One-Sided Invertibility of Toeplitz Operators on the Space of Real Analytic Functions on the Real Line

We show that a Toeplitz operator on the space of real analytic functions on the real line is left invertible if and only if it is an injective Fredholm operator, it is right invertible if and only if it is a surjective Fredholm operator. The characterizations are given in terms of the winding number of the symbol of the operator. Our results imply that the range of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite codimension. Similarly, the kernel of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite dimension.


Introduction
In this paper we study Toeplitz operators on the space of real analytic functions on the real line. We start by presenting the classical background for our investigation. The presentation is based on the beautiful books by Gelfond [21] and Markushevich [34]. In the next section we discuss our main results.
A fundamental result in function theory says that a holomorphic function locally develops into a power series. There are however also other series representations of holomorphic functions. Some holomorphic functions develop into Newton series where (z n ) is an appropriately chosen sequence of complex numbers. Such representations are deeply connected with interpolation theory and difference equations (actually the whole Gelfond's book [21] is devoted to the study of this relation). We briefly recall now the construction of Newton interpolation polynomials in order to provide motivation for our study. This leads to representations of form (1). Let z 0 , . . . , z n ∈ C be pairwise different. A function f : C → C can be written in the following way: where R n (z) := [z, z 0 , . . . , z n ](z − z 0 ) . . . (z − z n ).
Here [z 0 ] = f (z 0 ) and [z 0 , . . . , z k ], k = 1, . . . , n are the divided differences One easily notices that P n is a polynomial such that P n (z k ) = f (z k ) for k = 0, . . . , n. This polynomial is called a Newton interpolation polynomial (actually it is rather the Newton form of interpolation polynomial). It is worth mentioning that it appears for the first time in Newton's Principia Mathematica (Book III, Lemma V, Case 1).
Given a function f and a sequence (z n ), a natural question is whether the polynomials P n converge yielding representation of the form (1) with a n n! = [z 0 , . . . , z n−1 ].
One immediately notices a similarity between Newton series and Taylor series. In fact, the series (1) with coefficients defined by divided differences (3) can be considered as a discrete analog of Taylor series. The divided differences play the role of discrete analogs of the consecutive derivatives, the polynomials (z − z 0 ) . . . (z − z n−1 ) of the functions z n . The question, which we formulated above, that is, whether the polynomials P n converge is deep and does not have an easy answer. Some answers are known for entire functions (see [21], Satz 1, p. 43, Satz 1, p. 47 and also Chapter II, §2 and §3). Crucial for our study is the fact that the operators connected with Newton series turn out to be (appropriately defined) Toeplitz operators, the object of our investigation. Namely, it follows easily from the residue theorem for holomorphic functions that [z, z 0 , . . . , when f is (for simplicity) entire and γ is a C ∞ smooth Jordan curve such that the points z, z 0 , . . . , z n are contained in I(γ), the interior of the curve γ (recall Jordan's theorem). This implies that the rest term R n in (2) is, as we shall explain shortly, a composition of Toeplitz operators. Let us reformulate this observation: convergence properties of Newton series are governed by Toeplitz operators. Also, divided differences [z 0 , . . . , z k ], which are the coefficient functionals in (1), are just the evaluations of the value of a Toeplitz operator at one of the points z 0 , . . . , z k . An important situation occurs when z k = k, k = 0, . . . , n. Then n k (−1) k f (k) (5) and the corresponding Cauchy integral (4) is usually called Nörlund-Rice integral. The method of estimating it is considered as 'one of the basic asymptotic techniques of the analysis of algorithms'. We refer the reader to [19] and [38] for an enlightening discussion of the applications to theoretical computer science and discrete mathematics.
Observe that if f can be represented by Newton series (1) then f (z k ) is given by a finite sum a 0 + k n=1 an n! (z k −z 0 ) . . . (z k −z k−1 ). This property was used by Pólya to prove his result which characterizes functions holomorphic in some half-space z > α, satisfying some growth condition which vanish at integers, as polynomials ( [21], p. 133, Satz 10). Even more importantly, the fact that the function e αz develops into a Newton series is one of the key elements in the proof of the theorem which says that α and e α cannot be simultaneously algebraic numbers ( [21], p. 167, Satz 9). This in particular implies that the numbers e and π are transcendental. This result contains as a special case Lindemann classical theorem. We mention that Newton series representations draw some attention also in combinatorics in the so called umbral calculus [41]. There is an ongoing interest in Dirichlet series. Therefore it is worthy of mentioning that there is a relation between Newton series and Dirichlet series (see [21], p. 113, Satz 3).
Roughly speaking a divided difference considered as a function of one its arguments z → [z, z 0 , . . . , z n ] is well-defined and formula (4) holds true for functions which develop into power series. This is why we consider in this paper the case of real analytic functions. Recall that these are the functions f : R → C which develop locally, around each point of R into convergent power series. This means that a real analytic function on R is a germ on R of a holomorphic function. To sum this up, we study in this paper a class of operators, which includes the operators defined by divided differences (4), on functions which locally develop into power series. Observe that the assumption that f is real analytic is precisely what is needed to consider Nörlund-Rice formulae (4) and (5).
The space of all real analytic functions on R will be denoted by A(R). This space is not a Banach space, it is not even metrizable. It carries however a natural locally convex topology and such fundamental tools of functional 6 Page 4 of 51 M. Jasiczak, A. Golińska IEOT analysis as the Hahn-Banach theorem, the uniform boundedness principle and the open mapping/closed graph theorem are available. This will play some role in our proofs. Somewhat imprecise one may say that we consider here the space of all holomorphic functions on which we impose only one condition: their domain must contain the real line. One should compare this for instance with the Hardy, Bergman or Fock spaces in which cases the domain of a holomorphic function is fixed and common for all elements in the space. There are excellent books concerning Toeplitz operators on these spaces [4,37,47]. Real analytic functions play a significant role in mathematics. It suffices to mention here such fundamental results as the Cauchy-Kovalevskaya Theorem or Holmgren's Uniqueness Theorem (cf. [40], Theorems 2.21 and 2.26), which are formulated for this class of functions. Importantly, Hörmander characterized in [25] partial differential operators with constant coefficients which are surjective on the space of real analytic functions on a domain in R n . In [32] Langenbruch solved the right inverse problem for convolution operators on the space of real analytic functions. In this paper we consider similar problems for the class of Toeplitz operators, which we introduced in [9]. Our approach is however completely different.
A Toeplitz operator T F on A(R) is an operator of the form The symbol F of the operator T F belongs to the symbol space X (R), which is defined as the inductive limit of the spaces H(U \K)-the spaces of all functions holomorphic in U \K, Here the sets U run through complex neighborhoods of R and the sets K run through compact subsets of R.
If U is an open neighborhood of R, K is a compact subset of R and F is holomorphic in U \K, thenF determines an element in X (R), which is the equivalence class [F ] ∼ in limind H(U \K) of the functionF . Two such is the operator of multiplication by F and C : X (R) → A(R) is the Cauchy transform. We remark here that C turns out be a projection onto the space of real analytic functions A(R). We recall the details in Sect. 3.
Observe that if z 0 , . . . , z n are real then the operator given by (4) is precisely an operator T F with F which is the equivalence class in X (R) of the function 1 n k=0 (ζ − z k ) , which belongs to H(C\{z 0 , . . . , z n }). A fundamental result of Brown and Halmos [5] says that a Toeplitz operator on the Hardy space H 2 (T) is of the form P M φ , where M φ is the operator of multiplication by φ ∈ L ∞ (T) and P : L 2 (T) → H 2 (T) is the Riesz projections (which is also the Cauchy transform in the unit disk). Observe a similarity between this characterization of Toeplitz operators and formula (6). The similarity is deeper and it motivated our research. We mention here that in ( [9], Theorem 2) we proved a characterization of Fredholm Toeplitz operators on A(R) which strongly resembles the classical result for Hardy spaces. We shall recall this result in the next section, since it is one of the main tools in this paper.
Toeplitz operators on H 2 (T) are defined by a well-known matrix condition. Such a characterization holds true also in the case of the space of real analytic functions A(R). Any operator of the form T F , F ∈ X (R) satisfies locally near zero for complex numbers Here,F ∈ H(U \K) is a representative of F ∈ X (R). The set U is an open simply connected neighborhood of R and K is a compact subset of R. The symbol γ stands for (any) C ∞ smooth Jordan curve in U \K which contains the origin and the set K in its interior I(γ). Condition (7) says that if we consider Taylor series expansions around zero of the real analytic functions T F (x n ), n ∈ N 0 and create an infinite matrix M TF by putting the Taylor coefficients of the functions T F (x n ) in the consecutive columns of the matrix M TF , then this matrix is an (infinite) Toeplitz matrix Naturally such a matrix M T can be created for any operator T : A(R) → A(R), not only for the Toeplitz operators T F . We say that the matrix M T is associated with the operator T . It was proved in ( [9], Theorem 1) that if the matrix associated with a continuous linear operator T : A(R) → A(R) is a Toeplitz matrix, then T is a Toeplitz operator. That is, T is necessarily an operator T F defined in (6) for some symbol F ∈ X (R). This is an analog of the classical Brown-Halmos result. We emphasize that one should be careful with the matrix interpretation of a Toeplitz operator since monomials do not form a Schauder basis in A(R). In fact, as shown by Domański and Vogt [17] the space A(R) has no Schauder basis at all. Note however that condition (7) determines the operator, since polynomials are dense in A(R).
Our primary motivation in this project has been to build a theory of continuous linear operators on the space A(R) in a way similar to the known 6 Page 6 of 51 M. Jasiczak, A. Golińska IEOT theories of operators on Hilbert and Banach spaces. There is an extensive literature on linear continuous operators on these spaces. Much less is known about both concrete and abstract operators on locally convex spaces. There are however prominent exceptions such as differential or convolution operators. Nonetheless the theory of linear continuous operators on locally convex spaces is less developed than the Hilbert or Banach space counterpart. The basic object in our considerations is a matrix associated with an operator. The idea to consider operators determined by matrices associated with them comes from the work of Domański and Langenbruch. In a series of papers [10][11][12][13] they created the theory of the so-called Hadamard multipliers. These are the operators the associated matrix of which is just diagonal. This research was continued by Domański et al. [15] and also by Vogt in [44,45] for spaces of distributions, in [43] for spaces of smooth functions and by Trybu la in [42] for spaces of holomorphic functions. In [14] and [16] Domański and Langenbruch showed that this theory provides the correct framework to study Euler's equation. This equation on temperate distributions was studied in [46] by Vogt. Golińska in [23] and [24] studies operators defined by Hankel matrices. The paper is divided into four sections. In the next one we present our main results. Section three provides background for our study. We recall there basic facts concerning real analytic functions and Toeplitz operators. In that section we also prove an analog of Wiener-Hopf factorization. In section four we prove our main results. This is divided into two subsections. First we discuss left invertibility of Toeplitz operators. The second subsection is devoted to right invertibility of Toeplitz operators.

Main Results
We shall say that F ∈ X(R) does not vanish if F is represented byF ∈ H(U \K), which does not vanish in U \K.
Our main results are the following theorems:  of R and K is a compact subset of R. We may assume that U is simply connected. Then The symbol γ stands for a C ∞ smooth Jordan curve in U \K such that K is contained in the interior I(γ) of the curve γ. One easily observes that the definition is correct. This follows from Cauchy's theorem. A crucial tool in the proofs of our main results is the following theorem: It is worth mentioning that Theorems 1 and 2 are closely analogous to the classical Hardy space theory of Toeplitz operators with continuous symbols (see [22], Theorem 7.1 in accordance with [3], Lemma 6.14). We find it rather surprising. The methods are however completely different, since the Banach algebra techniques are not available in our setting.
The proofs of both Theorems 1 and 2 are based on the characterization of semi-Fredholm operators obtained in [28]. In order to present the idea of the proof we recall now these results.
Let F ∈ X(R) and let a holomorphic functionF ∈ H(U \K) be its representative: F = [F ] ∼ (here K ⊂ R is compact and U ⊃ R is open).
(i) We will say that F has real zeros going to infinity if there are x n ∈ R such that lim |x n | = ∞ andF (x n ) = 0. (ii) We will say that F has non-real zeros accumulating at a real point if the functionF has zeros z n / ∈ R whose limit lim z n exists and belongs to R.
We observe that these two properties depend only on the germ F ∈ X (R) and do not depend on the choice of K, U and the representativeF ∈ H(U \K) of F . The set of zeros of a functionF in H(U \K) is discrete in U \K. It follows that, given a non-zero germ F ∈ X (R), one can find U, K and a representativẽ F ∈ H(U \K) of F that has no zeros in U \K if and only if F ∈ X (R) neither has real zeros going to infinity nor non-real zeros accumulating at a real point. (iii) If F has real zeros going to infinity, but has no non-real zeros accumulating at a real point, then the operator T F is injective and the range of T F is a closed subspace of infinite codimension; (iv) If F has both real zeros going to infinity and non-real zeros accumulating at a real point, then the operator T F is injective and the range of T F is a dense subspace of infinite codimension. For any F ∈ X (R) one of the above four cases holds true.
We remark that we may assume that one of the four cases concerning the zeros ofF with [F ] ∼ = F ∈ X (R) holds since we may always shrink the open set U and enlarge the compact set K without affecting the equivalence class ofF in the symbol space X (R) = limind H(U \K) or the operator T F .
If condition (ii) of Theorem 2.2 holds then the operator T F is not injective. Hence it cannot be left invertible. Recall that if S is a left inverse of T F , ST F = I, then T F • S is a continuous projection onto the range of T F . Thus the range of T F is a closed complemented subspace of A(R). This implies that under condition (iv) of Theorem 2.2 the operator T F cannot be left invertible. The next result, which is essentially our first main result, deals with condition (iii).

Theorem 3. Assume that F ∈ X (R) has real zeros going to infinity, but has no non-real zeros accumulating at a real point. The range of the operator T F is not a complemented subspace of A(R).
Theorem 3 implies that the operator T F can be left invertible only when (i) of Theorem 2.2 holds. That is, only when T F is a Fredholm operator. This is the main ingredient of the proof of Theorem 1.
In the proof of Theorem 3 we obtain precise information on the range of the operator T F (see Proposition 4.3). This seems to be interesting per se. We explicitly construct a sequence of functionals ξ n ∈ A(R) such that Then we argue that if the range was complemented, then a natural projection should be in a reasonable way continuous. We show that this is however not the case. The proof is constructive and we show in particular how to obtain by means of Toeplitz operators functions dual to the functionals ξ n . In order to show that an injective Fredholm operator is left invertible one simply uses the closed graph theorem, which, let us emphasize this, works in A(R). This time there is also a constructive argument. Namely, we complete the picture of Fredholm Toeplitz operators on A(R) and prove a Wiener-Hopf type factorization in this space (Theorem 3.2 below) actually by mimicking the classical argument. With this tool at hand we may write down a left inverse explicitly.
As we have already remarked, our characterization of one sided invertible Toeplitz operators on A(R) resembles closely the Hardy space results. There is however a difference. An operator on a Hilbert space fails to be left invertible in two situations: it is not injective or the range is not closed. Note that the reason for T F not to be left invertible may be more involved. That is, the range of the operator may not be complemented. It seems worth mentioning that the study when the range of an operator is complemented drew some attention also in other natural cases [7,29]. By a well-known result of Lindenstrauss and Tzafriri [33] a Banach space is isomorphic to a Hilbert space provided every closed subspace is complemented. Theorems 2.2 and 3 immediately imply the following result:

The range of T F is a non-trivial complemented subspace of A(R) if and only if it is a subspace of non-zero finite codimension. This holds if and only if F ∈ X (R) does not vanish and
winding F > 0.
The codimension of the range of T F is equal to winding F . Theorem 3 means that there are plenty of closed subspaces of A(R), which are not complemented. By a trivial argument the subspace of even (odd) real analytic functions is complemented in A(R). Hence there are complemented subspaces in A(R) of infinite codimension. This also means that is not a Toeplitz operator, which follows immediately from the matrix representation (8) (this operator is however an Hadamard multiplier). Our results provide therefore a method to construct an abundance of subspaces of A(R) and also A(R) b , which are not complemented.
If an operator T F : A(R) → A(R) is right invertible, then it is surjective. It follows therefore from Theorem 2.2 that this happens either if T F is a surjective Fredholm operator or F = [F ] ∼ forF ∈ H(U \K), which vanishes only on a sequence (z n ) ⊂ C\R, which accumulates only at points of K. In the next result, which is essentially our second main result, we rule out the second possibility. Note that if T F is right invertible, then T F : is left invertible and, as a result, its range in A(R) b is complemented.

Theorem 4. Assume that F ∈ X (R) has non-real zeros accumulating at a real point but no real zeros going to infinity. The range of the operator T F is not a complemented subspace of
In order to prove the theorem we again prove a representation of the range of the adjoint operator T F of the form (9).
This theorem implies the following analog of Corollary 2.3:

The kernel of T F is a non-trivial complemented subspace of A(R) if and only if it is a subspace of non-zero finite dimension. This holds if and only if
F ∈ X (R) does not vanish and The dimension of the kernel of T F is equal to − winding F . The statements concerning the codimension of the range of T F in Corollary 2.3 and the dimension of the kernel of T F in Corollary 2.4 are immediate consequences of the Coburn-Simonenko theorem for Toeplitz operators on A(R), which we proved in ( [28], Theorem 1.2). This says, as in the Hardy space case, that either ker T F or ker T F is trivial.
Our work relies heavily on the previous results concerning Fredholm Toeplitz operators [9,27] and semi-Fredholm operators [28]. We also use extensively the Köthe-Grothendieck-da Silva duality. This will be recalled in Sect. 3. A basic tool in our arguments is Weierstrass factorization theory of entire functions. For this we refer the reader to [6].

The Space of Real Analytic Functions
For the reader's convenience we recall here basic properties of the space of real analytic functions on the real line and define Toeplitz operators on this space. For more details we refer to our previous works [9,27] and, especially, [28]. We refer the reader to [26,36] and [30] for the functional analytic background. Projective and injective limits of locally convex spaces are also studied in detail in [20]. The article [8] gives an instructive introduction to real analytic functions. There is also an interesting book [31], however of less functional analytic flavor. In our presentation of the space A(R) here we essentially follow [18]. The space of real analytic functions on the real line, denoted A(R), consists of all functions f : R → C which develop at every point x ∈ R into a power series convergent in a neighborhood of x to f . This implies that a real analytic function extends to a holomorphic function on some neighborhood of R and two such extensions define the same element of A(R) if they coincide on an open set which contains R. In other words, a function f ∈ A(R) is a germ on R of a holomorphic function and as sets where the inductive limit runs through all open neighborhoods of R and H(U ) is the Fréchet space of all holomorphic functions on U with the topology of uniform convergence on compact subsets. Equality (10) is used to equip the space A(R) with the so-called inductive topology. This is the strongest locally convex topology such that every restriction is continuous.
On the other hand, if K ⊂ R is compact then the restriction of f to K determines the germ in H(K), the space of all germs of holomorphic functions over K (we refer the reader to [2], p. 64 for detailed information on this space). Therefore, we may equip A(R) with the projective topology given by where the sets K run through all compact subsets of R. One easily observes that instead of taking in (11) all compact subsets K ⊂ R one can take any A fundamental result of Martineau [35] says that the inductive and the projective topology coincide. In [18] the authors give a simpler proof of this fact. Other proofs are discussed in [8]. With this topology the space A(R) is nuclear, complete and the polynomials are dense. It is also ultrabornological and webbed (see [36], Chapter 24 for definitions). This implies that the open mapping and the closed graph theorems hold for linear maps T : A(R) → A(R) (see [36], Theorems 24.30 and 24.31). Furthermore, the space A(R) is reflexive.
A key element of the proofs of Theorems 1 and 2 is a description of the dual space of A(R). The so-called Grothendieck-Köthe-Silva (see [30], pp. 372-378, also [1], Theorem 1.3.5) duality says that The symbol H 0 (C ∞ \[−n, n]) stands for the space of all holomorphic functions in C\[−n, n], which vanish at infinity with the topology of uniform convergence on compact subsets. The index b in H([−n, n]) b denotes that the dual space is equipped with the strong topology, i.e. the topology of uniform convergence on bounded sets in H(K) (we refer the reader to [36], Chapter 23, especially p. 269 for a presentation of topologies of the dual space). Naturally the image of A(R) under the map of restriction to K is dense in H(K) for any K ⊂⊂ R. It follows therefore from ( [20], §26, Satz 1.6) that algebraically where {K n } n∈N is a compact exhaustion of the real line, for instance K n = [−n, n], n ∈ R. Isomorphism (13) holds also topologically when the dual space of the space A(R) is considered with its strong topology and limind H(K n ) is equipped with the corresponding inductive topology (cf. [18], Proposition 1.7). Thus we have Since we will work with continuous linear functionals on A(R) it is important to write down the isomorphism (14) explicitly. Namely, let ξ be a continuous linear functional on A(R). Then there exists an interval [−n, n] and a function f ∈ H 0 (C ∞ \[−n, n]) such that for any g ∈ A(R), The function G is holomorphic in some open simply connected neighborhood of R and restricted to R is equal to g. The C ∞ smooth Jordan curve γ lies in the common domain of holomorphy of f and G, and surrounds [−n, n], i.e.
[−n, n] ⊂ I(γ). One easily observes that the formula (15) is correct, i.e. it does not depend on γ and G used to compute ξ(g). It also does not depend on the representative of the class of f in limind H 0 (C ∞ \[−n, n]).

Symbol Space and Toeplitz Operators
The symbol space X (R) is defined as the inductive limit of the spaces H(U \K), where U runs through all open neighborhoods of R and K through all compact subsets of R. Definition (16) makes X (R) a locally convex space (this was discussed in detail in [9], Sect. 3). As we have already written in the Introduction, any functionF ∈ H(U \K) determines the equivalence class . We call such a function a symbol function, its equivalence class is a symbol. Two such One easily notices that the space X (R) is an algebra. Thus for any F ∈ X (R) the operator of multiplication We recall now the definition of the Cauchy transform on the space X (R) following the presentation in [28]. Let F ∈ X (R) and let F be the equivalence class of a functionF ∈ H(U \K), i.e. F = [F ] ∼ , where U is an open neighborhood of R and K is a compact subset of R. As usual we may assume that U is simply connected. Let now z ∈ U and choose a C ∞ smooth Jordan curve in U \K such that both z and K are contained in I(γ). We put Naturally, for any z ∈ U we can find a C ∞ smooth Jordan curve γ ⊂ U \K such that z and K are in I(γ). By Cauchy's Theorem the value of (CF )(z) does not depend on γ. Hence (17) defines a function holomorphic in U . Its equivalence class [CF ] ∼ in X (R) is by definition the Cauchy transform of F . We use the same symbol, that is CF , to denote this object. One easily shows that CF ∈ X (R) does not depend on the representativeF chosen to compute (CF )(z) in (17). As we have already stated, (CF )(z) in (17) is holomorphic on some neighborhood of R. Hence its restriction to the real line belongs to A(R). By Cauchy's integral formula, for any F ∈ A(R) it holds that CF = F . That is, we have C 2 = C and C is a continuous linear projection onto A(R). The existence of a continuous projection onto A(R) readily implies that A(R) is a closed subspace of X (R). Also, I −C is a projection onto H 0 (C\R). Hence this space is also a closed subspace of X (R). Furthermore, it is essentially a consequence of Cauchy's integral formula that We again refer the reader to ( [9], Sect. 3) for the details. For any F ∈ X (R) we consider the operator and call it a Toeplitz operator. As we explained in the Introduction in (7) and (8), the matrix associated with the operator T F is a Toeplitz matrix. This justifies the name of this class of operators. We now formulate explicitly the formula for the operator T F , F ∈ X (R). Assume that f ∈ A(R) and letf ∈ H(V ) be an extension of f . Assume that For any z ∈ W such a curve exists. The function is therefore holomorphic in W -by Cauchy's theorem the definition does not depend on γ. Its restriction to R is a real analytic function and is precisely equal to (CM F )f . We slightly abuse the notation and denote the holomorphic function defined by (18) also by T F f . That is, T F f is now both an element of A(R) and its holomorphic extension to a simply connected neighborhood of R defined by (18).
In ( [28], Sect. 3) there is provided a detailed discussion of the definition of Toeplitz operators (we emphasize however that therein (18) was used as a definition of a Toeplitz operator). In particular, it was shown that the formula is correct (see [28], Proposition 3.1). That is, it does not depend on the extensionf ∈ H(V ) of f ∈ A(R) and the representativeF ∈ H(U \K). It is also an immediate consequence of the definitions that T F : A(R) → A(R) is a continuous linear operator on the space of real analytic functions (see [9], Sect. 3 for the details).
We remark that sometimes to simplify the notation we will use the same symbol for a function f ∈ A(R) and its holomorphic extension on some open neighborhood of the line R. Similarly, we use the same symbol to denote a class F ∈ X (R) and its representative in a certain space H(U \K).
As we have already mentioned, in the previous work [9] the authors characterized Fredholm Toeplitz operators (Theorem 2.1 above). Also, in [27] there was proved an analog of the Coburn-Simenenko Theorem for Toeplitz operators on the space of real analytic functions. This result will also be important in our arguments below.

Wiener-Hopf Factorization
Here we prove a result which is an analog of the Wiener-Hopf factorization. For information on this important result for Hardy spaces of Carleson curves we refer the reader to ( [3], Chapter 6.12). The proof which we will give below is actually similar to the classical case ( [3], Theorem 6.32). We include it for completeness. That is We used the symbol H(C ∞ \R) to denote the space where K runs through compact subsets of R.
It follows from Theorem 3.1 that both T G and T G −1 are invertible. Let Q := I − C be the complementary projection to the Cauchy projection. One easily shows that the operators are invertible. Let ϕ, ψ ∈ X (R) be the solutions to the equations We put Finally we put Then F −1 then F does not vanish and, as a result, T F is a Fredholm operator. Obviously, This follows from the fact that We infer that also T F− is invertible and, as a result, is a Fredholm operator of index 0.

Left Invertibility of Toeplitz Operators
We prove here our first main result. In order to explain and motivate our methods we shall deal first with the multiplication operator on A(R) by a function in A(R), which is naturally an example of a Toeplitz operator. is not complemented in A(R).
As we remarked above, this theorem is given to motivate the general method. This is why we only sketch the proof.
Proof. For each N ∈ N define the space Let also X := X N ⊂ A(R). Choose functions f n ∈ A(R) such that f n (a m ) = δ nm . For f ∈ X the sum f (a n )f n is well-defined, since it is a finite sum. Also for any f ∈ X the function f − f (a n )f n belongs to Y . Assume that Y is complemented and let P : A(R) → Y be a continuous projection onto Y . Since for any f ∈ X the function f − f (a n )f n is in Y , we have Recall that the sum is finite. This shows that if Y is complemented, then there exist functions ϕ n ∈ A(R) such that for any f ∈ X, Note that no function ϕ n vanishes identically. Indeed, if ϕ n ≡ 0, then f n = P f n , which implies that f n ∈ Y . This is impossible, since f n (a n ) = 1. There exists therefore a pointz ∈ R such that ϕ n (z) = 0 for all n ∈ N. Choose a function f ∈ H(C) such that f (a n ) = 1 ϕ n (z) , n ∈ N.

Such a function exists by the Weierstrass Theorem. Let
On the other hand, (I − P )(g N | R ) = g N (a n )ϕ n , since g N | R ∈ X. We have (I − P )(g N | R )(z) = f (a n )F N (a n )ϕ n (z) = 1 ϕ n (z) F N (a n )ϕ n (z) = ∞ n=1 F N (a n ) = F N (a 1 ) + · · · + F N (a N ), which diverges, since F N converges to 1 uniformly on compact subsets of the plane. This is a contradiction, since the evaluation atz is a continuous functional on A(R). The formal argument requires a precise choice of the numbers p n . We shall refrain from writing down the details here. This will be done in the proof of Theorem 3.
Our aim now is to generalize the arguments from the proof of Theorem 4.1 for Toeplitz operators. We consider the operator T F with F ∈ X (R), where F is the equivalence class ofF ∈ H(U \K), with U an open neighborhood of R and K a compact subset of R. We assume thatF vanishes only on a sequence z n ∈ R which accumulates only at ± ∞.
According to Theorem 2.2 the operator T F is injective and has closed range. We prove that the range of T F is not a complemented subspace of A(R). The first step is a factorization of the symbol functionF . Let m n be the multiplicity of z n , n ∈ N. We assume that Note that we can always assume that 0 ∈ K and z 1 = 0. Using Weierstrass theory we writeF where the sequence (p n ) is now only assumed to be chosen in such a way that for any In order to complete the proof of Theorem 3, a more precise choice of the sequence (p n ) will be needed. This will be done at the end of the proof. Observe that the function F 0 ∈ H(U \K) does not vanish in U \K.
Consider the operator T F0 . To simplify the notation we write T F0 for the Toeplitz operator corresponding to the symbol [F 0 ] ∼ ∈ X (R). We apply this convention also in the next results.
It follows from Theorem 2.1 that the operator T F0 is a Fredholm operator and Our first goal is to construct a sequence of continuous linear functionals ξ n ∈ A(R) , n ∈ N such that Any ξ is ξ ϕ (see (15)) for a function ϕ ∈ H 0 (C ∞ \L), L a compact subset of R. In order to simplify the notation we shall identify the functional ξ ϕ with the function ϕ. With this convention (20) takes the form for some sequence ϕ n ∈ H 0 (C ∞ \L n ), L n ⊂⊂ R. The following fact is a key to many results proved below. Roughly speaking continuous linear functionals cannot distinguish between a multiplication operator and a general Toeplitz operator.

Lemma 4.2. Assume that
Then where γ is a C ∞ smooth Jordan curve in W \(K ∪ L) such that K ∪ L ⊂ I(γ). The symbol W stands for a simply connected neighborhood of R contained in U ∩ V .  (18) is holomorphic in W -recall our convention concerning denoting Toeplitz operators. Let δ be a C ∞ smooth Jordan curve contained in W and such that L ⊂ I(δ). Then Choose a C ∞ smooth Jordan curve γ in W such that K ⊂ I(γ) and also δ ⊂ I(γ). We have by Cauchy's integral formula, since ϕ vanishes at ∞.
The next proposition is a key technical ingredient in the proof of Theorem 3. Let us recall that F 0 is holomorphic in U \K. We may assume without loss of generality that U is simply connected.
If index T F0 = k ≥ 0, then there exists a function ϕ ∈ H 0 (C ∞ \L), L a compact subset of R, and natural numbers l, μ l , μ l ≤ m l , such that If index T F0 = k < −1, then there exists a function ϕ ∈ H 0 (C ∞ \L), L a compact subset of R, and natural numbers l, μ l such that for z ∈ U \(K ∪ L).
If index T F0 ≥ 0, then there exists a function a ∈ H(U ) such that for z ∈ U \(K ∪ L) and z = z 1 , . . . , z l . To prepare the proof of Proposition 4.3 we need some lemmas. The next lemma in accordance with Lemma 4.2 explains the reason why (21) and also (22) hold true.

Proof. By Cauchy's integral formula
for sufficiently large R > 0, by Cauchy's theorem. This implies that G −n = 0 for n ∈ N. Hence, G = G + .
Our method to prove Proposition 4.3 is to reduce the proof to the finite dimensional case of Fredholm operators. We formulate now three auxiliary Lemmas.
In other words, if ξ ψ vanishes on R(T G ), then Proof. There is a finite number of zeros of the symbol function G in U \K, all of them are real. There is therefore a compact setK ⊂ R such that G does not vanish in U \K. It follows therefore from Theorem 2.1 that T G is a Fredholm operator. Furthermore, by the argument principle we have As in the proof of Lemma 4.5 it follows from the Coburn-Simonenko theorem that dim ker T G = 0 and dim coker T G = N −1 n=1 m n + l + 1.

Consider the family of functions
Each of these functions defines a continuous linear functional on A(R), since they belong to H 0 (C ∞ \ (L ∪ {z 1 , . . . , z N })) and z 1 , . . . , z N are real. The number of the functions is equal to dim coker T G . We claim that the continuous linear functionals defined by functions in G all vanish on R(T G ), i.e.

R(T
The curve γ is a C ∞ smooth Jordan curve contained in the intersection of the domains of F 0 and f , which may be assumed to be simply connected, and such that the sets K, L and the points z 1 , . . . , z N are contained in I(γ). Hence, for every f ∈ A(R) and every ψ ∈ G, since ξ ϕ vanishes on the range of T F0 . In order to complete the proof of the lemma it suffices to show that the functions in G are linearly independent. This is however elementary.
We now consider the case Find a number l ∈ N and μ l with 0 ≤ μ l ≤ m l such that with the convention that m 0 = 0. Consider the symbol function It follows from Theorem 2.1 that In view of Lemma 4.5 there exists ϕ ∈ H 0 (C ∞ \L) with a compact L ⊂ R such that Lemma 4.7. With the above notation consider the symbol function where 0 ≤ ν l ≤ m l − μ l . Then In other words, if ξ ψ vanishes on R(T H ), then Similarly, consider the symbol function where N > l and 0 ≤ ν N ≤ m N . Then In other words, if ξ ψ vanishes on R(T H ), then Proof. In order to prove the lemma, it suffices to apply Lemma 4.6 with G instead of F 0 . Indeed, there is a finite number of real zeros of G in U \K. Hence, there is a compact subsetK ⊂ R such that G does not vanish in U \K. Also, index T G = −1, we may therefore invoke Lemma 4.6. For instance when H is defined by (26) we have Now we investigate the case index T F0 = k < −1. Choose l ∈ N and 0 ≤ μ l ≤ m l such that Consider the symbol function Since the denominator vanishes only on a finite number of real zeros, G is a symbol function, i.e. it is holomorphic in some set of the form U \K. In this set the function G does not vanish, since F 0 does not vanish. It follows therefore from Theorem 2.1 that T G is a Fredholm operator and There exists therefore ϕ ∈ H 0 (C ∞ \L) such that Lemma 4.8. With the notation introduced above consider the symbol  (28) and to the functions in the family

Then the range of T H is the intersection of the kernels of the functionals which correspond to the functions in the family
, . . . , Consider the symbol for some 1 ≤ j ≤ l. The range of the operator T H is the intersection of the kernels of the functionals which corresponds to the functions in the family , . . . , Consider the same symbol as in (29) for j > l. Then the range of the operator T H is the intersection of the kernels of the functionals which correspond to the functions in the family , . . . , Proof. Again in order to prove the lemma, it suffices to apply Lemma 4.6 with G instead of F 0 .
We are now ready to prove Proposition 4.3.
We show that The same argument as in Lemma 4.6 shows the inclusion ⊂. Indeed, write F = E · F 0 , then . We use as before the same symbol f for an extension of f ∈ A(R) to some holomorphic function in H(V ), V an open neighborhood of R. As usual γ is a C ∞ smooth Jordan curve in W \(K ∪ L) with K ∪ L ⊂ I(γ). The symbol W denotes, as in Lemma 4.2, a simply connected neighborhood of R contained in the intersections of the open sets U and V . We use this convention below as well and also we write F to denote not only the symbol but also the symbol function, which represents F .
Assume that ξ = ξ ψ vanishes on the range of T F , i.e. for any f ∈ A(R) In other words, for every f ∈ A(R). We have F ∈ H(U \K) and ψ ∈ H 0 (C ∞ \L), hence F ψ ∈ H(U \(K ∪ L)). It follows from Lemma 4.4 that there exists a function a ∈ H(U ) such that For such points z we may write This means that a(z) vanishes for z n ∈ (K ∪ L) c . Hence, for a different function a ∈ H(U ) we have for z ∈ U \(K ∪ L). For z ∈ U \(K ∪ L) we can therefore write Consider the symbol function For any f ∈ A(R) we have This means that Indeed, assume that the equality does not hold. We showed in (31) that R(T F ) ⊂ K and we also know from Theorem 2.2 that R(T F ) is closed. Consider the locally convex space A(R)/R(T F ). By the Hahn-Banach theorem there exists ξ ∈ A(R) , which vanishes on R(T F ) and ξ(x) = 0 for some x ∈ K\R(T F ). By (33) such a functional ξ belongs to Hence it vanishes on K, which is a contradiction. We infer that R(T F ) = K. This proves the first assertion of the proposition.
Consider now the case index T F0 = k ≥ 0. Let now ξ ψ vanish on R(T F ), i.e.
T F f, ξ ψ = 0 for every f ∈ A(R). We again conclude that there exists a ∈ H(U ) such that for z ∈ U \(K ∪ L) and, as a result, We have by Cauchy's theorem, since the integrand is holomorphic in W ⊂ U ∩ V . It follows from Lemma 4.7 that As in the first part of the proof, it follows from the Hahn-Banach theorem that Now we consider the case index F 0 = k < −1. This time the proof follows from Lemma 4.8.
We now prove representations (21) and (22). Assume first that Then, as in (30), for some a ∈ H(U ) and z ∈ U \(K ∪ L). Since F 0 does not vanish, we have Assume that index T F0 = k < −1. The function ϕ is now chosen in such a way that for f ∈ A(R), where, let us recall, The numbers l and μ l were chosen in (27). For some a ∈ H(U ) we have for z ∈ U \(K ∪ L), z = z 1 , . . . , z l . Hence, similarly as before, The equality extends to U \(K ∪L). We therefore also have the representation for some a ∈ H(U ). Assume now that index T F0 = k ≥ 0. The function ϕ is now chosen in such a way that The numbers l and μ l are chosen in (24). We therefore have for some a ∈ H(U ) and z ∈ U \(K ∪ L). Hence for z ∈ U \(K ∪ L) and z = z 1 , . . . , z l Recall that we are guided by the proof of Theorem 4.1. This is why our next goal is a construction of dual functions in each case singled out in Proposition 4.3, i.e. functions f i ∈ A(R), i ∈ N such that ξ j (f i ) = δ ij . By ξ j , j ∈ N we denoted the functionals for which it holds that We treat all three cases together. Observe that in each case there are j ∈ N and numbers ν n ∈ N such that For simplicity we assume that j = 1 and ν n = m n for n ∈ N. The first step is of a different nature. Namely, there are two possibilities: either or the equality does not hold. In the latter case there exists a non-zero func- The function f/ξ ϕ (f ) is the first element of the construction in this case. Otherwise it suffices to construct functions f ∈ A(R) dual to functionals of the form ϕ (z−zn) k , n ∈ N, 1 ≤ k ≤ m n , which is also the second step in the second situation.
So assume that ξ is defined by a function ϕ (z−zn) k for some n ∈ N and 1 ≤ k ≤ m n with ϕ of the form postulated by Proposition 4.3. We seek for a function f ∈ A(R) such that for i = n and 1 ≤ j ≤ m i and also for j = k. Such a function will be constructed by means of Toeplitz operators with appropriately chosen symbols. Let us recall that either for some function a ∈ H(U ) or Our consideration must be subordinate to these two cases. Denote Observe that A is finite. This follows from (36) and (37) since a(z) = 0 implies ϕ(z) = 0. Recall that ϕ ∈ H 0 (C ∞ \L). Thus zeros of ϕ cannot accumulate at ± ∞. For n ∈ A let α n be the multiplicity of z n . Write for a function b ∈ H(U ). Assume first that index T F0 ≤ −1. Consider the symbol function Observe that F n is indeed a symbol function, since as we noted, A is finite. Let now g ∈ A(R) be arbitrary. For ι = n and 1 ≤ κ ≤ m ι it holds that by Cauchy's theorem, since the integrand is holomorphic. We used Proposition 4.3, formula (21). If index T F0 ≥ 0 we set for n > l. The number l was chosen in (24). It follows from Proposition 4.3 that if index T F0 ≥ 0, then we need to find functions dual to the functions If n > l and ι = n then for every g ∈ A(R) we have, dz by Proposition 4.3. Observe that either ι > l or ι = l and κ + μ l ≤ m l . We infer that also in this situation Consider now the case n = l. Then ι > n = l and we have Observe that in every case considered above we have T Fn g, ϕ = 0 (resp. T F l g, ϕ = 0, T Fn g, ϕ = 0) for every g ∈ A(R). This means that functions of the form T Fn g, g ∈ A(R) (resp. T F l g, T Fn g) are good candidates for f in formulas (34). Indeed, for every g ∈ A(R) conditions (34) are satisfied. In order to have dual functions we need therefore to choose functions g n,k ∈ A(R), n ∈ N, k = 1, . . . , m n such that If index T F0 ≥ 0 we have n > l or n = l and then j, k = 1, . . . , m l − μ l .
We treat the case index T F0 ≤ −1 first. By Cauchy's integral formula Thus the function g n,k ∈ A(R) must be chosen in such a way that From now on we consider only those natural numbers N ∈ N which satisfy |z N | < |z N +1 |. It follows from (19) that these numbers constitute a subsequence of N. For such a number N ∈ N we choose numbers p N n , n > N such that for |z| ≤ |z N | + δ N , where the numbers δ N > 0 are chosen in such a way that |z N | + δ N < |z N +1 |. We first determine the numbers ε N > 0 and then show that appropriate sequences (p N n ) can also be chosen. Denote By Cauchy's integral formula for k > 0 for |z| < |z N | + δ N , in particular for z = z 1 , . . . , z N . Hence for |z| ≤ |z N |, if ε N satisfies (44). For any i = 1, . . . , N such that m i ≥ 2 the number N > 0 must therefore be chosen in such a way that ε N < 1 and for j = 2, . . . , m i , where ψ M corresponds to ϕ (z−zi) j . This guarantees that the second term in (43) is uniformly small. If m i = 1, then this condition is void. In particular, if m 1 = · · · = m N = 1, then one can simply take ε N = 1. In general, for a fixed number N the number ε N must satisfy a finite number of conditions (45) corresponding to i = 1, . . . , N and j = 2, . . . , m i . Thus such a number can be chosen. Let With such a choice of ε N , under the assumption that a(z n ) = 0, we have The second term is ≤ 1 3 by (45). From the choice of the function f in (41) and (44) it follows that the first does not exceed 1 3 , as well. If j = 1, then the estimate reduces just to (44). Thus, it follows from (40) that which shows that as N → ∞. This estimate is valid under the assumption that a does not vanish on any z n . If this is not the case, the estimate differs by a constant. The conclusion is however the same, namely, (I − P )(T FN f )(z) diverges. Thus to complete the proof we need to choose the numbers p N n in such a way that (44) holds true. Fix N . There exists a number η N > 0 such that if |z| ≤ η N , then |e z − 1| ≤ ε N /3. Also, We can now complete the proof of Theorem 1.
Proof of Theorem 1. The proof is standard and elementary (Note however that we use such a powerful tool as the closed graph theorem). We include it for completeness.
Assume that T F is an injective Fredholm operator. We shall show that T F is left invertible. If T F is additionally surjective then it is invertible. Thus assume that the classes of the functions g 1 , . . . , g n ∈ A(R) in A(R)/R(T F ) form a basis of A(R)/R(T F ). For any f ∈ A(R) there are unique numbers α 1 , . . . , α n and a unique function g ∈ A(R) such that f = α 1 g 1 + · · · + α n g n + T F g.
The uniqueness follows from the fact that T F is injective by assumption. We now define a left inverse operator of T F . If f is represented by (47) we put Sf = g. The definition is correct, since the representation (47) is unique. By the same reason S is linear. It is clear that We show that S is continuous. We apply the closed graph theorem ( [36], Theorem 24.31). Observe that the space of real analytic functions is ultrabornological, i.e. it has the inductive topology of a family of Banach spaces and has a web (see [36], p. 287). The second statement follows from the fact that A(R) carries the projective topology of a sequence of (DF S)-spaces (see [36], Lemma 24.28,Corollary 24.29). Assume that f ν → 0 and Sf ν → h in A(R). We need to show that h ≡ 0. We have f ν = α ν 1 g 1 + · · · α ν n g n + T F g ν . It follows that the classes of f ν in A(R)/R(T F ) tend to zero in the quotient space. Hence, since the space A(R)/R(T F ) is finite dimensional we have α ν i → 0 for 1 ≤ i ≤ n. Also, Sf ν = g ν → h by assumption. Hence, since T F is continuous and also f ν → 0. Hence T F h = 0, which implies that h = 0, since T F is injective. We infer that S is a continuous linear left inverse of T F .
If T F is left invertible, then the range R(T F ) is a closed complemented subspace of A(R). It follows from Theorems 2.2 and 3 that F = [F ] in X (R) = limind H(U \K) for someF ∈ H(U \K) which does not vanish in U \K. It follows from Theorem 2.1 that T F is a Fredholm operator. As a left invertible operator it must be injective.
It is a consequence of Theorem 3.1 that T F can only be injective if windingF ≥ 0.
It follows from the Wiener-Hopf factorization (Theorem 3.2 above) that we can also give a constructive proof of left invertibility of injective Fredholm Toeplitz operators. First of all observe that where Q denotes the complementary projection, i.e. Q := I − C. It follows that if G ∈ A(R) or F ∈ H 0 (C ∞ \R) then CM F QM G = 0 on A(R). As a result, if F ∈ H 0 (C ∞ \R) or G ∈ A(R), then Since any F ∈ H(C ∞ \R) can be written as F (∞) + F 0 with F 0 ∈ H 0 (C ∞ \R) we also have (48) when F ∈ H(C ∞ \R) or G ∈ A(R). Assume now that T F is a Fredholm operator and k := index T F ≤ 0. That is, T F is an injective Fredholm operator. Then it follows from Theorem 3.2 that F can be factored as and F + ∈ A(R) and both symbols are invertible in the corresponding spaces. Now we have In other words, if T F is an injective Fredholm operator, then T z k F −1 is a left inverse. One notices that this part of the theory is indeed very similar to the classical Hardy space case.

Right Invertibility of Toeplitz Operators
Our goal now is to prove Theorem 4. Assume that T F : A(R) → A(R) is right invertible, Then T F : A(R) b → A(R) b is left invertible. Hence, its range in A(R) b is complemented. Also, if T F is right invertible then it is surjective. We show that if F has non-real zeros accumulating at a real point but no real zeros going to infinity then the range of T F is not complemented in A(R) b . As a result, it follows from Theorem 4 that T F can be right invertible only if T F is a surjective Fredholm operator.
We start with the following easy observation. In view of Theorem 2.2 this Proposition will be proved once we have shown the following Lemma. Proof. Since ker T = (R(T )) • we have (ker T ) • = (R(T )) •• ⊃ R(T ). We show that the equality holds, which completes the proof since (ker T ) • is closed in the strong topology of A(R) . To show that (ker T ) • ⊂ R(T ) we define the map T 0 : A(R)/ ker T → A(R), T 0 (x + ker T ) := T x. The map T 0