Toeplitz operators and Wiener-Hopf factorisation: an introduction

Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces $H^p$ of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in $L^p(\mathbb R)$.


Introduction
Wiener-Hopf factorisation, which was developed mainly in connection with the study of singular integral equations, convolution type operators and Riemann-Hilbert problems that are important in a variety of areas in Mathematics, Physics and Engineering, plays a prominent role in the theory of Toeplitz operators. Indeed many spectral properties, Fredholmness, invertibility and formulae for the inverses, when they exist, can be expressed in terms of a Wiener-Hopf (WH for short) factorisation of their symbols.
Several monographs have appeared in the last four decades ( [3,8,10,12,13,15] for instance) intending to present, as systematically and completely as possible, the myriad of results that kept appearing on WH factorisation and its relations with singular integral equations, boundary value problems and Toeplitz operators. However, the enormous quantity of results and the extension of most monographs make it difficult for someone not familiar with this topic to come to grips with the vast amount of information contained there and quickly cut through to get to the heart of the matter.
The present article is based on a mini-course given in the context of the 13th Advanced Course in Operator Theory and Complex Analysis held in Lyon in 2016. Its aim is to introduce the reader to a set of concepts and results enabling one to rapidly understand the essential ideas behind the notion of WH factorisation and its usefulness in the study of Toeplitz operators.
It would be clearly impossible to cover in a single article this vast domain, so the purpose of this paper required choices regarding which results to present, their setting, how to show their interrelations and how they build upon each other. This was done based on the author's own experience in learning the subject, as well as the author's research interests.
First, it should be explicitly noticed that only scalar-valued symbols will be considered. The factorisation of matrix-valued functions and the study of Toeplitz operators with matrix symbols involve considerably more difficulties that would overshadow the exposition in a first approach.
Secondly, we study here Toeplitz operators defined in the Hardy spaces of the (upper) half-plane, H p (C + ), with 1 < p < ∞. The existing literature overwhelmingly puts its emphasis on factorisation relative to a closed contour such as the unit circle and on Toeplitz operators defined in the Hardy spaces of the disk, whether in the context of H 2 or H p , 1 < p < ∞. However, the natural setting in which many problems appear, as for instance those formulated in terms of finite-interval convolution equations, is the real line. Translating the results presented here from the context of the unit circle to that of the real line, for p = 2, would require considering weighted Hardy spaces of the disk (see [4,13], for instance). By considering the halfplane setting from the start, on the one hand, we avoid this; on the other hand, it allows for a better understanding of the reasoning and the techniques involved when the real line is considered, provides a direct approach to problems that are naturally formulated in the context of R, and allows the use of tools such as the Fourier transform.
The paper is organised as follows. In Section 2 the fundamental function spaces used in the paper and their main properties are described. Toeplitz operators and some of their basic properties are presented in Section 3, and their relation to paired operators is described in Section 4. In Section 5 the Fredholm properties and invertibility of a Toeplitz operator are characterised in terms of a WH factorisation of its symbol. Piecewise continuous symbols are studied in Section 6, where criteria for existence of a WH factorisation of those functions are presented and explicit formulae for the factors are obtained in a particular case. The results of the previous sections are used in Section 7 to study the spectrum of the singular integral operator S J in three cases, J = R , J = R + , J = [0, 1], and to obtain an expression for the resolvent operator ( The main references for Section 2 are [6,7,14,17]; for Section 3, [3,10,17]; for Section 4, [3,9,15]; for Section 5, [6,13]; for Sections 6 and 7, [6,17].

Spaces of functions
For 0 < p < ∞, let L p denote the Lebesgue space of all complex Lebesgue measurable functions f which are p-integrable on R, with the norm We denote by H + p the Hardy space of all functions f which are analytic in the upper half plane C + such that, for all y > 0, |f (x + iy)| p is integrable over R and there is a constant M ∈ R + such that R |f (x + iy)| p dx < M for all y > 0, with the norm 7]). If f + ∈ H + p then the nontangential boundary functioñ is defined a.e. on R and belongs to L p , with ||f + || p = ||f + || H + p . In what follows, we assume that 1 ≤ p < ∞. We have 1 2πi Rf Conversely, iff + ∈ L p and Rf t−z dt = 0 for all z ∈ C + , then f + defined by belongs to H + p and its boundary value on R isf + . We define H − p similarly for the lower half plane C − and analogous results hold for f − ∈ H − p . In particular, iff − is the (nontangential) boundary value of f − on R then .

(2.2)
Identifying each function in H ± p with its boundary value on R, we have that H + p and H − p are closed subspaces of L p and the following holds: and for p, r ∈]1, ∞[ By L ∞ we denote the space of all essentially bounded functions on R, with the norm ||f || ∞ = ess sup x∈R |f (x)| , and by H ± ∞ the space of all functions which are analytic and bounded in C ± . We identify H ± ∞ with the subspaces of L ∞ consisting of their (nontangential) boundary functions on R. We have We also define, for 1 ≤ p < ∞ , ( [17]) and we have By the Luzin-Privalov theorem, if f is meromorphic in C + (resp. C − ) and has nontangential boundary value 0 on a set E ⊂ R with positive measure, then f (z) = 0 for all z ∈ C + (resp. C − ). Thus if a function f belonging to H ± p or to H ± p (1 ≤ p ≤ ∞) vanishes on a set of positive measure E ⊂ R then f is identically zero.
By C(R ∞ ) we denote the space of all continuous functions on R ∞ = R∪{∞}, with the supremum norm, and by R the set of all rational functions without poles on R ∞ . The set of all piecewise continuous functions on R ∞ , i.e., f ∈ L ∞ (R) such that f is continuous on R ∞ with the possible exception of finitely many points and with finite limits f (±∞) , f (x ± ) for all x ∈ R, is denoted by P C(R ∞ ).
If A is an algebra, we denote by GA the group of all invertible elements in A.

Toeplitz operators in H
where the integral is understood as a Cauchy principal value, defines, by a classical theorem of M. Riesz ([16]), a bounded operator in L p . We have We associate with S R two complementary projections By the Sokhotski-Plemelj formulae, P ± f are the nontangential boundary values on R of the analytic functions respectively. Thus P ± L p = H ± p and ker P ± = H ∓ p . In terms of these projections, (2.6) can be expressed by It follows from (3.3) and (2.4) that, if C denotes complex conjugation (C f = f ) then CP ± = P ∓ C , i.e., C P ± C = P ± (3.8) and, from (3.4), that P ± a − aP ± is compact in L p if a ∈ C(R ∞ ).
The Toeplitz operator with symbol g ∈ L ∞ , T g , is defined in H + p by Whenever we want to make the domain H + p explicit, we will use the notation T g,p . We have ||g|| ∞ ≤ ||T g || ≤ C ||g|| ∞ ( [6]), where C = 1 if p = 2. Thus T g = 0 if and only if g = 0.
The following properties hold: Note that in general (T g ) −1 = T g −1 .
For f + ∈ H + p we have that where the equality holds on R. Defining r by it follows from (3.10) that, for k ≥ 0, On the other hand from (P2) we have, for g ∈ L ∞ , k ∈ Z, From (3.10)-(3.14) we conclude the following.
Proposition 3.1. Let k ∈ Z and let r be given by (3.11).
Recall that an operator T is Fredholm if and only if its kernel and the kernel of its adjoint are finite dimensional, and the range of T is closed; the Fredholm index of T is Ind T = dim ker T − dim ker T * . We will see in the next section that, in a certain sense, all Toeplitz operators which are Fredholm can be reduced, through an appropriate factorisation of their symbol, to a Toeplitz operator with a simple rational symbol such as the one considered in Proposition 3.1. From the latter we see moreover that: (i) either dim ker T r k = 0 or dim ker T * r k = 0 ; (ii) T r k is always one sided invertible. We will also show in the next theorem that property (i) is shared by all non-zero Toeplitz operators, while (ii) holds for all Toeplitz operators which are Fredholm. We have: Thus Since the left hand side of (3.15) represents a function in H − 1 and the right hand side represents a function in H + 1 , we conclude that both are zero. By the Luzin-Privalov theorem either f + or h − must be zero, which is impossible since f + , h + = 0.
Coburn's Lemma can also be stated as follows: a non-zero Toeplitz operator has a trivial kernel or a dense range.
A necessary and sufficient condition for an operator A ∈ L(X, Y ), where X and Y are Banach spaces, to have a left (resp. right) inverse is that ker A = {0} and Im A is closed and complemented in Y (resp., Im A = Y and ker A is complemented in X) ( [9]). Therefore we have: By the Hartman-Wintner theorem ( [3]), if T g is semi-Fredholm, i.e., Im T g is closed and ker T g or ker T * g are finite-dimensional, then g ∈ GL ∞ . Therefore we also have:

Toeplitz operators and paired operators
It is clear that Therefore Toeplitz operators belong to the class of operators of the form where P ∈ L(X) is a projection in the Banach space X and A ∈ L(X). These operators, which are called operators of Wiener-Hopf (WH) type, are closely connected with operators in X of the form P AP + Q (where Q = I − P ) and to the paired operators AP + Q and P A + Q as follows: In this case we say that T * ∼ S ( [2]) . From Proposition 3.9 we see that On the other hand, taking for instance the operator AP + Q, we can write are invertible; thus each one of the three operators in (4.2) is equivalent after extension to P A P |Im P . In particular for P = P + , Q = P − we see that Operators which are equivalent after extension share many properties, namely the following ( [2]). It follows from (4.3) and from Theorem 4.2 that we can reduce the study of many properties of Toeplitz operators to the corresponding study for paired operators. We make use of this in the following theorem, which will be used later.
Proof. Taking Theorem 4.2 into account, it is enough to prove that (gP

Fredholmness, invertibility and Wiener-Hopf factorisation
It is easy to see, using properties (P1)-(P3) in Section 3, that the invertibility properties of T r k (cf. Proposition 3.1) also hold for any Toeplitz operator whose symbol g can be represented as a product In that case we say that (5.1) is a bounded WH factorisation and we have a corresponding factorisation for the Toeplitz operator T g : The integer k is called the index of the factorisation (5.1) and the latter is said to be canonical when k = 0. Since T g ± are invertible, and taking Proposition 3.1 into account, we have the following.
We can prove (iii) analogously.
Left and right inverses are not unique; for k = 0 another one sided inverse (left or right depending on whether k > 0 or k < 0) is given by T g −1 [6,15]) Whenever we associate an integer k to an operator A, and A is invertible if k = 0, only left invertible if k ∈ Z + and only right invertible when k ∈ Z − , we say that the invertibility of A corresponds to the integer k.
A class of symbols that always admit a bounded WH factorisation is GR, the set of all rational functions that belong to GL ∞ . It is easy to see that, for is the winding number of R around the origin, ind R, also called the index of R with respect to the origin ([1]), which is given by the difference between the number of zeroes and the number of poles of R in C + . Thus, for example, .
As a consequence of Theorem 5.1 and Corollary 3.5 we have thus: A necessary and sufficient condition for T R to be Fredholm in H + p is that inf x∈R |R(x)| > 0. In that case the invertibility of T R corresponds to k = ind R and dim ker Other classes of continuous functions, called decomposing algebras with the factorisation property ( [5,10,13]), such as the Wiener algebra on the real line W (R ∞ ) = C + FL 1 and the algebra C µ (R ∞ ) of Hőlder continuous functions with exponent µ ∈]0, 1[, also possess the property that every invertible element of the algebra admits a bounded WH factorisation (5.1). As a consequence, the result of Corollary 5.3 also holds if we replace R by W (R ∞ ) or C µ (R ∞ ) and Theorem 5.1 provides formulae for the inverses, or one sided inverses.
Using rational approximation, we can show that the results of Corollary 5.3 still hold when we replace R by C(R ∞ ) and we can also obtain formulae for the inverses, or one sided inverses, of the operator T g . However, in contrast to (5.3)-(5.5), these formulae now involve infinite series of operators ( [5,10]). Extending the previous results to more general classes of symbols in such a way that we can obtain similarly simple explicit formulae requires generalising the concept of factorisation. This is done in such a way that the main properties of the factorisation which are used to garantee invertibility of T g , when g admits a canonical bounded WH factorisation g = g − g + , hold at least in a dense subset of H + p . We choose the latter to be R + 0 , consisting of the rational functions belonging to R ∩ H + p . Thus we look for a factorisation of the form g = g − r k g + , where the factors are no longer required to be bounded but the equalities (2.6)). However, unlike the case when g ±1 + and g ±1 − are bounded in C + and C − respectively, this is not enough to garantee the invertibility of T g when g = g − g + . Having in mind that ( is a bounded factorisation, we will also require now that the operator g −1 is well defined and admits a bounded extension to H + p . Definition 5.4. A Wiener-Hopf (or generalised) factorisation of a function g ∈ L ∞ with respect to L p (WH p-factorisation for short) is a representation g = g − r k g + with k ∈ Z and The integer k is called the index of the factorisation, and the latter is said to be canonical when k = 0. If a WH p-factorisation exists, then it is unique up to a constant factor: Theorem 5.5. Let g ∈ L ∞ and let g = g − r k g + and g =g − rkg + be two WH p-factorisations. Then k =k and there is C ∈ C \ {0} such that Proof. We start by proving the uniqueness of the index. If k >k, then r k−k g +g −1 . Since k −k − 1 ≥ 0, the left hand side belongs to H + 1 , the right hand side to H − 1 ; thus both would be zero, which is impossible. We can see analogously that we cannot have k <k. It follows that g − g + =g −g+ , which is equivalent to g −g −1 − = g −1 +g + . Again, since the left hand side belongs to (x − i) H − 1 and the right hand side to (x + i) H + 1 , we conclude that both sides are equal to a non-zero constant (cf. (2.12)).
We now want to prove that T g is Fredholm in H + p if and only if g admits a WH p-factorisation, and invertible if and only if this factorisation is canonical. We will need the following lemmas.
Proof. Let r 0 (x) = 1 (x+z 0 ) n with z 0 ∈ C + , n ∈ N. We have Note that it follows from Lemma 5.6 that g −1 + P + (g −1 − r 0 ) ∈ H + p r 0 ∈ R + 0 and therefore g −1 Proof. We prove this by induction considering, without loss of generality, that r 0 (x) = 1 (x+z 0 ) n with z 0 ∈ C + , n ∈ N. For n = 1 we have since both terms on the right hand side belong to H − p , Theorem 5.8. Let g ∈ L ∞ . The operator T g is invertible in H + p if and only if g admits a canonical WH p-factorisation Proof. (i) First we prove that if g admits a factorisation (5.9) then T g is invertible. Assume thus that (5.9) is a canonical WH p-factorisation. Then T g is injective because we conclude that f + = 0. On the other hand T g is surjective. To prove this, let T 0 = g −1 + P + g −1 − I + : R + 0 → H + p and let r 0 ∈ R + 0 . Then, using Lemmas 5.6 and 5.7, Now take any ϕ + ∈ H + p and let (r + n ), with r + n ∈ R + 0 for all n ∈ N, be a sequence such that r + n → ϕ + in H + p . Let moreover T be the continuous extension of T 0 to H + p . We have T g T r + n = T g T 0 r + n = r + n and it follows that T g T ϕ + = ϕ + , i.e., ϕ + ∈ Im T g . (ii) Conversely, assume that T g is invertible in H + p . By Theorem 4.3 this is equivalent to T g −1 being invertible in H + p ′ , since we must have g ∈ GL ∞ by Corollary 3.5. Let then u + ∈ H + p , v + ∈ H + p ′ be the unique solutions of where λ + is defined in (2.8). Then we have Multiplying these two equations we get Since the left hand side is in H + 1 and the right hand side is in H − 1 we conclude that both are zero, therefore The first equality in (5.11) implies that (λ + u + )(λ + v + ) = 1; thus, defining g + = λ + v + , we have that g + ∈ H + p ′ , g −1 Let r o ∈ R + 0 and let f + = (T g ) −1 r 0 . Then P + (gf + ) = r 0 , i.e., gf + = r 0 + P − (gf + ). Now The left hand side of the last equality belongs to H + 1 , while the right hand side belongs to H − 1 , so both are equal to zero and we conclude that Taking these relations into account, we get and it follows that g −1 Remark 5.9. Remark that (5.8) was not needed to prove the injectivity of T g and the relation R + 0 ⊂ Im T g in the first part of the proof above.
Theorem 5.10. Let g ∈ L ∞ . The operator T g is Fredholm in H + p if and only if g admits a WH p-factorisation g = g − r k g + and, in that case, its Fredholm index is Ind T g = −k.
Proof. (i) Assume that T g is Fredholm and let −k be its Fredholm index. Then if k ≤ 0 we have r −k ∈ H + ∞ and Ind T r −k = k. Therefore T g T r −k = T gr −k is a Toeplitz operator which is Fredholm with index zero. By Corollary 3.5, T gr −k is invertible. Therefore gr −k admits a canonical WH p-factorisation gr −k = g − g + and it follows that g = g − r k g + is a WH p-factorisation. If k ≥ 0 we have r −k ∈ H − ∞ and Ind T r −k = k, so T r −k T g = T gr −k is a Toeplitz operator which is Fredholm with index zero, therefore invertible, and we conclude analogously that g admits a canonical WH p-factorisation.
(ii) Conversely, let us now assume that g = g − r k g + is a WH p-factorisation. If k ≥ 0 then T g = T g − g + T r k ; if k ≤ 0 then T g = T r k T g − g + . Since T g − g + is invertible by Theorem 5.8 and T r k is Fredholm with index −k, we conclude that T g is Fredholm with index −k.
Corollary 5.11. If g ∈ L ∞ admits a WH p-factorisation g = g − r k g + , then the invertibility of T g corresponds to the integer k. For k > 0 the operator g −1 + P + g −1 − T r −k is a left inverse for T g and dim ker T * g = k; for k < 0 the operator T r −k (g −1 + P + g −1 − I + ) is a right inverse for T g and dim ker T g = |k|.
It is clear from Definition 5.4 that g ∈ L ∞ admits a WH p-factorisation g = g − r k g + if and only if g 0 = r −k g admits a canonical WH p-factorisation.
In that case T g 0 is invertible by Theorem 5.8. Therefore the factors g ± satisfy (5.6), (5.7) and the range of T g 0 = T g − g + is closed in H + p , i.e., T r −k g = T g − g + has closed range. (5.13) We can see from the proof of Theorem 5.8 that the converse is also true and thus condition (5.8) in Definition 5.5 can be replaced by (5.13). In fact, from the first part of the proof of Theorem 5.8 we see that (5.6) and (5.7) imply that T g 0 is injective and R + 0 ⊂ Im T g 0 . Since R + 0 is dense in H + p , by adding (5.13) to (5.6) and (5.7) we conclude that T g 0 is surjective, therefore invertible. Thus we have the following.
This means that we can replace (5.8) by another condition which is easy to verify for a large class of functions in L ∞ , including all piecewise continuous functions ( [6,10]), as we show in section 6.
We can now ask when does a function g ∈ L ∞ admit a WH p-factorisation and, if it exists, how to obtain its factors and, in particular, its index.We address this question in the following section.

Piecewise continuous symbols
There are several classes of functions for which simple criteria for existence of a WH p-factorization can be established. (see for instance [3,5,13]). We will consider here only the class P C(R ∞ ) of all piecewise continuous functions with finite limits at ±∞.
If g ∈ C(R ∞ ) then g p (x, w) = g(x) for all x ∈ R ∞ and w ∈ R ±∞ . It is also easy to see that if p = 2 then The image of g p in the complex plane is a closed curve Γ obtained by adding to the image of g(x), with x ∈ R ∞ such that g is continuous at x, certain arcs of a circle (or line segments, when p = 2) connecting the points corresponding to g(x − ) and g(x + ) whenever these two values are different.
then we can associate with g p an integer, designated by ind g p , defined as the winding number of Γ around the origin. If g ∈ C(R ∞ ) then condition (6.3) means that g ∈ GC(R ∞ ) and in that case ind g p = ind g.
Definition 6.1. We say that g ∈ P C(R ∞ ) is p-nonsingular, or p-regular, if and only if (6.3) holds.
Note that the product of two p-regular functions may be p-singular, as shown in an example presented below in this section. However, if two functions g , h ∈ P C(R ∞ ) have no common points of discontinuity, then (gh) p (x, w) = g p (x, w) h p (x, w) and, if gh is p-regular, then ind (gh) p = ind g p ind h p ( [6]).
For Toeplitz operators with piecewise continuous symbols we have the following.
Theorem 6.2. ( [6,17]) Let g ∈ P C(R ∞ ). The operator T g has closed range in H + p if and only if g is p-regular. In this case T g is Fredholm, the invertibility of T g corresponds to k = ind g p and the Fredholm index of T g is Ind T g = −k. Corollary 6.3. If g ∈ P C(R ∞ ), then g admits a WH p-factorisation if and only if g is p-regular and, in this case, g = g − r k g + with k = ind g p .
In the case p = 2 we see from the previous corollary that g ∈ P C(R ∞ ) is 2-regular if and only if g is locally sectorial, i.e., for all x ∈ R ∞ we have g(x ± ) = 0 (6.4) Condition (6.5) means precisely that the discontinuities are such that the line segment connecting the points g(x − ) and g(x + ) in the complex plane does not include the origin. Now, having considered the Fredholm properties of T r k with k ∈ Z, in Proposition 4.2, it is natural to consider also the case when the exponent k is not an integer, to illustrate the previous results. Let p = 2 , α ∈ R \ Z with |α| ≤ 1/2, and let g The image of (g (α,∞) ) 2 (x, w) in the complex plane consists of the upper half of the unit circle and a line segment connecting the points 1 and −1, when α = 1/2; it consists of an arc of the unit circle in the upper half-plane connecting the points e 2iπα and 1, and a line segment from 1 to e 2iπα , when 0 < α < 1/2. It follows from Corollary 6.3 that g (α,∞) admits a WH 2factorisation if and only if |α| < 1/2 and, in that case, the factorisation is canonical.
More generally, defining for c ∈ R , α ∈ C \ Z, where the branch cut connecting 0 to ∞ intersects the unit circle at the point is continuous for all points of R ∞ except the point c. If the discontinuity points of g ∈ P C(R ∞ ) are c 1 , c 2 , ...c n ∈ R and (eventually) ∞, then g can be represented in the form ( [6]) where g 0 ∈ C(R ∞ ) and , for j = 1, 2, ..., n , α 0 = 1 2πi log g(+∞) g(−∞) .
Although there are explicit formulae for the factors in a WH p-factorisation of g ∈ P C(R ∞ ), once its index is known ( [17]), the factors can in some cases be obtained by inspection. For example, if −1/2 < α < 1/2 we can write choosing appropriate branches such that (x − i) ±α ∈ H − 2 , 1 x+i ±α ∈ H + 2 . Since the range of T g , with g given by the left hand side of (6.7), is closed by Theorem 6.2, it follows from Corollary 5.12 that (6.7) is a canonical WH 2-factorisation of g.
7 The spectrum of the singular integral operator S J in L 2 (J) Let S J , where J ⊂ R is an interval, denote the singular integral operator In this section we study the spectrum of S J in three cases, J = R , J = R + , J = [0, 1], using the results of the previous sections. The spectrum of S J in these three cases was described in [17] using a slightly different approach. Furthermore, using the factorisation of the symbol of an associated Toeplitz operator, we obtain expressions for the resolvent operator (S R + − λI R + ) −1 when λ / ∈ σ(S R + ). Here I R + denotes the identity operator in L 2 (R + ).
Second case: J = R + . For ϕ ∈ L 2 (R + ) we have Let χ ± denote the characteristic functions of R ± , respectively. Identifying L 2 (R + ) with a subspace of L 2 (R), L 2 (R + ) = χ + L 2 (R), we have We use now the Fourier transform, a natural tool to deal with convolution integrals, which is an isometric isomorphism from χ + L 2 onto H + 2 . We have Thus we can reduce the study of the invertibility of S R + − λI R + to the study of the invertibility of the Toeplitz operator where sign (x) = χ + (x) − χ − (x), since, for all f + ∈ H + 2 .