Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators

Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on $L^2(\mathbb T) \oplus L^2(\mathbb T)$ and, if their symbols are invertible in $L^\infty(\mathbb T)$, to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson's corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operators.


Introduction
Toeplitz operators have been for a long time one of the most studied classes of nonselfadjoint operators ( [3]). They are defined as compressions of multiplication operators on L 2 (T), to the Hardy space of the unit disk H 2 (D). Dual Toeplitz operators are analogously defined on the orthogonal complement of H 2 (D), identified as usual with a subspace of L 2 (T), as multiplication operators followed by projection onto L 2 (T) ⊖ H 2 (D). Although they differ in various ways from Toeplitz operators, they also share many properties, which is not surprising given that they are anti-unitarily equivalent. The algebraic and spectral properties of dual Toeplitz operators, and the extent to which their properties are parallel to those of Toeplitz operators on H 2 (D), were studied in [20].
Truncated Toelitz operators, defined as compressions of multiplication operators to closed subspaces of H 2 (D) which are invariant for the backward shift S * , called model spaces, have also generated great interest, partly motivated by Sarason's paper [16]. Their study, as well as that of asymmetric truncated Toeplitz operators later introduced in [7], raised many interesting questions and has led to new and sometimes surprising results, see for example [2,5,7,9]. It is natural to consider dual truncated Toeplitz operators, defined analogously as compressions of multiplication operators to the orthogonal complement of a model space in L 2 (T). These operators were very recently introduced and studied in [14,15,12]. It turns out that, in this case, they behave very differently from truncated Toeplitz operators. For instance, the symbol of a dual truncated Toeplitz operator is unique and the only compact operator of that kind is the zero operator, in sharp contrast with what happens with truncated Toeplitz operators on model spaces.
In this paper we study the kernels and various spectral properties, such as Fredholmness and invertibility, of dual truncated Toeplitz operators. The results are applied to describe the spectra of dual truncated Toeplitz operators in several classes including, as particular cases, the dual truncated shift and its adjoint. We do this by using a novel approach to dual truncated Toeplitz operators and their asymmetric analogues, defined similarly between the orthogonal complements of two possibly different model spaces. This involves proving their equivalence after extension to paired operators in L 2 (T) ⊕ L 2 (T), defined in Section 2, and establishing connections with the corona theorem. This allows moreover to show that, whenever their symbol is invertible in L ∞ (T), dual truncated Toeplitz operators are in fact equivalent after extension to truncated Toeplitz operators with the inverse symbol.
The paper is organized as follows. In Section 1 we introduce asymmetric dual truncated Toeplitz operators and present some basic properties, while in Section 2 we recall the concepts of paired operator and equivalence after extension between two Banach spaces. In Section 3 we study the solvability of certain equations involving asymmetric dual truncated Toeplitz operators in connection with equations involving paired operators. In Section 5 we show that dual truncated Toeplitz operators are equivalent after extension to truncated Toeplitz operators with the inverse symbol, if the latter is invertible in L ∞ (T). In Section 6 we study the kernels of asymmetric dual truncated Toeplitz operators in terms of explicitly defined isomorphisms with kernels of other operators. We show in particular that the kernels of a dual truncated Toeplitz operator and its adjoint are isomorphic and related by the usual conjugation on a model space. In Section 7 we present sufficient conditions for a dual truncated Toeplitz operator to be injective or invertible in terms of certain corona pairs, i.e., pairs of functions satisfying the hypotheses of Carleson's corona theorem ( [17,11]). We use the previous results to study the Fredholmness, invertibility and spectra of several classes of dual truncated Toeplitz operators.

Elementary properties
Let P + , P − be the orthogonal projections from L 2 onto H 2 and H 2 − =zH 2 , respectively. We have For an inner function θ define the model space K θ = H 2 ⊖ θH 2 and let P θ and Q θ be the orthogonal projections from L 2 = L 2 (T) onto the model space K θ and its orthogonal complement Let α, θ be inner functions and let ϕ ∈ L 2 . An asymmetric truncated Toeplitz operator A θ,α ϕ is defined by [7,5]), whereas the asymmetric dual truncated Toeplitz operator D θ,α ϕ is defined by have a bounded extension to K θ or (K θ ) ⊥ , respectively, we denote them also by A θ,α ϕ and D θ,α ϕ , respectively. When α = θ instead of A θ,θ ϕ and D θ,θ ϕ we write A θ ϕ and D θ ϕ , respectively. We start with some elementary properties of asymmetric dual truncated Toeplitz operators. These properties were proved in [14] for α = θ.
Proof. Assume that D θ,α ϕ is compact and let f n ∈ (K θ ) ⊥ , with f n weakly convergent to 0 (f n ⇀ 0). Then D θ,α ϕ f n → 0. Note that for f n = θf n withf n ∈ H 2 , we have f n ⇀ 0 (in θH 2 ) if and only iff n ⇀ 0 (in H 2 ). It follows that iff n ⇀ 0, then D θ,α ϕ (θf n ) → 0. Since Tᾱ θϕfn D θ,α ϕ (θf n ) (see the proof of Proposition 1.1), we have that Tᾱ θϕfn → 0 wheneverf n ⇀ 0. So, if D θ,α ϕ is compact, then Tᾱ θϕ is also compact and therefore ϕ = 0. Remark 1.3. Proposition 1.2 implies, in particular, that the only symbol for the zero asymmetric dual truncated Toeplitz operator is ϕ = 0. Since the question of D θ,α ϕ being the zero operator is equivalent to ϕ being a multiplier from (K θ ) ⊥ into K α , we conclude also that there are no non-trivial L ∞ -multipliers from (K θ ) ⊥ into K α . In contrast with this, the question of whether there are non-trivial multipliers from K α into (K θ ) ⊥ , which is equivalent to A α,θ ϕ being the zero operator, has a positive answer ( [7,5]).

Paired operators and equivalence after extension
For a Banach space X denote by L(X) the space of all bounded linear operators A : X → X. Let P ∈ L(X) be a projection and let Q = I − P be its complementary projection. An operator of the form AP + BQ or P A + QB, where A, B ∈ L(X), is called a paired operator ( [13]).
Paired operators are closely connected with operators of the form P CP | Im P and QCQ| Im Q , where C ∈ L(X), which are called general Wiener-Hopf operators or operators of Wiener-Hopf type ( [13,18]). To understand this relation, it will be useful to introduce here the concept of equivalence after extension for operators. Definition 2.1 (Equivalence after extension, [1]). Let X,X, Y ,Ỹ be Banach spaces and let us use the term operator to mean a bounded linear operator.
The operators T : X →X and S : Y →Ỹ are said to be (algebraically and topologically) equivalent if and only if T = ESF where E and F are invertible operators; in that case we use the notation T ∼ S.
The operators T and S are equivalent after extension (T ⋆ ∼ S) if and only if there exist (possibly trivial) Banach spaces X 0 , Y 0 , called extension spaces, and invertible operators E : Operators that are equivalent after extension share many properties. In particular we have the following.
. Let T and S be two operators, T : X →X, S : Y →Ỹ , and assume that T ⋆ ∼ S. Then (1) ker T is isomorphic to ker S, i.e., ker T ≃ ker S; (2) Im T is closed if and only if Im S is closed and, in that case,

18]), then the other is generalized (left, right) invertible too; (4) T is Fredholm if and only if S is Fredholm and, in that case,
dim ker T = dim ker S, codim Im T = codim Im S.
It is not difficult to see that P CP | Im P (respectively, QCQ| Im Q ) is equivalent after extension to CP + Q (respectively, P + CQ) and (and analogously P + CQ ∼ P + QC ∼ P + QCQ).
As an example of two operators which are equivalent after extension we have the following.
Equivalence after extension for two operators T and S implies that there is a strong connection between the solvability of the equations T ϕ = ψ and Sx = y, in particular as regards the existence and uniqueness of solutions.
In the next section we study the relations between the solutions of the equations as a first step to establishing the equivalence after extension of D θ,α ϕ to a paired operator

Solvability relations
Let ϕ ∈ L ∞ and let α, θ be inner functions. We define This last system of equations can be written in matrix form as .
Now the result for Φ follows from the fact that can be written as a system of equations which is equivalent to Moreover, the above is equivalent to The first equation in the system above implies that and the second equation gives The relations in Theorem 3.1 imply that D θ,α ϕ and AP + + BP − share many properties. Indeed, in the next section we show that the former is equivalent after extension to the latter. 4. Equivalence after extension of D θ,α ϕ to a paired operator Let us introduce some notations (see [6]). If H, K 1 , K 2 are Hilbert spaces and A 1 : In what follows, α and θ are inner functions, ϕ ∈ L ∞ and as at the beginning of Section 3.
Proposition 4.1. Let ϕ ∈ L ∞ and let α, θ be inner functions. Then Proof. An easy computation shows that Clearly, F 1 and E 1 are invertible.
Proof. Given the result of Proposition 4.1 and the fact that ⋆ ∼ is an equivalence relation and thus transitive, we only have to prove that To this end, we note that Q α ϕQ θ ⊞ P α is obviously equivalent after extension to Q α ϕQ θ ⊞ P α 0 0 Using the relations from Section 3 and rewriting them appropriately, we get (as can be verified independently) , The operators F and E are invertible by Lemmas 4.3 and 4.4 below.
The proofs of the following two lemmas are straightforward. , In what follows GL ∞ will denote the set of all invertible elements of the algebra L ∞ . Proof. The equivalence is a consequence of Theorem 2.2 and Theorem 4.2. To prove that ϕ is invertible note that | det A| = | det B| = |ϕ|. Since a necessary condition for the operator AP + + BP − to be semi-Fedholm is that In particular, we conclude that if D θ ϕ is invertible, then ϕ ∈ GL ∞ ([14, Proposition 2.4]) and then, denoting by ϕ(T) the essential range of ϕ ∈ L ∞ , we have [15,Theorem 4.1]).

Equivalence relations between D θ,α ϕ and truncated Toeplitz operators
In view of Corollary 4.6, the case where ϕ is an invertible element of L ∞ becomes particularly interesting. Assume then that This implies that, for A, B defined by (4.1), we have A, B ∈ G(L ∞ ) 2×2 with In that case where B (identified with multiplication by B on L 2 ⊕ L 2 ) is invertible, as well as I + P − B −1 AP + because It follows from Theorem 4.2 that D θ,α ϕ is equivalent after extension to P + CP + + P − . It is easy to see that P + CP + + P − ∼ P + GP + + P − , where G = 0 1 1 0 C 0 1 −1 0 = ᾱ 0 ϕ −1 θ (since G and C differ by constant factors). Therefore we have that where we took Theorem 2.3 into account. We have thus proved the following: Conversely, if ϕ ∈ GL ∞ and A θ ϕ −1 is Fredholm, then Corollary 5.2 implies that D θ ϕ is also Fredholm. The proof for invertibility is analogous.

Kernel Isomorphisms
By Theorem 2.2, the kernels of two operators that are equivalent after extension, are isomorphic. Using the relations from Section 2, we describe here several of those isomorphisms. We use the same notation as in Section 4.
A conjugation on a Hilbert space H is an antilinear isometric involution (see for instance [10]). In what follows let C θ : L 2 → L 2 denote the conjugation defined as The conjugation C θ preserves both the model space K θ and its orthogonal complement (K θ ) ⊥ , (i.e., C θ P θ = P θ C θ ), and therefore induces a conjugation in K θ and in (K θ ) ⊥ , which we also denote by C θ . This conjugation plays an important role in the study of truncated Toeplitz operators.
Theorem 6.3. The map The above means that ). Thus N D is an isomorphism and we have if and only if f ∈ (K θ ) ⊥ and ϕf ∈ K α . In other words, g = ϕf ∈ K α and ϕ −1 g = f ∈ (K θ ) ⊥ , that is, g ∈ ker A α,θ ϕ −1 .

Dual truncated Toeplitz operators and the corona theorem
Corona problems, seen as left invertibility problems, have a strong connection with the invertibility and Fredholmness of block Toeplitz operators (see for instance [4] and references in it). In this section we extend some of those connections to paired operators and apply them to the study of injectivity and invertibility of dual truncated Toeplitz operators.
Let CP ± denote the sets of corona pairs, i.e., pairs of functions satisfying the so called corona conditions in D and C \ (D ∪ T), denoted here by D ± , respectively: By the corona theorem h ± ∈ CP ± if and only if there existh ± with with Ah + (t) = 0 a. e. on T, then the operator So, using (7.2), we can write Taking determinants on both sides, we get which, since ϕ ∈ GL ∞ , is equivalent to Moreover, since the left hand side of the above equality represents a function in H 2 while the right hand side represents a function in H 2 − , both sides must be zero. It follows that det h ± Φ ± = 0, so there exist non-zero linear combinations, with δ 1± and δ 2± defined a. e. on T, δ 1± h ± + δ 2± Φ ± = 0. Multiplying on the left by (h ± ) T , by (7.1) we have Therefore there exist δ ± = 0 a. e. on T such that 2 − and δ + = δ − , we have δ ± = 0 and therefore Φ ± = 0.
Proof. If h ± satisfy (7.5), then Therefore, by [4,Theorem 4.5] and the proof given there, we have that the Toeplitz is invertible, only injective, or only surjective if and only if k = 0, k > 0 or k < 0, respectively. We conclude that the same holds for AP + + BP − because (by (2.1)), Taking Theorem 4.2 into account, we also have the following.
Dual truncated Toeplitz operators can also be related to corona problems by using Corollary 5.2 and the known relations between truncated Toeplitz operators and the corona theorem ( [7,8]), as in the following theorem which will be used in the next section to describe the spectrum of a class of dual truncated Toeplitz operators with analytic symbols.
Theorem 7.4. Let θ be an inner function. If ϕ ∈ GL ∞ and there exist Proof. In this case D θ ϕ is invertible if and only if A θ ϕ −1 is invertible, and this is equivalent to the Toeplitz operator T G being invertible. The result now follows from Theorem 3.11 in [8].

Fredholmness, inwertibility and spectra of dual truncated Toeplitz operators
In this section we apply the previous results to study several classes of dual truncated Toeplitz operators. In what follows θ is always an inner function. 8.1. Analytic symbols. We start by using the results of Theorem 7.4 to describe the spectrum of dual truncated Toeplitz operators with analytic symbols of a particular type.
Several other results regarding analytic symbols will be obtained from the properties studied below.
We can obtain further results if ϕ is rational and continuous on T, i.e., ϕ ∈ R. We start by describing the kernels of dual truncated Toeplitz operators with rational symbols.
Theorem 8.10. Let R ∈ R and R = P/Q, where P and Q are polynomials without common zeros. Then f ∈ ker D θ R if and only if there is a decomposition f = f − + θf + , f − ∈ H 2 − ,f + ∈ H 2 such that there are polynomials P 1 , P 2 with deg(P 2 ) max{deg(P ), deg(Q)} − 1 such that Then f ∈ ker D θ R if and only if R(f − + θf + ) ∈ K θ . The latter happens if and only if there exist k + ∈ H 2 and k − ∈ H 2 − such that P Q f − + P Q θf + = k + , θk + = k − , or equivalently, By a generalization of Liouville's theorem, both sides of the first equation in (8.2) must be equal to a polynomial P 1 such that P 1 P ∈ H 2 − and, analogously, both sides of the second equation in (8.2) must be equal to a polynomial P 2 such that P 2 P ∈ H 2 and P 2 = Qk − − Pθf − . So the degree of P 2 is appropriate and k + = P 1 Q + θ P Qf + = P 1 Q + θ P 2 Q ∈ K θ . Conversely, if P 1 and P 2 are polynomials satisfying desired conditions, then for f − = P 1 P ,f + = P 2 P we have that f − ∈ H 2 − ,f + ∈ H 2 and R(f − + θf + ) = P Q P 1 P + θ P 2 P = P 1 Q + θ P 2 Q ∈ K θ , so that f = f − + θf + ∈ ker D θ R .
The previous theorem enables us to characterize the points λ ∈ σ(D θ R ) for R ∈ R, as follows. Theorem 8.11. If R ∈ R, then σ(D θ R ) = R(T) ∪ σ p (D θ R ) = σ e (D θ R ) ∪ σ p (D θ R ). Proof. From Theorem 8.8 we have σ e (D θ R ) = R(T) ⊂ σ(D θ R ). If λ / ∈ R(T), then D θ R−λ is Fredholm, so it is invertible if and only if it is injective by Corollary 6.5. Therefore λ ∈ σ(D θ R ) if and only if ker D θ ϕ−λ = {0}, i.e., λ is an eigenvalue of D θ ϕ . As an application we study the spectra of the dual truncated shift D θ z and its adjoint -the dual truncated backward shift D θ z . We start by studying their kernels. The next result is a consequence of Theorems 8.10 and 8.11.