On extended eigenvalues and extended eigenvectors of truncated shift

We give a complete description of the set of extended eigenvectors of truncated shifts defined on the model spaces K_u := H^2\ominus uH^2, in the case of u is a Blaschke product.


Introduction and preliminaries
Let H be a complex Hilbert space, and denote by L(H) the algebra of all bounded linear operators on H. If T is an operator in L(H), then a complex number λ is an extended eigenvalue of T if there is a nonzero operator X such that T X = λXT . We denote by the symbol σ ext (T ) the set of extended eigenvalues of T . The set of all extended eigenvectors corresponding to λ will be denoted as E ext (λ). Obviously 1 ∈ σ ext (T ) for any operator T . Indeed, one can take X being the identity operator.
Moreover, when H is finite dimensional, in [2] the set of extended eigenvalues has been characterized by the following theorem Proof. First we consider the case when T is not invertible. In this situation both T and T * have nontrivial kernels. Let X ′ be a nonzero operator from kernel of T * to kernel of T . Define X = X ′ P where P denotes the orthogonal projection on kernel of T * . Clearly, X = 0, and T X = 0 = λXT for any λ ∈ C. Consequently, σ ext (T ) = C. On the other hand, since T is not invertible, for any complex number λ, 0 ∈ σ(T ) ∩ σ(λT ). Thus Now assume that T is invertible so that 0 / ∈ σ(T ). In view of (1.1) it suffices to show that {λ ∈ C : σ(T ) ∩ σ(λT )} ⊂ σ ext (T ). So suppose that α is a (necessarily nonzero) complex number such that α ∈ σ(T ) and α ∈ σ(λT ). Since α ∈ σ(T ) there exists a vector a such that T a = αa. On the other hand, α ∈ σ(λT ) implies that λ = 0 so α/λ ∈ σ(T ). Therefore, (α/λ) ∈ σ(T * ) and there is a vector b such that Then T X = λXT and consequently λ ∈ σ(T ).
From this theorem it derives the following consequences (3) σ ext (T ) = C if and only if 0 ∈ σ(T ). Moreover, this assertion remains available in infinite dimensional Hilbert spaces if 0 ∈ σ p (T ) ∩ σ p (T * ).
The next section contains the needed background on the spaces K 2 u .
2. background on K 2 u Nothing in the section is new, and the bulk of it can be found in standard sources, for example [3], [1], [6] and [5].
2.1. Basic notation, model spaces and kernel functions. Let H 2 be the standard Hardy space, the Hilbert space of holomorphic functions in the open unit disk D ⊂ C having square-summable Taylor coefficients at the origin. We let S denote the unilateral shift operator on H 2 . Its adjoint, the backward shift, is given by For the remainder of the paper, u will denote a non-constant inner function. the subspace K 2 u = H 2 ⊖ uH 2 is a proper nontrivial invariant subspace of S * , the most general one by the well-known theorem of A. Beurling. The compression of S to K 2 u will be denoted by S u . Its adjoint, S * u , is the restriction of S * to K 2 u . For λ in D, the kernel function in H 2 for the functional of evaluation at λ will be denoted by k λ ; it is given explicitly by Letting P u denote the orthogonal projection from L 2 onto K 2 u . The kernel function in K 2 u for the functional of evaluation at λ will be denoted by k u λ . It is natural that k u λ equals P u k λ , i.e., In the general case, if B is an infinite Blaschke product defined by then the following Cauchy kernels In particular, if p i = 1 for i in N * , then e i,0 will be denoted by e i , i.e., [5]) the dual set of {e i,l : i ≥ 1, l = 0, ..., p i − 1}, (i.e., the set of kernels verifying where ., . denotes the inner product in L 2 , and δ ij denotes the well-known Kronecker δ−symbol), then we have the following lemma otherwise.
Proof. For the first equality, if l = 0, then Otherwise, For the second equality, it is sufficient to use the first one together with the fact that ., e * i,pi−1 } To complete the proof, we shall show that (S B − α i I) l+1 is injective on span{E * j : j ≥ 1 and j = i}. But the subspaces E * j are invariant of (S B − α i I) l+1 . Thus, it is sufficient to show that (S B −α i I) l+1 is injective on E * j for any j = i. To do so, suppose to the contrary that (S B − α i I) l+1 x = 0 for x ∈ E * j and j = i, then (S B − α i I) l x ∈ span{e * i,pi−1 }, which contradicts the fact that E * j is invariant of (S B − α i I) l .
Biswas and Petrovic determine in [2] the extended spectrum of truncated shift. Our main result, that is Theorem 3. i ⊗ e j ∈ E ext (α i /α j ). It is natural to ask weather this eigenvector is unique or not. The following theorem answers this question affirmatively.
then there are a family of complex numbers {a ij } n i,j=1 such that Since S B is invertible and {E ij := e * i ⊗ e j } n i,j=1 is a Riesz basis for L(K 2 B ), α k α l a kl = α i α j a kl , ∀k, l = 1, ..., n, that is why we have said that this solution is unique. Now, let B be an infinite Blaschke product as in (2.5), and let {γ i } i∈I be the set of limit points of {α i } i≥1 on the circle T = {z ∈ C : |z| = 1}. By (1.1), we have The following theorem shows that this inclusion is proper, more precisely Theorem 3.3. If B is an infinite Blaschke product defined by (2.5), then and for any i, j ≥ 1, we have Proof. Let λ ∈ C and X ∈ L(K 2 B ) be such that then by Lemma 2.1, for all j ≥ 1 we have if l = 0, ..., p j − 2.
So, in this case, X has the same behavior like the e * j,l case, i.e., X = X m,n is a solution of (3.2).
Thus, we have exactly described the solution of (3.2) on a set which spans the space as desired.

Concluding remarks
We finish this paper with some remarks which are summarized in the following. First, it is clear that Theorem 3.1 is a particular case of last theorem, nevertheless we have proved it as a direct result of Theorem 1.1.
In addition, if the set of zeros {α i } i≥1 satisfies the well-known Carleson condition (see [3]), then the set {e * i,l } forms a Riesz basis for K 2 B , and the solution of 3.2 is given in terms of this basis and the dual Riesz basis {e i,l }.
Also, if we suppose that α 0 = 0 is a zero of B, then by using the proof of Theorem 1.1, we have that σ ext (S B ) = C. Indeed, the operator X = e * 0,p0−1 ⊗ e 0,0 satisfies that S B X = 0 = λXS B , for all λ in C.
And finally, as a direct result of (2) in Corollary 1. c k−r (l + r − k)!(n − r − 1)! (l − k)!(n − 1)! e * α,n−r−1 ) ⊗ e α,l−k , l = 0, ..., n − 1}. Lastly, this paper gives a complete description of the set of extended eigenvectors of S u in the case of u is a Blaschke product, and this leads naturally to the following question Problem 1. What is the set of extended eigenvectors of S u in the case of u is a singular inner function?