Nearly invariant subspaces and applications to truncated Toeplitz operators

In this paper we first study the structure of vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We discover that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This has far reaching applications, in particular we show that there is an all encompassing approach to the study of the kernels of many variations of the truncated Toeplitz operator.


Introduction
The purpose of this paper is to study vector and scalar-valued nearly S * -invariant subspaces of the Hardy space defined on the unit disc. We first produce some results on the structure of nearly S * -invariant subspaces with a finite defect, in particular we produce a powerful tool which allows us to relate the vector-valued nearly S * -invariant subspaces to scalar-valued nearly S * -invariant subspaces with a finite defect. These results then allow us to adopt a previously unknown universal approach to the study of the kernel of: the Toeplitz operator, the truncated Toeplitz operator, the dual truncated Toeplitz operator and the truncated Toeplitz operator on the multiband space (all to be defined later).
We denote T to be the unit circle and D to be the open unit disc. The vectorvalued Hardy space is denoted H 2 (D, C n ) and is the Hilbert space defined to be a column vector of length n with each coordinate taking values in H 2 ; background theory on the classical Hardy space H 2 can be found in [16,12]. The backwards shift on the space H 2 (D, C n ) is defined by . . .
If we denote H 2 0 = {f : f ∈ H 2 , f (0) = 0}, then it is readily checked that H 2 0 is the orthogonal complement of H 2 in L 2 (T). Then in the scalar case (i.e. when n = 1) using Beurling's Theorem one can then deduce that all non-trivial S * -invariant subspaces are of the form K θ = θH 2 0 ∩ H 2 for some inner function θ. We call K θ a model space and further information on model spaces can be found in [7]. One can further check that for distinct λ i ∈ D if θ = i z−λ i 1−λ i z , then K θ is the span of Cauchy kernels k λ i (z) = ∞ n=0 (λ i z) n . The Cauchy kernel k λ i is the eigenvector of the backwards shift with eigenvalue λ i . The concept of (scalar) nearly backward shift invariant subspaces was first introduced by Hitt in [15] as a generalisation to Hayashi's results concerning Toeplitz kernels in [14]. These spaces were then studied further by Sarason [18]. The study of nearly backwards shift invariant subspaces was then generalised to the vectorial case in [5], and generalised to include a finite defect in [6]. Kernels of Toeplitz operators are the prototypical example of nearly S * -invariant subspaces. Truncated Toeplitz operators were introduced in [19], and over the past decade there have been many further publications studying their properties. Although truncated Toeplitz operators share many properties with the classical Toeplitz operator it is easily checked that the kernel of a truncated Toeplitz operator is not nearly S * -invariant. This motivates our study for section 2 where we show under certain conditions the kernel of a truncated Toeplitz operator is in fact nearly S * -invariant with defect 1. In many cases the study of Toeplitz operators becomes greatly simplified when the operator has an invertible symbol; in section 2 we also show that the symbol of a truncated Toeplitz operator may be chosen to be invertible in L ∞ .
In section 3 we prove a powerful result that shows for any i ∈ {1 . . . n} the first i coordinates of a vector-valued nearly S * -invariant subspace is a nearly S * -invariant subspace with a finite defect. We then generalise Theorem 3.2 in [5] and Corollary 4.5 in [6] to find a Hitt-style decomposition for the vector-valued nearly S * -invariant subspaces with a finite defect.
In section 4 we show that in all cases the kernel of a truncated Toeplitz operator is a nearly S * -invariant subspace with defect 1; this then allows us to decompose the kernel in to an isometric image of a model space. The approach of decomposing a kernel in to an isometric image of a model space much resembles the works of Hayashi [14] and Hitt [15] for the classical Toeplitz operator. We also make the observation that we can decompose the kernel of a truncated Toeplitz operator in to a nearly S * -invariant subspace multiplied by a power of z (where z ∈ D is the independent variable). Then using the results of [15], this observation also gives us a second method to decompose the kernel in to a isometric image of a model space. Furthermore we show that in general our two choices of decomposition of the kernel of a truncated Toeplitz operator yield different results.
In section 5 we study the kernel of dual truncated Toeplitz operator. Dual truncated Toeplitz operators have been studied in both [11,9] as well as many other sources. The kernel of a dual truncated Toeplitz operator has been studied in [8]. Although the domain of the dual truncated Toeplitz operator is not a subspace of H 2 we still can use similar recursive techniques used in previous sections to decompose the the kernel in to a fixed function multiplied by a S * -invariant subspace.
In section 6 we study the truncated Toeplitz operator on the multiband space. We show every truncated Toeplitz operator on a multiband space is unitarily equivalent to an operator which has kernel nearly S * -invariant with defect 2. This allows us to apply our previously developed theory to give a decomposition for the kernel of the truncated Toeplitz operator on a multiband space in terms of S * -invariant subspaces.

Notations and convention
• From section 3 onward we assume the symbol of any Toeplitz operator (denoted g) is bounded and hence the Toeplitz operator is bounded.
• Throughout we let θ be an arbitrary inner function.
• We use the notation f i /f o to denote the inner/outer factor of f .
• GCD stands for greatest common divisor, and the greatest common divisor of two inner functions is always taken to be an inner function.
• All limits are taken in the H 2 (D, C n ) sense unless otherwise stated.
• All subspaces of H 2 (D, C n ) are assumed closed unless otherwise stated.

Preliminary results
Using orthogonal decomposition we can write L 2 = H 2 0 ⊕ K θ ⊕ θH 2 . We define P θ : L 2 → K θ to be the orthogonal projection. The truncated Toeplitz operator A θ g : K θ → K θ having symbol g ∈ L 2 is the densely defined operator Theorem 2.1. For any g ∈ L 2 we write g = g − +g + where g − ∈ H 2 0 and g + ∈ H 2 . If g − is not cyclic for the backwards shift then there exists ag ∈ L 2 such that Proof. Theorem 3.1 of [19] shows that A θ g 1 = A θ g 2 if and only if g 1 − g 2 ∈ θH 2 + θH 2 , so we may initially assume without loss of generality that g ∈ K θ ⊕ K θ . Using Lemma 2.1 in [17] we can construct an outer function u such that |u|= 2|g|+1, furthermore u ∈ L 2 so u ∈ H 2 . Then it follows that for any inner function α has the property that |g − αu| |u|−|g|> |g|+1 > 0 almost everywhere on T, and so (g − αu) −1 ∈ L ∞ . Our construction of u shows | 1 u | 1 and as the reciprocal of an outer function in is outer, we have 1 u is outer and in L ∞ , so 1 u ∈ H ∞ . Furthermore by Corollary 4.9 in [17] we can say 1 u ∈ H 2 is non-cyclic for S * and hence must lie in a model space K Φ . Defineg := (g − Φθu), then as previously statedg −1 ∈ L ∞ . We now showg −1 = ∞ k=0 (−1)g k (Φθ 1 u ) k+1 where the limit is taken in the sense of uniform convergence. We writeg −1 N to be By our construction of u this is less than ||g −1 || ∞ ( 1 2 ) N , which clearly converges to 0. Now our choice of Φ ensures that Φ 1 u ∈ H ∞ , we also have θg ∈ H 2 . This means (−1)g k (Φθ 1 u ) k+1 ∈ H 2 and is bounded by 1 so must actually lie in H ∞ , sog −1 (being the uniform limit of a sequence in H ∞ ) must also be in H ∞ .
Examining the first part of the above proof we can also deduce the following proposition.
Proof. In (1) if we set α to equal θ, keep our construction of u the same and definẽ This has an interesting relation to Sarason's question posed in [19]; which is whether every bounded truncated Toeplitz operator has a bounded symbol. This was first shown to have a negative answer as Theorem 5.3 in [2], and further results in [1] characterise the inner functions θ which have the property that every bounded truncated Toeplitz operator on K θ has a bounded symbol.
These results suggest that under certain circumstances ker A θ g may be a nearly invariant subspace with a finite defect. This is because f ∈ ker A θ g if and only if f ∈ K θ and gf ∈ H 2 0 ⊕ θH 2 , so if f (0) = 0 and f ∈ ker A θ g then we must have This may lead us to believe that ker A θ g is a nearly S * -invariant subspace with a defect given by g −1 span{S * (θ)}, but the issue here is g −1 S * (θ) need not necessarily lie in K θ or even H 2 . Theorem 2.1 shows us that under some weak restrictions we can choose our non-unique symbol g so that g −1 S * (θ) ∈ H 2 , but to fully understand ker A θ g as a nearly invariant subspace with a defect we must study vector-valued nearly invariant subspaces with a defect.

Vector-valued nearly invariant subspaces with a defect
Let M ⊆ H 2 (D, C n ) be a nearly invariant subspace for the backwards shift with a finite defect space D and let dim D = m. If not all functions in M vanish at 0 then we define W := M ⊖ (M ∩ zH 2 (D, C n )) and Corollary 4.3 in [5] shows that r := dim W n, in this case we let W 1 . . . W r be an orthonormal basis of W . For i = 1 . . . n we let P i : H 2 (D, C n ) → H 2 (D, C i ) be the projection on to the first i coordinates.
Proof. We first consider the case when not all functions in M vanish at 0. Let In the case when all functions in M vanish at 0 then W = {0} and we would just to be the zero vector.
To further build on this result we will now give a Hitt style decomposition for a vector-valued nearly invariant subspace with a finite defect. This style of decomposition was first introduced by Hitt in [15] when he decomposed the nearly S * -invariant subspaces. This was then generalised to the vectorial case as Corollary 4.5 in [5]. This style of proof was then adapted to produce a similar result for the (scalar) nearly invariant subspace with a defect, which is Theorem 3.2 in [6].
For a Hilbert space H and x, y ∈ H we define x ⊗ y(f ) = f, y x. We say an operator T on H belongs to the class C .0 if for all x ∈ H, lim n→∞ ||(T * ) n x||= 0. Consider a subspace M which is nearly S * -invariant with defect 1, so that D = span{e 1 }, say, where ||e 1 ||= 1.
Suppose first that not all functions in M vanish at 0, then 1 r = dim W n. Let F 0 be the matrix with columns W 1 . . . W r , and let P W be the orthogonal projection on to W . For each F ∈ M we may write ..r forms an orthonormal basis of W , we obtain the following identity of norms: We may now repeat this process on G 1 to obtain We iterate this process to obtain We now argue ||G n ||→ 0 as n → ∞. We can write G n = P 1 S * P 2 (G n−1 ), where P 1 is the projection with kernel e 1 and P 2 is the projection with kernel As e 1 is orthogonal to W we have and so the adjoint of R is We now apply the second assertion of Proposition 2.1 from [5] to show the adjoint of R is of class C .0 , and so R n−1 applied to S * P 2 (G 1 ) converges to 0; now from (3) we see ||G n+1 ||→ 0. As a consequence taking limits in (2) we may write We denote a n (z) = F 0 (z) (A 0 + A 1 z + . . . A n−1 z n−1 ), and a 0 (z) = F 0 ∞ k=0 A k z k , where ∞ k=0 A k z k is taken in the H 2 (D, C n ) sense (this is defined by the equality of norms given immediately after (2)). Then in the H 1 (D, C n ) norm we must have For each i ∈ {1 . . . n} we define C i to equal the maximum H 2 norm of each coordinate of W i multiplied by n, then we apply Hölder's inequality on each coordinate to obtain Thus in the H 1 (D, C n ) norm we have a n → a 0 , a similar computation shows (β 1 z + . . . + β n z n ) e 1 (z) converges to ( ∞ k=1 β k z k )e 1 in the H 1 (D, C n ) norm, so the H 1 (D, C n ) limit of and furthermore by taking limits in the equality of norms immediately after (2) we know We may alternatively express this as saying F ∈ M if and only if where (k 0 , k 1 ) lies in a subspace K ⊆ H 2 (D, C r ) × H 2 which is identified with H 2 (D, C r+1 ). By virtue of (4) we can see that K is the image of a isometric mapping, and hence closed. We now argue K is invariant under the backwards shift (on H 2 (D, C r+1 )). Since in the algorithm we have k 0 (0) = A 0 and k 1 (0) = β 1 we can write F as is a closed subspace of H 2 (D, C n ), where K is a S * -invariant subspace of H 2 (D, C r+1 ), then M is nearly S * -invariant with defect 1. To show this we first need a lemma. Proof. If W k (0) = i =k λ i W i (0) this would mean W k − i =k λ i W i vanishes at 0 and therefore lies in zH 2 (D, C n ).
If F ∈ M and F (0) = 0 then we must have F 0 (0)k 0 (0) is equal to the zero vector. We now add n − r vectors X 1 . . . X n−r which are linearly independent from W 1 (0), . . . W r (0) as extra columns to the matrix F 0 (0) to obtain a matrix F ′ 0 (0) = [W 1 , . . . W r , X 1 , . . . , X n−r ]. We now add n − r extra 0's to the end of the column vector k 0 (0) and label this k ′ 0 (0). As F 0 (0)k 0 (0) is equal to the zero vector, then F ′ 0 (0)k ′ 0 (0) must also be equal to the zero vector. We can now invert F ′ 0 (0) to obtain k ′ 0 (0) is equal to the zero vector and hence k 0 (0) must be zero. This allows us to write and as K is S * -invariant this is clearly an element of M ⊕ span{e 1 }.
If all functions in M vanish at 0 then there is no non-trivial reproducing kernel at 0, but we may now write with G 1 ∈ M and β 1 ∈ C, and furthermore We can then iterate on G 1 as we have previously done to obtain F (z) = β 1 ze 1 + β 2 z 2 e 1 + . . . .

Application to truncated Toeplitz operators
Throughout this section our symbol g is bounded and so the truncated Toeplitz operator A θ g : K θ → K θ is defined by where P θ is the orthogonal projection L 2 → K θ . It was originally observed in [3] that the kernel of a truncated Toeplitz operator is the first coordinate of the kernel of the matricial Toeplitz operator with symbol Scalar-type Toeplitz kernels (first introduced in [10]) are vector-valued Toeplitz kernels which can be expressed as the product of a space of scalar functions by a fixed vector function. A maximal function for ker T G is an element f ∈ ker T G such that if f ∈ ker T H for any other bounded matricial symbol H, then ker T G ⊆ ker T H . By Corollary 3.9 in [10] ker T G is of scalar type, it is also easily checked that ker T G is not shift invariant and so by Theorem 3.7 in [10] we must have that ker T G has a maximal function. Now by Theorem 3.10 of [17] we can deduce that W = ker T G ⊖ (ker T G ∩ zH 2 (D, C n )) has dimension 1. If we denote w 1 w 2 to be the normalised element of W then using Corollary 4.5 from [5] we can write where Φ is an inner function. We now can write We can describe Φ with the following proposition.
Proof. We first show that up to multiplication by a unitary constant there can only be one inner function Φ satisfying where GCD(p i 1 , p i 2 ) = 1. Suppose there are two inner functions Φ 1 , Φ 2 such that where both GCD(p i 1 , p i 2 ) = 1 and GCD(q i 1 , q i 2 ) = 1. This would then imply that and so Φ 1 divides Φ 2 . A similar computation shows Φ 2 divides Φ 1 , and so we must have Φ 1 is a unitary constant multiple of Φ 2 . We now show that Φ is such that If it is the case that α = GCD(p i 1 , p i 2 ) = 1 then it would follow that w 1 w 2 Φα ∈ ker T G , which would be a contradiction as Φα / ∈ K zΦ .
It is easily checked that ker T G is nearly S * -invariant, and because ker A θ g = P 1 (ker T G ) we can use Corollary 3.2 to deduce the kernel of a truncated Toeplitz operator is nearly S * -invariant with a defect given by span{ w 1 z } ∩ H 2 . With this information we can use the following result given as Theorem 3.2 in [6] (or equivalently Theorem 3.4 with n = 1) to study ker A θ g . Conversely if a closed subspace M ⊆ H 2 has a representation as in 1 or 2, the it is a nearly S * -invariant subspace with defect m.
To use the desired result we have to assume that our defect space is orthogonal to ker A θ g ; we consider two separate cases. We first assume that all functions in ker A θ g vanish at 0. We set O := ker A θ g + span{ w 1 z }, E := O ⊖ ker A θ g , we let e be P E ( w 1 z ) and then e is orthogonal to ker A θ g . In this construction e = 0 as this would imply w 1 z ∈ ker A θ g = w 1 K zΦ which is clearly a contradiction. Theorem 4.2 now yields where multiplication by ez is an isometry from K Ψ to ker A θ g . This expression for ker A θ g is more familiar than w 1 K zΦ as in this case the multiplication is an isometry as opposed to a contraction. We can also relate this expression to nearly S * -invariant subspaces. If we let n be the greatest natural number such that e z n ∈ H 2 then ker A θ g z n+1 = e z n K zΨ , now e z n (0) = 0 so ker A θ g z n+1 = e z n K zΨ is a nearly S * -invariant subspace. We can conclude the following theorem in this case. We now turn our attention to the case when not all functions in ker A θ g vanish at 0. In this case it must also follow that w 1 (0) = 0 as otherwise w 1 K zΦ (0) = 0, so using Corollary 3.2 we must have the defect space for ker A θ g = 0 so can conclude the following theorem. When ker A θ g is nearly S * -invariant we may proceed by using Proposition 3 of the paper of Hitt [15] to show ker A θ g = uK zψ where u ∈ ker A θ g ⊖ (ker A θ g ∩ zH 2 ) is an isometric multiplier. As was noted in [13] we can call ψ the associated inner function to u, and it is easily checked (similar to the approach in Proposition 4.1) this is an inner function such that guψ = zp 1 + θp 2 where p 1 is outer.
In fact using (7) we can view these two theorems as specialisations of the following theorem.
Theorem 4.5. If f ∈ H 2 , and I is an inner function such that f K I is a closed subspace of H 2 , then if f (0) = 0 then f K I is a nearly invariant subspace. If f (0) = 0 then f K I is both a nearly invariant subspace multiplied by a power of z and a nearly invariant subspace with a 1-dimensional defect space f z (K I ⊖ (K I ∩ zH 2 )). Proof. The only non-trivial statement to prove is if f (0) = 0 then f K I is a nearly invariant subspace with a defect space f z (K I ⊖ (K I ∩ zH 2 )), but this follows from So under the assumptions f ∈ H 2 and I is an inner function such that f K I is a closed subspace of H 2 , if f (0) = 0 then Theorem 4.5 gives us two possible approaches to decomposing f K I .
1. Divide f K I by z n where n ∈ N is chosen such that f z n (0) = 0, then use the Hitt decomposition given in [15]. Then we could write f K I as z n u multiplied by some model space, where u ∈ f K I z n ⊖ ( f K I z n ∩ zH 2 ) .
2. Use Theorem 3.2 in [6] with f z (K I ⊖ (K I ∩ zH 2 )) as the defect space. Then we could write f K I as ze multiplied by some model space , where e is chosen to be an element of f z (K I ⊖ (K I ∩ zH 2 )) + f K I orthogonal to f K I . In both of these cases we obtain a model space multiplied by an isometric multiplier.
Due to the similarities in the way these two decompositions are developed one might expect that the two possible ways of decomposing f K I might actually yield the same result. We show this is not the case and in general we have two different expressions with an example. ). So we conclude where multiplication by z 3

Application to dual truncated Toeplitz operators
It is easily checked that in L 2 we have K ⊥ θ = H 2 0 ⊕ θH 2 . We denote Q to be the orthogonal projection Q : L 2 → (K θ ) ⊥ . Throughout this section we assume g ∈ L ∞ . The dual truncated Toeplitz operator D θ g : Theorem 6.6 in [8] shows that for a symbol g that is invertible in L ∞ we have ker D θ g = g −1 ker A θ g −1 , so given our observation (7) under the condition that g is invertible in L ∞ we can write ker D θ g as an L 2 function multiplied by a model space. We now aim to use similar recursive methods that were used to prove Theorem 3.4 to obtain a decomposition theorem for ker D θ g .
Throughout this section we assume that ker D θ g is finite dimensional.
Proof. Suppose we have a non-zero f ∈ ker D θ g ⊆ C, then by construction of C we must have f z ∈ ker D θ g ⊆ C. Iterating this we can obtain f z n ∈ ker D θ g for all n ∈ N, which can't be true as given n sufficiently large gf z n / ∈ H 2 .
Proof. If ker D θ g = {0} then Lemma 5.1 shows that 1 dim(ker D θ g ⊖ C). Let F 1 be the orthogonal projection of gk 0 on to ker D θ g and F 2 be the orthogonal projection of θk 0 on to ker D θ g , where k 0 is the reproducing kernel at 0, then ker D θ g ⊖ C is generated by F 1 , F 2 . Indeed if f ∈ ker D θ g and f is orthogonal to F 1 , F 2 then f, F 1 = gf, k 0 = 0, so f ∈ A, and f, F 2 = θf, k 0 = 0, so we also have P (θf ) ⊆ zH 2 , so f ∈ C.
Consider g ker D θ g = gC ⊕ (g ker D θ g ⊖ gC), by Corollary 5.2 we must have g ker D θ g ⊖ gC is at most 2-dimensional. If g ker D θ g ⊖ gC is 2-dimensional then we denote its orthonormal basis elements by gf 0 , gh 0 . Then for all f ∈ ker D θ g using orthogonal projections and the observation that C z ⊆ ker D θ g we can write where gf 1 ∈ g ker D θ g , and furthermore In a similar process to Theorem 3.4 we can iterate this process starting with gf 1 to obtain Following the argument laid out in section 3 to deduce (3) we can deduce that in the H 2 norm ||gf N +1 ||→ 0 as N → ∞, then ||gf N +1 || must also converge to 0 in the L 1 norm , and so in the L 1 norm we must have Now two applications of Hölder's inequality shows the and furthermore by taking limits in (8) we can deduce Mimicking the argument from section 3 between (5) and (6) we can say f ∈ ker D θ g if and only if where k 0 k 1 lies in a closed S * -invariant subspace of H 2 (D, C 2 ). With obvious modifications for when dim ker D θ g ⊖ C = 1 we can deduce the following theorem. Theorem 5.3.
Cancelling the g and using the same notation as the previous theorem we obtain the following.

Application to truncated Toeplitz operators on multiband spaces
Truncated Toeplitz operators on multiband spaces were introduced in [4] and they are defined (on the unit circle) as follows. Let g ∈ L ∞ , let φ, ψ be unimodular functions in L ∞ such that φK θ ⊥ ψK θ , we define the multiband space M := φK θ ⊕ ψK θ . The truncated Toeplitz operator on M denoted A M g : M → M is defined by where P M is the orthogonal projection on to M. We write K θ (D, C n ) ⊆ H 2 (D, C n ) to mean the vectors of length n with each coordinate taking entries in K θ . To study truncated Toeplitz operators on multiband spaces we first consider the truncated Toeplitz operator A θ G acting on K θ (D, C 2 ), where G = g 11 g 12 g 21 g 22 , has each entry in L ∞ . A result of [4] shows that any truncated Toeplitz operator on a multiband space is unitarily equivalent to A θ G for a certain choice of G, thus we turn our attention to studying ker A θ G . 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. More generally T and S are equivalent after extension if and only if there exists (possibly trivial) Banach spaces X 0 , Y 0 , called extension spaces and invertible linear operators E :Ỹ ⊕ Y 0 →X ⊕ X 0 and F : X ⊕ X 0 → Y ⊕ Y 0 , such that We now consider the case when dim W = 3, in this case W (0) is a 3-dimensional subspace of C 4 . We again have a correspondence [W 1 , W 2 , W 3 ] → [W 1 (0), W 2 (0), W 3 (0)].
In this case we can perform column operations to W 1 (0), W 2 (0), W 3 (0) to obtain a matrix which takes one of the following four forms ( here we denote x 1 , x 2 , x 3 to be some unknown unspecified values in C ),