Bergman spaces. In this paper we investigate the Fréchet space of all holomorphic functions. We present now our main results.
Our first goal is to provide a characterization of the operators which we study. Recall that a classical result of Brown and Halmos [14] says that a continuous linear operator on the Hardy space H 2 (T), the matrix of which is a Toeplitz matrix is of the form where P + is the Riesz projection and M φ is the operator of multiplication by φ ∈ L ∞ . We obtain a similar characterization of the class of operators which we study. An easy to formulate is actually a corollary of our first main result, which concerns the unit disk.
Observe that condition (2) says that the matrix of the operator T is a Toeplitz matrix. We refrain from formulating here the corresponding main results (Theorems 4.2, 4.3, 4.4) , since it requires some rather technical preparation. We shall provide here only some intuition. In the case of a general domain the boundary of which consists of n + 1, C ∞ smooth Jordan curves, it is natural to define a Toeplitz operator as composition of multiplication by a symbol and the Cauchy transform. One cannot however expect that the matrix of such operator with respect to some basis is a Toeplitz matrix. There is however a substitute of this condition. Namely, in this case one considers not one matrix, as in the unit disk case, but n + 1 matrices. One matrix corresponds to every boundary curve. For the class of operators which we consider each of these matrices turns out to be a Toeplitz matrix. We emphasize that these are not the matrices with respect to some Schauder basis, unlike in the unit disk case. In order to characterize these operators one needs also some compatibility condition. We refer the reader to Sect. 4 for the explanation. We stress here however that the characterization which we obtain is analogous to (1). The operators are compositions of some projections and multiplication by symbols. We explain this now.
Assume that D is bounded by some C ∞ smooth Jordan curves γ 0 , γ 1 , . . . , γ n , γ 0 is the outer boundary, γ 1 , . . . , γ n form the inner boundary. Such a domain will be denoted by D(γ 0 ; γ 1 , . . . , γ n )-the details are given in Sect. 3, see also Definition 1 below. A function which is holomorphic in the intersection of some neighborhood of the boundary and the domain D defines a symbol-we shall say that F is holomorphic in a neighborhood of γ in D. Any such function F is of the form F = F 0 ⊕ F 1 ⊕ · · · ⊕ F n , where F i are holomorphic in a neighborhood in D of the curve γ i , i = 0, 1, . . . , n. The operators which we study are of the form Here, (γ i ) ε , ε > 0 are appropriately defined dilatations of the curves γ i . Naturally, only the values of F close to γ = γ 0 + γ 1 + · · · + γ n matter. This is why the symbol space is where U i run through open neighborhoods of γ i . This is explained and studied in Sect. 4. While reading the Introduction the reader should have in mind operators given by (3) for some F ∈ S(D), which can be thought as holomorphic in D close to bD. Apart from the domains D(γ 0 ; γ 1 , . . . , γ n ) and the corresponding spaces of all holomorphic functions H (D(γ 0 ; γ 1 , . . . , γ n )) in D(γ 0 ; γ 1 , . . . , γ n ) we also consider the unbounded domains the boundary of which consists of some curves γ 1 , . . . , γ n . Such domains are denoted by D(γ 1 , . . . , γ n ). In this case we consider two situations: the space of all functions holomorphic in D(γ 1 , . . . , γ n ) which vanish at ∞ and the space of all functions which are just holomorphic in D(γ 1 , . . . , γ n ). Naturally, in the first case the functions extend holomorphically to ∞ by 0. The symbol spaces are defined as appropriate inductive limits, just as in (4). In order to formulate our main results we need some definitions, which summarize the remarks given so far. If γ is a C ∞ smooth Jordan curve we denote by I (γ ) the interior of the curve γ and by E(γ ) the exterior-recall Jordan's theorem. Let γ 1 , . . . γ n ⊂ C be C ∞ smooth Jordan curves such that I (γ i ) ∩ I (γ j ) = ∅, i = j. Then and X denotes either the Fréchet space H 0 (D) of all functions holomorphic in D which vanish at ∞ or the space H (D) of all functions holomorphic in D (in general with no value at ∞).
We develop the theory of our Toeplitz operators on the spaces H (D) just as in the Hardy space case. First, we investigate the Fredholm property. Observe that the symbols are defined only up to the equivalence relation of equality close to the boundary bD. Thus only asymptotic behavior of the symbol close to the boundary may matter while studying the properties of the Toeplitz operators T F . Assume that D = D(γ 0 ; γ 1 , . . . , γ n )-in the other two cases the definitions are similar and we postpone their formulation to Sect. 5. Although we formulate the definitions only in the case D = D(γ 0 ; γ 1 , . . . , γ n ), the main results are presented for all the three cases.
We shall say that F is non-degenerate if F = F 0 ⊕ F 1 ⊕· · ·⊕ F n and none F i is identically equal to zero. Otherwise, we say that F is degenerate. If F i ∈ lim ind H (U i ∩ D) then we say that F i does not vanish, if F i is the equivalence class of a function (denoted by the same symbol) F i such that F i (z) = 0 for z in the domain of definition. Equivalently, one may say that the zeros of F i do not accumulate on γ i . We say that F = F 0 ⊕ F 1 ⊕ · · · ⊕ F n , F = 0 does not vanish if every F i is either identically equal to zero (in which case F is degenerate) or F i does not vanish. We emphasize that with this convention there are symbols which are both degenerate and do not vanish.
Our second main result is the following theorem-here and below we use the notation of Definition 1.

Main Theorem 1
The Toeplitz operator T F : X → X , with F ∈ S(D) the corresponding symbol, is a Fredholm operator if and only if F is non-degenerate and does not vanish.
Next we deal with the other three cases. This results allows us to characterize semi-Fredholm operators.

Corollary 1.1
Consider the Toeplitz operator T F : X → X with the symbol F ∈ S(D).

(i) The operator T F is a + -operator if and only if F is non-degenerate. (ii) The operator T F is a − -operator if and only if F is non-degenerate and does not vanish.
The next step is a characterization of invertible Toeplitz operator. This is done just as in the Hardy space case and we obtain first an analog of the classical Coburn-Simonenko Theorem [15,41], see also [13,Theorem 2.38,p 69]).

Main Theorem 3
Consider the Toeplitz operator T F : X → X with the symbol F ∈ S(D).
(i) If F is non-degenerate then either ker T F is trivial or ker T F is trivial.

(ii) The operator T F is invertible if and only is F is a Fredholm operator of index zero.
We remark here that we were also able to characterize one-sided invertible Toeplitz operators. This will be presented in a separate publication.
We now gather information which we obtain in the unit disk case. There is now a growing interest in operator theory on spaces which can be named nonstandard. This includes locally convex spaces, especially Fréchet spaces, in particular those spaces of holomorphic functions. We only mention the articles which have had some impact on our research. This includes the results by Albanese, Bonet and Ricker [2][3][4][5], Bonet, Lusky and Taskinen [10,11], Bonet and Domański [9], Bonet and Taskinen [12] and also Trybuła [43]. Some of this research concerns Toeplitz operators on weighted sup-norm spaces. We believe that our results may be an interesting element of this line of investigation. Naturally our results should be compared with the classical theory of Toeplitz operators. There are excellent and fundamental sources of information on this subject-the Hardy space case is treated in [13,37], the Fock space case in [50], some information on the general theory of the Bergman space case can be found in [44]. The case of the Hardy spaces on multiply connected domains is studied in [1].
The motivation for this research was the investigation initiated by Domański and Langenbruch in [17,19,20]. The authors therein considered the operators on the space of real analytic functions, called the Hadamard multipliers, the eigenvalues of which are just the monomials. They developed a very interesting theory of this class of operators and showed how to apply it to the Euler equation [22]. This was further studied in [21,24]. Domański, Langenbruch and Vogt studied the several variables case in [23]. Vogt studied the case of smooth functions in [45], distributions in [46,47] and surjectivity of Euler operators on temperate distributions in [48]. It is just one step from the operators, which are in some sense diagonal, to the operators, the associated matrices of which are Toeplitz and Hankel. We made this step in [18] and also in [28][29][30] and investigated Toeplitz operators on the space of all real analytic functions on the real line A(R). Golińska in [25] studied the Hankel case. As it has already been written, the methods which we have developed can be applied to the other spaces of holomorphic functions. In [31] we investigated the space of all entire functions. This paper concerns the space of all holomorphic functions on finitely connected domains on the Riemann sphere.
The paper is divided into six sections. In the next we discuss the Riemann-Hilbert problem in the space of all holomorphic functions. The preliminaries are presented in Sect. 3. We characterize the Toeplitz operators in Sect. 4. The Fredholm property is studied in Sect. 5. We discuss invertibility of Toeplitz operators in Sect. 6.

The Riemann-Hilbert problem in H(D)
This section serves as a motivation to study Toeplitz operators on the Fréchet spaces X introduced in Definition 1. The reader may prefer to skip this section first and then return to it when the necessary notation is introduced and the properties of the Toeplitz operators are established.
We follow [36, p 87] and call the following problem the homogeneous Hilbert problem: To find a sectionally holomorphic function (z) of finite degree at infinity, under the boundary condition where G(t) is a non-vanishing function of the point t ∈ bD, satisfying the Hölder condition.
The statement that is sectionally holomorphic means that is holomorphic on the complement of the cycle γ := bD and is continuous on γ . The problem formulated above is often called the Riemann problem, but the Author in [36, p 87] claims that it is should be attributed to Hilbert. Today's formulation of the problem uses the theory of the Hardy spaces H p and their boundary value properties. We refer the reader to [33] for every sufficiently small ε > 0, then dim H − (F) = 0 and for every sufficiently small ε > 0, then dim H + (F) = 0 and Proof For a given F ∈ S(D) consider the Toeplitz operator T F : X → X . According to Cauchy's theorem, where 0 < ε 1 < ε 2 and z ∈ D is the extension to D c , which vanishes at ∞. On the other hand, if F · f extends to D c and vanishes at ∞ then f ∈ ker T F (see the proof of Theorem 5.6). Thus H + (F) = ker T F . Also, which is justified in Theorem 6.1. Thus, if h ∈ ker T F then (7) gives the extension to D of the germ F · h. We argue similarly that indeed H − (F) = ker T F . According to Theorem 6.1 either ker T F = {0} or ker T F = {0}. It follows from Main Theorem 1 that equality (5) holds true. We need to justify equality (6).
First of all, notice that it follows from Main Theorem 1 that T F is a Fredholm operator. Thus according to Proposition 5.1 the range of the operator T F is closed in H (D). We infer that the range of the operator T F is a Fréchet space, hence ultrabornological. We can apply the Open mapping theorem [34,Theorem 24.30] to conclude that the operator T F is open onto its image. Such an operator is called a topological endomorphism. Also, since ker T F is finite dimensional one easily shows that there exists a continuous projection onto the kernel of the operator T F . Similarly, one shows that there exists a continuous projection onto the range, which is of finite codimension and closed. Such operators are called in [39] -transformations (see [

Background
We give now definitions of the objects which we consider. Some information will be repeated.
The symbol C ∞ stands for the Riemann sphere, which topologically is the one point compactification of the complex plane C ∞ : } is homeomorphic to S 1 and γ does not vanish, γ (t) and all its derivatives agree at the endpoints t = 0 and t = 1. While talking about curves we identify the map γ with the set γ ([0, 1]), both are denoted by γ . Thus γ ⊂ C is a meaningful statement. In this case Jordan's Theorem says that the curve γ divides the plane into two domains, denoted I (γ ) and E(γ ), with the common boundary γ . The former domain is bounded and is called the interior of the curve γ . The domain E(γ ) is called the exterior of γ .
The spaces H (D) (and also H 0 (D)) are Fréchet spaces, i.e. locally convex spaces, which are metrizable and complete. If {K n } n∈N is a compact exhaustion of D, then a fundamental system of seminorms { p K n } n∈N on H (D) is given by A particular and also a very important example of the spaces H (D) is the space of all holomorphic functions in the unit disk H (D). We refer the reader to the fundamental book [34] for the functional analysis background. The other classic sources of information as far as functional analysis is concerned are [27,40,42] and the recent book [49].
For complex analysis background we refer the reader to [7,16]. Assume that D = D(γ 0 ; γ 1 , . . . , γ n ) be fixed. Let also σ be a 1-cycle homologous to 0 in D. This will be denoted by σ ≈ 0. That is, for every z ∈ C \ D. Here, n(σ i , z) is the index of the curve σ i with respect to the point z, Assumption (9) is equivalent [7, Proposition 1.9.13] to the fact that σ is 1-boundary in the sense of homology theory. This explains the terminology. Under the assumption that σ ≈ 0, we have Cauchy's integral formula for every f holomorphic in D and z ∈ D \ σ i . Note that this formula remains valid for functions which vanish at ∞ when D = D(γ 1 , . . . , γ n ), that is, when D is unbounded. Cauchy's integral formula (and Liouville's theorem) implies that In particular, These decompositions hold not only algebraically but also in the category of locally convex spaces. They will play an important role later on. This is why, we discuss it in detail here. We concentrate on the case of the general domains D(γ 0 ; γ 1 , . . . , γ n ). Later we shall draw some conclusions also on the unbounded domains D(γ 1 , . . . , γ n ) and the spaces H 0 (D). The case of the spaces H (D) with D = D(γ 1 , . . . , γ n ) requires some additional arguments. We assume that −iγ j (t) is a complex number representing the direction of the outward pointing normal vector to the boundary at the point γ j (t)-that is, we assume the standard orientation. For γ := γ j the function γ ε (t) := γ + iεT (γ (t)), where T (γ (t)) stands for the unit tangent vector to γ = γ j at γ (t), parametrizes the curve obtained by allowing a point at a distance ε along the inward pointing normal to γ (t) ∈ bD to trace out a curve as γ (t) ranges over the boundary curve γ = γ j . We call the curves (γ 0 ) ε , . . . , (γ n ) ε the ε-dilatations of the curves γ 0 , . . . , γ n . Denote It follows from Cauchy's integral formula that by the formulas where z ∈ I (γ 0 ) and ε > 0 is chosen in such a way that z ∈ I ((γ 0 ) ε ) and where this time z ∈ E(γ i ) and ε > 0 is chosen in such a way that z ∈ E((γ i ) ε ). If ε > 0 is small enough the definitions are correct (in particular, they do not depend on the numbers ε) and indeed define projections onto the corresponding spaces. Furthermore, . . , n, P i P j = 0, i = j and P i = I . We remark that the formulas remain valid when D = D(γ 1 , . . . , γ n ) and we consider the functions from H 0 (D).
The dual space of H (D) is isomorphic to the space of all germs of holomorphic functions on the (closed) set D c which vanish at ∞. This is the fundamental Köthe-Grothendieck-da Silva duality [32, pp 372-378]. We recall this result here tailored to our needs.
Let F be a closed set in C ∞ . If U 1 , U 2 ⊃ F are open sets and g 1 , g 2 are holomorphic in U 1 and U 2 , respectively, then we write If g ∈ H (U ), open U ⊃ F, then we denote the equivalence class of g with respect to equivalence relation (11) by [g] ∼ F or just [g] ∼ , if no confusion arises. The object [g] ∼ F is called the germ of the function g on the set F. The (linear) space of all germs of holomorphic functions on the set F will be denoted by H (F). This space carries a natural inductive topology of the inductive family For basic information concerning inductive topologies we refer the reader to [8].
is well-defined. It is a fundamental result that with respect to this duality the dual space of with respect to the duality This implies (and also is a direct consequence of [32, pp 376-378]) that with respect to the duality . . , γ n )) and (γ ) ε is the ε-dilatation of the cycle γ 0 + γ 1 + · · · + γ n . In particular, with respect to the same duality with the curve γ 0 omitted.
for z ∈ I (γ 0 ). Consider the inductive family It follows from the above arguments that the projection P 0 extends to a linear map A comment is in order at this moment. Although the relation which defines the elements of lim ind {H (U ∩ D), r U ,V } resembles the one which defines germs, the elements of the inductive limit lim ind {H (U ∩ D), r U ,V } are not, strictly speaking, germs, since there is no common point in the intersection U ⊃γ 0 U ∩ D.
We remark here that lim ind {H (U ∩ D), r U ,V } is considered in the category of locally convex space. The fact that an inductive (Hausdorff) locally convex topology exists on lim ind {H (U ∩ D), r U ,V } is a consequence of the fact that as Fréchet spaces, for every open neighborhood U of the curve γ 0 . This is a consequence of Cauchy's integral formula and Liouville's theorem. Also, the evaluations at the points of I (γ 0 ) are continuous linear functionals on the first component in (16) as locally convex spaces. Also, with respect to this direct sum decomposition P 0 : The fact that P 0 here is continuous is a consequence of the fact that for every U ⊃ γ 0 the composition is a continuous map between Fréchet spaces (see [34,Proposition 24.7]). Here, r U maps f ∈ H (U ∩D) to its equivalence class in the inductive limit lim ind Similar arguments are applied to the other boundary curves. We infer that for i = 1, . . . , n, and the operator P i extends to a continuous projection defined on lim ind

Characterization
We assume that the C ∞ smooth Jordan curves γ 0 , γ 1 , . . . , γ n are fixed and satisfy condition (8) formulated in Sect. 3. We consider the domains D = D(γ 0 ; γ 1 , . . . , γ n ). We now define Toeplitz operators on the space of all holomorphic functions on the domain D.

Proposition 4.1 As locally convex spaces,
We assign to every F ∈ S(D) a continuous linear operator on the space H (D) and argue that it is reasonable to call this operator Toeplitz operator. To this end let F ∈ S(D). Thus This definition is correct.
The following fact is immediate.

Theorem 4.1 For every F ∈ S(D),
is a continuous linear operator.
Let us be more specific for a moment. Assume that F is a function, which is defined and holomorphic in some neighborhood of γ 0 ∪ γ 1 ∪ · · · ∪ γ n in D. We may assume that this neighborhood is a disjoint sum of neighborhoods U 0 , U 1 , . . . , U n of the curves γ 0 , γ 1 , . . . , γ n , respectively. Denote by F i the function F restricted to the set U i ∩D. Naturally, such a function F defines an element in S(D), which for simplicity, we shall denote by the same symbol.
where ε > 0 is chosen to guarantee that z ∈ D ε and each F i is holomorphic at any point of To give a basic example consider now the unit disk D and let F be defined and holomorphic in some neighborhood in D of the boundary T. Then we simply have for f ∈ H (D), for z ∈ D and r > |z| sufficiently close to 1.
Our goal now is to provide a characterization of the operators T F : H (D) → H (D), which justifies its name. Let us recall that it is a fundamental result of Brown-Halmos [14] that a continuous linear operator on the Hardy space H 2 (T), the matrix of which is a Toeplitz matrix, is necessarily of the form where M F is the operator of multiplication by F ∈ L ∞ and P + : (19) is also a composition of the Cauchy transform defined in a reasonable way on S(D) and multiplication by symbol. Thus it is of the form (20). We seek for a deeper characterization of the operators T F which would justify their name. Our strategy is to investigate properties of the operator T F and then show that the properties which we find actually characterize the operators of this class of operators. Let F ∈ S(D) be fixed. Let us consider first the operator P 0 T F . We have for f ∈ H (D), Observe that this operator, a priori defined on extends to a continuous linear operator on For simplicity we assume that a 0 := 0 ∈ D. For every m ∈ Z the function z → z m defines an element of S 0 (D) and P 0 T F (ζ m ) is a function holomorphic in I (γ 0 ). Hence, it is holomorphic in some small disk around 0. It is therefore meaningful to ask about its Taylor series expansion around 0. We have With this notation we have locally close to 0, where m ∈ Z. These coefficients can be put into a matrix ⎛ This matrix satisfies the Toeplitz condition. That is, it is constant on the diagonals. Consider now the inner curve γ 1 and the composition However, it extends to a continuous linear operator on ). It therefore develops into Laurent series centered at a 1 , which converges for |z − a 1 | > R, R sufficiently large. It is therefore meaningful to ask about the coefficients of these expansions. We have For l ∈ Z let us denote With this notation we have for |z − a 1 | > R, R large enough, This is again a Toeplitz matrix. We reach the same conclusion for the other boundary curves.
Observe that the coefficients of the matrices just introduced depend on the choice of the points a i , i = 0, 1, . . . , n. However the fact that they are Toeplitz matrices does not. This is a consequence of Theorems 4.2, 4.3 and 4.4.
The procedure of assigning matrices to operators can be applied to every continuous linear operator such that for every i = 0, 1, . . . , n, the composition P i T extends to a continuous linear operator on S i (D) with values in H (I (γ 0 )) or H 0 (E(γ i )), respectively. For such an operator the functions P 0 T F (ζ m ), m ∈ Z are holomorphic close to 0. Hence, they develop into Taylor series Naturally, the radius of convergence may depend on m. Observe, however, that it does not depend on m for the operators T F . Similarly, for every i = 1, . . . , n, the functions P i T ((ζ − a 1 ) m ), m ∈ Z are holomorphic for |z − a i | > R with R large enough, which may depend on m and i. Thus, We assign to every continuous linear operator We shall say that the matrices just defined are associated with the operator T . Please note that we do not claim that, in general, the functions z k , (z − a 1 ) k , . . . , (z − a n ) k , k ∈ Z form a Schauder basis in H (D). In fact, this may be false as one easily observes when γ 1 is an ellipse centered at 0, which is not a circle.  Proof Fix i between 1 and n. Choose F ∈ S i (D).
represented by a function, which we shall denote by the same symbol F − . The function is defined and holomorphic on an open set U ⊃ E(γ i ) c , which without loss of generality may be assumed simply connected. The functions (z − a i ) k , k ∈ N 0 are linearly dense in H (U ) by Runge's theorem. That is, there is a sequence of polynomials -strictly speaking Runge's theorem concerns functions holomorphic in C, not C ∞ but composition with 1 z allows to deal with this minor problem. To sum this up, for every The proof for i = 0 is analogous.
Proof We showed that (ii) implies (i) and, also, that if T = T F , then properties (23) are fulfilled. Assume now that T : H (D) → H (D) is a continuous linear operator such that (i) is satisfied. Consider the operator P 0 T . It follows from the assumption that P 0 T extends to H (I (γ 0 )) ⊕ H 0 (I (γ 0 ) c ). Consider the function P 0 T (1). This function belongs to H (D) by assumption. It follows from the assumption that it develops locally close to 0 into a Taylor series which converges in some neighborhood of 0. Consider the functional On the other hand, the function which represents the germ ϕ, denoted by the same symbol, can be developed into a Laurent series which converges for |z| large enough. We have ζ n , ϕ = c −(n+1) , n ∈ N 0 . We infer that a 0 −n = c −(n+1) , n ∈ N 0 . Set F 0 (z) := T (1) + zϕ − a 0 0 . This function defines an element of S 0 (D). Indeed, observe that T (1) ∈ H (I (γ 0 )) by assumption. Also, it follows from development (24) and the fact that ϕ ∈ H 0 (I (γ 0 ) c ) that zϕ −a 0 defines a germ in H 0 (I (γ 0 ) c ).
Consider the Toeplitz operator T F 0 on S 0 (D) with the just constructed symbol F 0 . It holds that for every m ∈ Z. Indeed, by assumption locally close to 0, and by Cauchy's theorem Thus it follows from (21) that T F 0 (ζ m ) also develops into (26). Thus, (25) holds in a neighborhood of 0, it must hold on the connected set D. It follows from Lemma 4.1 that the functions {ζ n } n∈Z are linearly dense in S 0 (D). Hence, as operators on S 0 (D), the operator P 0 T is equal to T F 0 . As a result, they are equal as operators on H (D).
Consider now an inner boundary curve γ i and the operator P i T , which by assumption extends to a continuous linear operator on H 0 (E(γ i )) ⊕ H (E(γ i ) c ). By assumption, is a function in H 0 (E(γ i )) and when |z − a i | is large enough,

Consider the functional on S i (D) defined by
which is equal to in the sense of duality (13). Note that the definition is correct, since

by assumption. Observe also that the constant function 1 defines a germ in H (E(γ i ) c ) = H (I (γ i )). Since ξ is a continuous linear functional on
The function which defines the germ ψ, denoted by the same symbol, develops close to a i into a Taylor series By Cauchy's theorem we have for m ∈ N, By assumption we also have Thus

This expression defines an element of S i (D). Consider the Toeplitz operator T F i on S i (D).
It holds that for m ∈ Z. It follows from Lemma 4.1 that the functions {(z − a i ) m } m∈Z are linearly dense in S i (D). This implies that as operators on S i the operators P i T and T F i are equal.
We apply this Theorem to the basic case of the unit disk. We slightly simplify its formulation.

(ii) There exists a function F holomorphic in some annulus
In particular Theorem 4.2 implies the following result: If T = T F for F = F 1 ⊕ · · · ⊕ F n ∈ S(D) then the coefficients of matrices (22) are the moments of the symbol. That is, Now we consider the spaces H (D) for D = D(γ 1 , . . . , γ n ). Then for f ∈ H (D) define where γ ∞ is (for example) the circle |ζ | = R for R > 0 large enough. The operator P ∞ is a continuous projection onto the space of all entire functions H (C). It follows from Cauchy's theorem that Denote The following fact is now immediate and was basic in [31]:

Proposition 4.2 As locally convex spaces,
The symbol H 0 (∞) stands for the space of all germs at ∞ of functions holomorphic at ∞, which vanish at ∞. The operator P ∞ extends to a continuous projection on S ∞ onto H (C) along H 0 (∞). The formula for the extension is again (28) with R > 0 large enough.
We again assign to every F ∈ S(D) ⊕ S ∞ a continuous linear operator on the space H (D) and then obtain a characterization of this class of operators. To this end let F be a symbol.
This definition is correct.

and the projection P i is defined and continuous on lim ind {H
For every m ∈ Z the function ζ → ζ m defines an element of S ∞ . For every f ∈ S ∞ it holds that P ∞ T F ( f ) ∈ H (C). Direct computation gives that  (22) and (32) satisfy the Toeplitz condition. That is, there exist sequences

. , n the composition P i T extends to a continuous linear operator on S i (D) with values in H 0 (E(γ i )) and also P ∞ T extends to a continuous linear operator from S ∞ with values in H (C). The matrices associated with T and defined in
If T = T F for F = F 1 ⊕ · · · ⊕ F n ⊕ F ∞ ∈ S(D) ⊕ S ∞ then the coefficients of matrices (22) and (32) are the moments of the symbol. That is,

Fredholm and semi-Fredholm operators
We characterize Fredholm Toeplitz operators T F : X → X . The spaces X were defined in the Introduction. First we repeat the key definitions. Assume that D = D(γ 0 ; γ 1 , . . . , γ n ) for some C ∞ smooth Jordan curves γ 0 , γ 1 , . . . , γ n which satisfy condition (8). Let F ∈ S(D) be a symbol as defined in Definition 2.

Definition 3
We shall say that F is non-degenerate if F = F 0 ⊕ F 1 ⊕ · · · ⊕ F n and none F i is identically equal to zero. Otherwise we say that F is degenerate.

Definition 4
If F i ∈ lim ind H (U i ∩ D) then we say that F i does not vanish, if F i is the equivalence class of a function (denoted by the same symbol) F i such that F i (z) = 0 for z in the domain of definition. Equivalently, one may say that the zeros of any function F i which represent the class F i do not accumulate on γ i .

Definition 5
We say that F = F 0 ⊕ F 1 ⊕ · · · ⊕ F n , F = 0 does not vanish if every F i is either identically equal to zero (in which case F is degenerate) or F i does not vanish.
We formulated the definitions in the case X = H (D), when D = D(γ 0 ; γ 1 , . . . , γ n ). They carry over in an obvious way to the spaces H 0 (D) when D = D(γ 1 , . . . , γ n ). For such a domain D, when X = H (D), and F = F 1 ⊕ · · · ⊕ F n ⊕ F ∞ we say that F is non-degenerate if F 1 ⊕ · · · ⊕ F n is non-degenerate (in the sense just defined) and F ∞ is not identically equal to zero. We shall say that F = F 1 ⊕ · · · ⊕ F n ⊕ F ∞ , F = 0 does not vanish if for every i = 1, . . . , n, ∞ either F i is identically equal to zero or some representative of F i is = 0 in the domain of definition. Some explanation is in order. Assume that We prove the following theorem.

Assume also that F ∈ S(D) is a symbol. The operator T F : H (D) → H (D) is a Fredholm operator if and only if F is non-degenerate and does not vanish.
In this case, for ε > 0 small enough to guarantee that (γ i ) δ ⊂ D for 0 < δ < and each F i (z) = 0 in this set.

D) for D unbounded) is a continuous linear operator such that im T is of finite codimension. Then im T is closed.
This can be proved as [18,

Proof of Theorem 5.1
The idea of the proof is similar to [18,Theorem 2]. We introduce the necessary modifications and also some simplifications. We assume that the symbol F ∈ S(D) is represented by a function (denoted by the same symbol) F such that F(z) = 0 in the domain of definition. To be precise, we assume that F is defined in the set {ζ : ζ = (γ i (t)) for some i, 0 < ε < ε 0 and t ∈ [0, 1]} and F does not take the value 0 in this set.
In the proof we use Hardy spaces on finitely connected domains in C. For ε > 0 small enough we consider the Hardy space H 2 (D ε ) on the domain D ε , which let us recall is defined as The Hardy spaces on such domains are studied in [6] (see in particular [6,Section 4]). We used this theory in [18,29]. We refer the reader for example to [18,Section 2.2] or [29,Section 4] for basic information on these spaces. In particular, we assume that the reader is familiar with the equivalence of the definition of the Hardy space as a subspace of L 2 (γ ) and the space of holomorphic functions in D satisfying a certain growth condition (this is explained in [6,Section 6]).
For F ∈ S(D) we define, when ε > 0 is small enough, the Toeplitz operator on the space Since F i restricted to (γ i ) ε , ε > 0 is bounded, the operators are well-defined and continuous. Since F| γ ε does not take the value 0, the operator T F,ε is a Fredholm operator and for every ε > 0 small enough This is proved for instance in [35,Proposition 4.1.6]. The Toeplitz operators considered therein are defined on L 2 spaces but the proof carries over to our setting without a difficulty. Formula (34) together with Cauchy's theorem, since F is non-degenerate and does not vanish, imply that the indices of the operators T F,ε are constant when 0 < ε < ε 0 . We argue that not only the indices of the operators T F,ε are constant but also constant are the dimensions of the kernels ker T F,ε : and, as a result, the cokernels Furthermore they are, what we call, globally generated. This will suffice to complete the proof. We sketch the idea of the proof and show how to simplify our previous argument in [18]. We refer the reader to [18] for technical details which we omit here.
Assume that f ∈ ker T F,ε is non-zero when 0 < ε < ε 0 . By a standard Hardy space argument and Cauchy's integral formula when z ∈ D ε 1 and ε 1 > ε is small enough. Take ε 1 < ε 2 < ε 0 . Then by Cauchy's integral formula This formula requires some explanation. Namely, for z close to γ i , it reads This means that the function f , a priori defined in D ε , extends to D, since F is holomorphic close to bD and F(z) = 0 there. This means that every function f ∈ H 2 (D ε ) which belongs to ker T F,ε is holomorphic in the whole domain D. By Cauchy's integral formula we have that f ∈ ker T F,δ for every 0 < δ < ε. This is what we called globally generated-there are functions, which are holomorphic in D and which span ker T F,ε for every ε > 0. We infer that as claimed. This also implies that In order to reach such a conclusion in our study of the space of real analytic functions A(R) in [18] we used a rather non-trivial result on the Cauchy transform [6,Theorem 3.4]. This, as we showed above, is not necessary-Cauchy's integral formula suffices. We now investigate the cokernels. Formulae (34) and (37) imply that dim H 2 (D ε )/im T F,ε = ν ≡ const for some ν ∈ N 0 and every 0 < ε < ε 0 . Choose a sequence ε 0 > ε 1 > · · · > 0 and consider the cover We shall solve a certain Cousin problem subordinate to this cover. First however we observe that if for some k ∈ N and f ∈ H 2 (D ε k ) it holds that T F,ε k f ∈ H 2 (D ε l ) for l > k, then f ∈ H 2 (D ε l ). This again follows from Cauchy's integral formula and a limit argument in the Hardy spaces. Indeed, for ε k > ε k and z ∈ D ε k \D ε k we have So the formula gives an extension of f ∈ H 2 (D ε k ) to a function in H 2 (D ε l ) (see formula (36)).
Assume now that the classes of (38) for some f ∈ H 2 (D ε 1 ) and z ∈ D ε 1 , then f ∈ H 2 (D ε k ). Furthermore, by Cauchy's theorem and a limit argument in the Hardy spaces equation (38) holds for z ∈ D ε k . This is however impossible. We infer that the classes of the restrictions of the functions f k 1 , . . . , f k ν to D ε 1 generate H 2 (D ε 1 )/im T F,ε 1 . Without loss of generality we can assume that the classes of f 1 i and f k i in H 2 (D ε 1 )/im T F,ε 1 are equal. For simplicity assume that i = 1. We shall extend the function f 1 1 . As we remarked, we may assume that for every k ∈ N there is a function

From the observation just proved we infer that if for some non-zero scalars
. We claim that the functions g lk = h l − h k form Cousin data. This is easy to prove, the details are given in [18]. The I Cousin problem is solvable in C (see [26,Theorem 1.4.5]). There are therefore function g j ∈ H (D ε j ) such that in D ε k and appropriate ε l > ε l and ε k > ε k . In this way we extend the function f 1 1 + T F,ε 1 g 1 to D, ε 1 > ε 1 . This procedure is applied to the other functions f 1 2 , . . . , f 1 ν . We claim that the classes of these functions span H (D)/im T F and also H 2 (D ε )/im T F,ε for ε > 0. This follows easily from the fact that the classes of f 1 1 , . . . , f 1 ν are linearly independent in We conclude that which completes the proof of the fact that if F does not vanish and is non-degenerate then is a Fredholm operator. D(γ 1 , . . . , γ n ) for some C ∞ smooth disjoint Jordan curves γ 1 , . . . , γ n and consider the space X = H 0 (D). Let F ∈ S(D) be a symbol. If F is non-degenerate and does not vanish, then the operator
Proof Let a 1 ∈ I (γ 1 ). Without loss of generality we may assume that a 1 = 0. Let φ(z) = 1 z be the inversion. For f ∈ H 0 (D), D = D(γ 1 , . . . , γ n ) consider the function The operator C is an isomorphism between the spaces H 0 (D) and H (D −1 ), where This is an n connected domain in C, the outer boundary of which is the curve γ −1 Here, M = D(γ 1 , . . . , γ n ) is a submanifold with boundary of the Riemann sphere C ∞ and f (z) = 1 z is an orientation preserving diffeomorphism, since it is a holomorphic invertible self-map of the sphere.
In particular, D −1 is a bounded domain. Assume that F ∈ S(D) is represented by a function denoted by the same symbol F. That is, F is holomorphic in some neighborhoods in D of the curves γ i , i = 1, . . . , n. Let G(z) := F 1 z . Then G is holomorphic in some neighborhoods in D −1 of the curves γ −1 i . Consider the symbol induced by the function G and denote it by the same symbol G ∈ S(D −1 ).
We claim that In order to prove this we need the following Lemma.
Assume that g is holomorphic in some domain which contains the sets: Proof The statement is trivial if g is holomorphic also on . Then (γ ) 1 and (δ) 1 are homotopic and the claim follows just from Cauchy's Theorem. However we do not assume that g is holomorphic on -we need to approximate g by functions which satisfy this condition.
Let R be a rational function with poles outside the setḠ ∪¯ such that sup z=(γ ) 1

Such a function exists by Runge's Theorem. Then
We infer that for any c > 0, Also, by the same arguments We emphasize that the orientation of the curves γ −1 , (γ −1 ) ε is the correct one. Altogether, we conclude, referring to Theorem 5.1, that We now consider the case of Toeplitz operators on the space H (D) when D =  D(γ 1 , . . . , γ n ) is unbounded. D(γ 1 , . . . , γ n ) for some C ∞ smooth disjoint Jordan curves γ 1 , . . . , γ n . Let F ∈ S(D) ⊕ S ∞ be a symbol. If F is non-degenerate and does not vanish, then the operator
Proof In order to prove the Theorem one considers the domains and repeats the proof of Theorem 5.1.

Theorem 5.4 Assume that F ∈ S(D) is degenerate or not and vanishes. Then the range of T F has infinite codimension in H (D). In particular, if F vanishes then T F is not a Fredholm operator.
Proof We give the details in the case X = H (D) for D = D(γ 0 ; γ 1 , . . . , γ n ). The other two cases can be treated similarly. Assume that F is represented by F 0 ⊕ F 1 ⊕ · · · ⊕ F n . Assume that F 1 (z k ) = 0 where (z k ) ⊂ D accumulates at a point p ∈ γ 1 (or F 1 is identically equal to zero). Without loss of generality we may assume that z k → p as k tends to ∞. Let 0 < ε k < ε 0 be chosen in such a way that for z ∈ {z = (γ (t)) ε : t ∈ [0, 1], ε k < ε < ε 0 }. In other words, The function is holomorphic in E((γ 1 ) ε 0 ). In particular, the limit lim k→∞ g(z k ) exists. We infer that exists and is finite. In other words, if F vanishes and F(z k ) = 0 for z k → p ∈ γ 1 then for every function h ∈ range T F the limit lim n→∞ h(z k ) exists and is finite. The space of all functions h ∈ H (D) with this property has an infinite codimension in H (D). Indeed, no finite non-trivial combination of the functions 1 (z− p) m , m ∈ N belongs to this space. We conclude that codim range T F = ∞.

Corollary 5.1 Assume that F is represented by a function F
: lim k→∞ f (z k ) exists and is finite .
We are now able to prove Main Theorem 1.

Proof of Main Theorem 1
If F is non-degenerate and does not vanish then it follows from Theorems 5.1, 5.2 and 5.3 that the operator T F : X → X is a Fredholm operator in each case of the space X in Definition 1. If F vanishes then it follows from Theorem 5.4 that the operator T F is not a Fredholm operator. This also covers the case when F is degenerate.
The proof of Theorem 5.4 reveals that, in particular, if there exists i ∈ {0, 1, . . . , n} such that F i ≡ 0 then the range of T F is of infinite codimension in H (D). This can be seen by a direct and simpler argument. If for example F 1 ≡ 0 then which is naturally of infinite codimension in H (D(γ 0 ; γ 1 , . . . , γ n )).
If F is degenerated we denote by d(F) the set of all indices i ∈ {0, 1, . . . , n} such that F i ≡ 0.

Theorem 5.5 If F ∈ S(D) vanishes (it is degenerate or not) then T F is injective.
Proof We keep to the notation of the proof of Theorem 5.4 and repeat the corresponding arguments. From (41) we have Thus, if f ∈ ker T F then As we have already observed, g is holomorphic in E((γ 1 ) ε 0 ). We infer that this function vanishes identically since z n → p ∈ E((γ 1 ) ε 0 ). Again from (41) it follows that F 1 · f vanishes identically in {z : z = (γ 1 (t)) ε , t ∈ [0, 1], 0 < ε < ε 0 }. Since F 1 is non-zero it must be that f ≡ 0. This completes the proof. The other two cases are proved in the same way.
Assume now that F ∈ S(D) is a symbol which, in general, may vanish and may be degenerate. We now give a description of the kernel of the operator T F .
Furthermore assume that F ∈ S(D) and consider the Toeplitz operator Proof Assume that 0 / ∈ d(F) and F 0 f extends to E((γ 0 ) ε 0 ) and vanishes at ∞. Also, assume that for i / ∈ d(F) the function F i f extends to I ((γ i ) ε 0 ). We have for 0 < ε < ε 0 small enough. By Cauchy's theorem for R > 0 large enough Since F 0 f vanishes at ∞, is holomorphic. Assume now that f ∈ ker T F and 0 / ∈ d(F). We show that F 0 · f extends to E((γ 0 ) ε ) and vanishes at infinity.
We have for all z ∈ D and ε > 0 sufficiently small. That is, By Cauchy's theorem for 0 < ε < ε 0 small enough. It follows from (42) that The right hand side of (43) is holomorphic in E((γ 0 ) ε 0 ) and vanishes at ∞. This function and F 0 · f coincide on an open set. Hence F 0 f extends to E((γ 0 ) ε 0 ) as claimed. The argument for the other i / ∈ d(F) is similar.
The previous theorem has an obvious analog for the space X = H 0 (D) when D = D(γ 1 , . . . , γ n ). For such a domain D let us also consider the space X = H (D). Let the symbol be equal to F ⊕ F ∞ with F ∞ not equal to zero (see formula (30) in Sect. 4). Then the functions f in the kernel of the operator T F⊕F ∞ satisfy, apart from the conditions formulated in Theorem 5.6, also: F ∞ f extends holomorphically to ∞. We now study Toeplitz operators with degenerate symbols. For simplicity we assume that 1 ∈ d(F). Proof Assume that 0 / ∈ d(F). It follows from the assumption and the previous Theorem that there exists a function f = 0 in H (D(γ 0 ; γ 2 , . . . , γ n )) such that F 0 f = g 0 , where g 0 is belongs to H 0 (E((γ 0 ) ε 0 )), and F i f = g i where g i ∈ H (I (γ i ) ε 0 ). Assume that a 1 ∈ I (γ 1 ).
In order to motivate the next result consider the Toeplitz operator T F : where D is the annulus with radii 1 and 1 2 and F ≡ 1 ⊕ 0. The symbol F induces also a Toeplitz operator on the space H (D). Then it is injective-this is just Cauchy's integral formula. However for any Laurent series f (z) := ∞ n=1 a n z n convergent in |z| > 1 2 we have T F f = 0. We expect that degenerate symbols generate Toeplitz operators with infinite dimensional kernels. This is indeed the case, as we now show.
It follows from the previous Theorem that for any k ∈ N Again the analog of Theorem 5.8 for the space X = H 0 (D) when D is unbounded is easy to formulate. We deal with the case of the space X = H (D) for such domains. Actually only one case requires a slightly different argument. Proof For simplicity we assume that F is non-degenerate. First we show that T F⊕0 is not injective. Then that the kernel is of infinite dimension. The operator T F is a Fredholm operator when considered on the space H 0 (D(γ 1 , . . . , γ n )) and Let a 1 ∈ I (γ 1 ) and consider the symbol since the orientation of the curve γ 1 is clockwise. There exists therefore f ∈ H 0 (D) such that We conclude that (ζ − a 1 ) m f ∈ H (D) belongs to the kernel of the operator T F⊕0 . For every k ∈ N the functions (ζ − a 1 ) m+k f belong to H (D) and satisfy the extension property postulated by (the analog for H (D)) Theorem 5.6. Theorem 5.10 Assume that F = 0 is degenerate but does not vanish. Then the range of the operator T F : X → X is a closed subspace of infinite codimension.
In view of Proposition 5.1 the range of this operator is closed in H (D(γ 0 ; γ m+1 , . . . , γ n )). Proposition 5.1 is formulated for operators on the space H (D). One however easily notices that its proof (the proof of [18, Proposition 5.1]) carries over to the more general setting which we consider now. This implies that the range of the operator T F as an operator on H (D(γ 0 ; γ 1 , . . . , γ n )) is closed (and also of infinite codimension). Indeed, it follows from decomposition (10)  The argument is the same if 0 ∈ d(F). Then one invokes Theorem 5.2 instead of Theorem 5.1.
If D = D(γ 1 , . . . , γ n ) then again Theorem 5.2 and the same arguments imply that the range of T F is closed and of infinite codimension if F = F 1 ⊕ · · · ⊕ F n is degenerate but does not vanish.

Theorem 5.11 If F is degenerate and vanishes then the range of the operator T F is not closed.
Proof Assume first that D = D(γ 0 ; γ 1 , . . . , γ n ), where the C ∞ smooth Jordan curves γ 0 , γ 1 , . . . , γ n satisfy assumptions (8). First we deal with the case of the space H (D) for these domains, then we provide comments on the other cases. Assume that F = F 0 ⊕ F 1 ⊕· · ·⊕ F n and F 1 = F 2 = · · · = F m ≡ 0 for m < n and F i are not identically equal to 0 for i = m + 1, . . . , n. Also, assume that there is a sequence (z k ) ⊂ D, z k → p ∈ γ m+1 such that F m+1 (z k ) = 0. That is, F m+1 vanishes.
We show that there exists a function g ∈ range T F which is not in the range of the operator T F . Let f be a function which is holomorphic in I (γ 0 ) except for a singularity s ∈ D. Thus, the function f does not belong to the space H (D). However, it defines an element of the direct sum Hence, the expression T F f is meaningful, for z ∈ D and ε > 0 small enough. Observe that g := T F f belongs to the space H (D). We claim g belongs to the closure of the range of the operator T F considered as an operator on the space H (D), however it does not belong to the range itself (recall that f is not in H (D)). This proves the theorem. In order to show that we need to show that for every compact set K ⊂ D and every ε > 0 there is a function h ∈ H (D) such that Fix a compact set K ⊂ D and a positive number ε. Choose δ > 0 such that and such that the singularity s does not belong to the closure We have This shows that in order to approximate T F f we need to approximate f only on the curves (γ i ) δ for i / ∈ d(F). It follows from Runge's theorem that there exists a rational function R with poles in a point b ∈ I (γ 1 ) and the infinity such that Set h := R and notice that h ∈ H (D) by the choice of the poles of R. It follows from (44) that so T F f belongs to the closure of This follows from the fact that F m+1 vanishes. We essentially have already proved this fact. We repeat the argument now. Assume that for z close to γ m+1 and every δ > 0 small enough. From Cauchy's integral formula for δ 1 > δ 2 > 0 such that z k ∈ I ((γ m+1 ) δ 1 ) ∩ E((γ m+1 ) δ 2 ). It follows from (45) that We infer that 1 for z ∈ I ((γ m+1 ) δ 1 ). Indeed, the integral defines a holomorphic function which vanishes on a sequence with an accumulation point in the domain of existence of the function. Thus,

Theorem 5.12 Assume that F is non-degenerate and vanishes then the range of the operator T F is closed.
Proof We start with the case D = D(γ 0 ; γ 1 , . . . , γ n ) for the C ∞ smooth Jordan curves γ 0 , γ 1 , . . . , γ n . Let F = F 0 ⊕ F 1 ⊕ · · · ⊕ F n , where no F i is identically equal to 0. Without loss of generality we may assume that there is a sequence (z k ) ⊂ D, z k → p such that F 0 (z k ) = 0, k ∈ N. Choose positive numbers ε k such that z 1 , . . . , z k ∈ D ε k := I ((γ 0 ) ε k ) ∩ E((γ 1 ) ε k ) ∩ · · · ∩ E((γ n ) ε k ) and no z l is on the curve (γ ) ε k . By the argument principle Consider the operators T F,ε k acting on the space H (D ε k ). Note that in the proof of Theorem 5.1 the symbol T F,ε k stands for an operator acting on the Hardy space H 2 (D ε k ), now the operator acts on the space H (D ε k ). It follows from Theorem 5.1 that T F,ε k is a Fredholm operator-the zeros of F do not accumulate on (γ ) ε k and index T F,ε k : for ε k > ε k close enough to ε k . By Cauchy's theorem this index is indeed equal to Assume that T F f l → g in H (D) as l → ∞ for some function g ∈ H (D). We prove that there is f ∈ H (D) such that g = T F f . We have that T F f l → g in H (D ε k ) for every k ∈ N when both the functions T F f l and the function g are restricted to D ε k . As we know the operator T F,ε k is Fredholm on the space H (D ε k ). Its range is therefore close in H (D ε k ). We infer that for every k ∈ N there is a function f k ∈ H (D ε k ) such that g| D ε k = T F,ε k f k . By (47) and (48) we infer that index T F,ε k < 0 for k ∈ N large enough. It follows from Theorem 6.1 (proved below) that the operator T F,ε k is injective. Thus, by Cauchy's theorem We conclude that if we define f = f k in D ε k then f ∈ H (D) and again by Cauchy's theorem This completes the proof in the case of the domain D = D(γ 0 ; γ 1 , . . . , γ n ).
In the case of the space H 0 (D) where D = D(γ 1 , . . . , γ n ) we use Theorem 5.2 instead of Theorem 5.1.

Proof of Main Theorem 2
We now gather the results proved so far. (i) If F is non-degenerate and vanishes then by Theorem 5.5 the operator T F is injective and it follows from Theorem 5.12 that the range of T F is closed. Assume that F is degenerate and does not vanish. It follows from Theorems 5.8 and 5.9 that the kernel of the operator T F is of infinite dimension. Theorem 5.10 says that the range of T F is closed and of infinite codimension. This proves (ii). If F is degenerate and vanishes then it is injective as we have already observed. Also, under this assumption it follows from Theorem 5.11 that the range is not closed. This completes the proof.
Consider the function f · F · h, a priori defined only in some neighborhood of γ in D. That is, for z ∈ D close to γ k , k = 0, 1, . . . , n.
We infer that f · F · h extends to an entire function which vanishes at ∞. We conclude that either f ≡ 0 or for every k = 0, 1, . . . , n we have h k ≡ 0 since no F k is identically equal to 0-it is here where the non-degeneracy comes into play.
The proof carries over to the case of the spaces H 0 (D) with D = D(γ 1 , . . . , γ n ) without any difficulty.
While considering the case H (D) for D = D(γ 1 , . . . , γ n ) one just needs to realize that H (D) ∼ = H (I (γ 1 )) ⊕ · · · ⊕ H (I (γ n )) ⊕ H 0 (∞), where H 0 (∞) stands for the space of all germs at ∞ of holomorphic functions at ∞ which vanish at 0. Data Availability There is no data data associated with this article.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.