The invariant subspaces of S ⊕ S*

Abstract Using the tools of Sz.-Nagy–Foias theory of contractions, we describe in detail the invariant subspaces of the operator S ⊕ S*, where S is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.


Introduction
The recent series of papers [2][3][4] explore the class of so-called dual truncated Toeplitz operators, which act on a subspace of the usual Lebesgue space L on the unit circle T. In the preprint [1] Câmara and Ross discuss the invariant subspaces of one of these operators, the dual of the compressed shift. In their investigation they encounter the problem of determining the invariant subspaces of the operator S ⊕ S * , where S is the usual unilateral shift operator, acting as multiplication by the variable on the Hardy-Hilbert space H , and they state it as an open question.
It turns out that the answer can be given through the Sz.Nagy-Foias theory of characteristic functions of contractions on a Hilbert space [6]. That theory includes a general result about the relation between invariant subspaces of a contraction and regular factorizations of its characteristic function. In particular, we may use it in order to obtain an explicit description of all invariant subspaces of S ⊕ S * , giving thus a complete answer to the open question in [1].
The plan of the paper is the following. After some preliminaries, in Section 3 we provide a short presentation of the relevant part of the Sz.Nagy-Foias theory. Section 4 contains the main result, the description of the invariant subspaces. Section 5 provides an example related to [1], while Section 6 details the most interesting class of invariant subspaces.

Preliminaries
We denote shortly L = L (T, dm), where m is Lebesgue measure on the unit circle T. Its subspace H is the Hardy-Hilbert space of functions that can be analytically extended to the unit disk D; then H − = L H . The orthogonal projections in L onto H and H − will be denoted by P+ and P− respectively. The map f →f , with f (z) = f (z) is an involution on H . An inner function θ ∈ H is characterized by |θ(e it )| = for almost all t. If θ is inner function, thenθ is also inner.
We will also use Lebesgue and Hardy spaces de ned on the unit circle with values in a Hilbert space E; they will be denoted with L (E) and H (E) respectively.
If S is the shift operator on H , de ned by (Sf )(z) = zf (z), Beurling's Theorem states that the invariant subspaces of S are { } and the spaces θH with θ inner. We denote K θ = H θH ; so the invariant subspaces for S * are H and K θ for θ inner. The map C θ de ned by C θ f = θzf is a conjugation on K θ ; in particular, (2.1) It will be convenient in the sequel to consider, rather than S * , the operator S * , acting on H − as the compression of multiplication by z to H − . This is unitarily equivalent to S * , and the precise unitary operator that implements this equivalence is J : Our purpose in this paper will be the determination of the invariant subspaces of S ⊕ S * ; we will see below that this is a model operator in the sense of Sz.Nagy and Foias. The invariant subspaces of S ⊕ S * are then immediately obtained by applying the operator J.
Some of these invariant subspaces of S⊕S * may easily be described; namely, the subspaces X ⊕ X , where X ⊂ H is invariant to S, while X ⊂ H − is invariant to S * . We will call them splitting invariant subspaces. The next lemma summarizes the above remarks. One may say that these are the obvious invariant subspaces of S ⊕ S * . There is, however, a large variety of nonsplitting invariant subspaces, for whose determination we will have to bring into play the Sz.-Nagy-Foias theory of contractions [6].
We end the preliminaries with a lemma that will be helpful.

Sz.Nagy-Foias theory of contractions and invariant subspaces
The general reference for this section is the monograph [6]. Suppose Θ : D → L(E, E * ) is an analytic function in the unit disc D with values in the algebra of bounded operators from E to E * , with Θ(z) ≤ for all z ∈ D; we will call it a contractive analytic function. Θ has boundary values almost everywhere on T, that will be denoted by Θ(e it ). A contractive analytic function is called pure if Θ( )x < x for any x ∈ E. Any contractive analytic function admits a decomposition in a direct sum Θ = Θp ⊕ Θu, where Θp is pure and Θu is a constant unitary operator; then Θp is called the pure part of Θ.
To a pure contractive analytic function corresponds a functional model, de ned as follows. Denote ∆(e it ) = (I − Θ(e it ) * Θ(e it )) / . Then the model space is If H is a Hilbert space, a completely nonunitary contraction T ∈ L(H) is a linear operator that satis es T ≤ , and there is no reducing subspace of T on which it is unitary. The defect of T is the operator D T = (I − T * T) / , and the defect space is D T = D T H. It is shown in [6] that any completely nonunitary contraction T is unitarily equivalent to S Θ T , where Θ T is the pure contractive analytic function with values in L(D T , D T * ) de ned by The invariant subspaces of S Θ are in correspondence with the regular factorizations of Θ, that we will de ne in the sequel. Suppose F is a third Hilbert space and Θ : is isometric, and may thus be completed to an isometry The relation between invariant subspaces and regular factorization is summed up in the next theorem, which follows from [6, Theorem VII.1.1], [6,Theorem VII.4.3], and the remark following it.
The characteristic function of S Θ |Y is the pure part of Θ . Conversely, any invariant subspace Y determines a regular factorization Θ = Θ Θ , such that Y is given by (3.2).
In general, the main di culty in the application of Theorem 3.1 is the identi cation of the regular factorizations of a given contractive analytic function. Fortunately, this can be done explicitely in the case that interests us.
Let us denote by m→n the zero contractive analytic function considered as acting from C m to C n . The functional model associated to Θ = → is the space

The invariant subspaces
We may now use the information provided by Corollary 3.2 and Lemma 3.3 in order to obtain the desired description of invariant subspaces. The next theorem is the main result of the paper.

Case (1)
We have dim F = , so ∆ = , and ∆ = I L , so Z : The invariant subspace Y is the second component (the space on which acts S * ); it is obviously splitting.

Case (2.1)
Here dim F = , ∆ = I L , ∆ = . Z is the same operator as in the previous case. We have

Case (3)
Here we have dim F = . From (3.5) it follows that Θ * Θ = Θ Θ * = I C a.e., so ∆ = , while ∆ is a projection a.e.; that is, ∆ = ∆ . Also, Θ unitary a.e. implies that ΘΘ * = I C a.e, which is equivalent to It follows that Z : Denoting u ∈ H (C ) by u = u u , we have, according to (3.3), We have again to discuss two cases.

Case (3.2)
The last case appears when θ and θ are not proportional, which leads us precisely to the invariant subspaces of type (II). To nish the proof, we have to show that these do not split. We already know the subspaces that split, so we must show that subspaces of type (II) do not coincide with any of them. By Corollary 3.2 (iii) any other factorization producing the same invariant subspace must be of type (3), with the associated invariant subspace given by (4.4). So we must have two inner functions θ, θ and a × unitary matrix a a a a such that θ θ It follows that θ = a θ, θ = a θ. So θ and θ are proportional, contrary to the assumption. The proof is thus nished.
The nontrivial case is the latter; then

An example
The following example exhibits a whole class of nonsplitting invariant subspaces, for which we may obtain a simpler form than that given by Theorem 4.1. By Theorem 4.1, we obtain the following nonsplitting subspace: There is a simpler way to write this subspace. First, P−u = . Secondly, K ba is a one dimensional space generated by the reproducing kernel ka(z) = −āz , and the orthogonal projection onto K ba has the formula If we denote u = ( − |α| ) / u + αba u , then u(a) = ( − |α| ) / u (a); moreover, if u , u are arbitrary functions in H , then u is also an arbitrary function in H . We may therefore write It is easy to see that when α ∈ D \ { },ᾱ ( −|a| ) ( −|α| ) / covers C \ { }. Let us denote β =ᾱ ( −|a| ) ( −|α| ) / ; the invariant subspace is then

Parametrization of nonsplitting subspaces
The nonsplitting subspaces are the most interesting ones, so it is worth to obtain a more detailed description of this class. Equations 3.5 de ne Θ and Θ in an implicit manner; we will determine in this last section a parametrization of these two functions.
We start with a pair of nonproportional functions θ , θ ∈ H ∞ that satisfy |θ | + |θ | = . First, if we denote by g , g the outer parts of θ , θ respectively, they satisfy |g | + |g | = . In fact, this means an outer function g bounded by 1 and subject to the condition ( − |g | ) > −∞, which is equivalent to g not being an extreme point of the unit ball of H ∞ (see [5]). Then g determines g up to a scalar of modulus 1.
Divisibility implies then that β = λβ and β = −λβ for some λ ∈ C, |λ| = . If we denote, for simplicity, β = β and β = β , we may write So the nonsplitting invariant subspace Y is determined by the following "free" objects: (i) An outer function g bounded by 1 that is not an extreme point of the unit ball of H ∞ . (ii) Two arbitrary inner functions α , α . (iii) Two arbitrary, but coprime inner functions β , β . (iv) A complex number λ of modulus 1. To obtain from these parameters θ ij , note rst that g determines up to a constant of modulus 1 an outer function g , such that |g | + |g | = . Then we have θ = α β g , θ = α β g , θ = −λα β g , θ = λα β g . (6.1) The condition for two parametrizations to produce the same invariant subspace follows from the last statement of Theorem 4.1. One sees that there is a remarkable richness of nonsplitting subspaces.