Compactness and hypercyclicity of co-analytic Toeplitz operators on de Branges-Rovnyak spaces

Abstract We study the compactness and the hypercyclicity of Toeplitz operators Tϕ¯,b {T_{\bar \varphi ,b}} with co-analytic and bounded symbols on de Branges-Rovnyak spaces ℋ(b). For the compactness of Tϕ¯,b {T_{\bar \varphi ,b}} , we will see that the result depends on the boundary spectrum of b. We will prove that there are non trivial compact operators of the form Tϕ¯,b {T_{\bar \varphi ,b}} , with ϕ ∈ H∞ ∩ C(𝕋), if and only if m(σ(b) ∩ 𝕋) = 0. We will also show that, when b is non-extreme, then Tϕ¯,b {T_{\bar \varphi ,b}} is hypercyclic if and only if ϕ is non-constant and ϕ(𝔻) ∩ 𝕋 ≠ ∅.


Introduction
We shall mostly be discussing co-analytic Toeplitz operators Tφ with symbolφ where φ ∈ H ∞ , that are naturally de ned on the de Branges-Rovnyak space into itself. These operators have been introduced by Lotto-Sarason in [13,Lemma 2.6], see also [14,Section II.7]. Some special cases have long ago appeared in literature for φ ∈ L ∞ (T), most notably as standard Toeplitz operators Tφ : H → H studied by A. Brown and P. Halmos in the paper [5] and as the adjoints of truncated Toeplitz operators A Θ φ on model spaces K Θ introduced by Sarason in [15]. We will consider Toeplitz operators with di erent domains and di erent ranges. To avoid confusion, we adopt di erent notations. We will denote by Tφ the Toeplitz operator de ned from H into itself, by Tφ ,b the Toeplitz operator de ned from H(b) into itself, and by Tφ ,b the Toeplitz operator de ned from H(b) into H .
It turns out that de Branges-Rovnyak spaces, which are a family of subspaces H(b) of the Hardy space H , parametrized by elements b of the closed unit ball of H ∞ are invariant under Tφ, where φ ∈ H ∞ . We shall give the precise de nition in section . In general H(b) is not closed in H , but it carries its own norm ||.|| H(b) making it a Hilbert space. The spaces H(b) were introduced by de Branges and Rovnyak in the appendix of [6] and further studied in their book [7].
The general theory of H(b)-spaces generally splits into two cases, according to whether b is an extreme point or a non-extreme point of the unit ball of H ∞ . The dichotomy b extreme/non-extreme will also often appear in this paper. The general idea is that the extreme case has many features that are not far from the case of b = Θ inner (the classical model space K Θ ),while the non-extreme case has several properties that are similar to the case where b = (the Hardy space H ).
This paper treats two properties related to the restricted Toeplitz operators Tφ ,b when φ ∈ H ∞ . One of these properties is based on the particular operator X b = Tz ,b that plays a central role in the theory and particularly in the model developed by de Branges and Rovnyak. Indeed, it serves as a model for a large class of contractions (see [8,Theorem 26.16]).
The rst one concerns the compactness of the Toeplitz operator Tφ ,b , More precisely, given an element b of the closed unit ball of H ∞ , that is extreme or non-extreme, we are interested in nding a necessary and su cient condition for the operator Tφ ,b to be compact. Let us mention that Brown and Halmos [5] have shown that there is no compact Toeplitz operators Tφ : H → H with symbol φ ∈ L ∞ (T) (except the trivial case where φ = ). It is not surprising that in the case where b is non-extreme, we reach a similar result for the restricted Toeplitz operator Tφ ,b with φ ∈ H ∞ .
On the other hand, Ahern and Clark [1] studied the compactness of truncated Toeplitz operators A Θ φ , with continuous symbol φ on T and they got a necessary and su cient condition based on the image of the spectrum of Θ intersected with the unit circle. Recently, Garcia, Ross and Wogen [9] recovered this result with another proof. Using their method, we generalize the result of Ahern and Clark to the operator Tφ ,b with φ ∈ C(T), and from this generalization we prove that there are non trivial compact operators of the form Tφ ,b , Our second problem is related to the hypercyclicity of the restricted Toeplitz operator Tφ ,b . Godefroy and Shapiro, using their Criterion [11], proved that if φ ∈ H ∞ , then Tφ is hypercyclic if and only if φ is non constant and φ(D) ∩ T ≠ ∅. On the other hand, it is obvious that there are no hypercyclic Toeplitz operators with analytic symbols. For general symbols φ ∈ L ∞ (T), few results are known; see a recent paper of A.Baranov and A. Lishanskii [2] and a paper of Shkarin [16] who studied the case where φ(z) = az + b + cz. We succeed to extend the result of Godefroy and Shapiro in our context of H(b) spaces, when b is non extreme. Whereas, when b is extreme, we give a necessary condition for the operator Tφ ,b to be non-hypercyclic. This condition is based on the point spectrum of the operator. It also turns out that this necessary condition is not su cient.
The structure of the paper is the following. After a preliminary section with generalities about de Branges-Rovnyak spaces, Toeplitz operators and de nitions of hypercyclic and frequently hypercyclic operators, we discuss compactness properties of the restricted Toeplitz operators Tφ ,b in Section 3. The last section is dedicated to the hypercyclicity of Tφ ,b .
If φ ∈ L ∞ (T) satis es ||φ||∞ ≤ , then the corresponding Toeplitz operator Tφ is a contraction on the Hilbert space H . The associated de Branges-Rovnyak space H(Tφ) is de ned by H(Tφ) = (I − Tφ Tφ) / H . For simplicity, we denote the complementary space H(Tφ) by H(φ) (see [8,Section 17.3]). Therefore, the de nition of an H(φ)-space uses the defect of the contraction Tφ [8]. Hence, no doubt, the Toeplitz operators are extremely important in this context. Our main concern is when φ is a nonconstant analytic function in the closed unit ball of H ∞ . In this case, by tradition, we use b instead of φ.
We recall an alternative and equivalent de nition based on reproducing kernels. Namely, H(b) is the Hilbert space of analytic functions on D whose reproducing kernel is given by That is, The space H(Θ) is also called the model space and is denoted by K Θ = H(Θ). By Beurling's theorem, the spaces K Θ correspond to the lattice of closed, non trivial, invariant subspaces for the backward shift operator S * = Tz on H .
In the general case, the spaces H(b) are Hilbert spaces that are contained contractively in H . Moreover, it is well-known that there are relations between the inner products of H(b) and its cousin H(b) since these relations are special cases of the Lotto-Sarason theorem [8, Theorem 16.18 and corollary 16.19]. For further reference, we restate this result below.
It is now a well-known fact that the general theory of H(b)-spaces splits into two cases, according to whether b is an extreme point or a non-extreme point of the unit ball of H ∞ (recall that, according to De Leeuw-Rudin's Theorem, b is a non-extreme point of the closed unit ball of H ∞ if and only if log( − |b|) ∈ L (T), in particular every inner function b = Θ is an extreme point). For example, (see [8,Theorem 23.23 and corollary 25.8]).
Furthermore from the above characterization of a non-extreme point it follows that, if b is non-extreme, then there is an outer function a such that a( ) > and |a| +|b| = a.e. on T [14]. The function a is uniquely determined by b. We shall call (a, b) an euclidian pair.The following result gives a useful characterization of H(b) in this case.

Theorem 2.2 ([8], Theorem 23.8). Let b be a non-extreme point of the closed unit ball of H ∞ , let (a, b) be an euclidian pair and let f ∈ H . Then f ∈ H(b) if and only if Tb f ∈ Tā(H ). In this case, for f
An important operator in the theory of model spaces is the compression of Toeplitz operators on K Θ : for φ ∈ L ∞ and Θ an inner function, one de nes the truncated Toeplitz operator A Θ φ by with P Θ the orthogonal projection of H to K Θ . It turns out that when φ is in H ∞ , then K Θ is invariant for T φ and the adjoint of the truncated Toeplitz operator with symbol φ is ( When b is a non-extreme point of the closed unit ball of H ∞ , it follows from (1) and (2) that (3)

Compactness of Tφ ,b .
Ahern and Clark [1] have given a necessary and su cient condition for the truncated Toeplitz operator A Θ φ to be compact, when the symbol φ is continuous on the boundary. See also an alternative proof by Garcia-Ross-Wogen in [9]. The characterization of Ahern-Clark involves the notion of the spectrum of an inner function.
Recall that the spectrum of a function b in the closed unit ball of H ∞ [8, Section 5.2 and 22.6], denoted by σ(b) is de ned as follows In particular if b = Θ is a non constant inner function, and since Θ is unimodular a.e. on T then where Z(Θ) = {λ ∈ D : Θ(λ) = } and ν is the measure representing the singular part of Θ.

Now Ahern and Clark's result says:
The compactness property of the operators Tφ ,b will depend on the boundary spectrum of b and it is a consequence of the generalization of Ahern and Clark's result. For this reason we begin by this generalization, and we study the compactness of the general operator Tφ ,b with φ ∈ C(T), using the same technique used by Garcia, Ross and Wogen [9] to prove the Ahern-Clark result on compactness of A Θ φ .
Recall that the notation Tφ ,b represents the Toeplitz operator de ned from H(b) into H .

Theorem 3.2. Let b be a point of the closed unit ball of H ∞ and let φ ∈ C(T). Then the operator,
Let ε > and pick ψ ∈ C(T); ψ = on an open set containing (σ(b)). And consider (fn)n a sequence of H(b) such that (fn)n weakly converges to zero. We know that for each ζ ∈ K, the function (see [8,Theorem 21.1]). In particular, since (fn)n weakly converges to zero in H(b), we have fn(ζ ) =< fn , k b ζ > b → , as n → ∞, and for every n ∈ N, ||fn|| b ≤ C.
Therefore, since b is analytic on a neighborhood of the compact set K we obtain By the dominated convergence theorem, and using (4) it follows that whence T ψ,b is compact and therefore T φ,b is compact.
which is the square of the absolute value of the normalized reproducing kernel for H(b). Observe that F λ (z) ≥ .
Since ζ ∈ σ(b) ∩ T then there is a sequence λn in D such that λn → ζ and |b(λn)| → c with c < ( by the de nition of the spectrum of b already mentioned). Suppose that ζ = e iα and note that if |t − α| ≥ δ, then for some absolute constant C δ > . Thus since |b(λn)| → c with c < , we get that The rst integral can be made small by the continuity of φ. Once δ > is xed the second term goes to zero since sup |t−α|≥δ F λn (e it ) → as n → ∞. In addition Furthermore, on one hand And on the other hand the sequence ( )n converges weakly to , because |λn| → and |b(λn)| → c with c < . Indeed, using that We deduce that for f ∈ H ∞ ∩ H(b), After all these computations, we see that Finally, φ(ζ ) = .
We now present consequences of this result.

. Compactness of Tφ ,b
Recall that the notation Tφ ,b represents the Toeplitz operator de ned from H(b) into itself.

Corollary 3.3. Let b be a point of the closed unit ball of H
Then the operator is compact if and only if φ = .
In particular, |b(ζ )| = and ζ ∈ T \ E. We deduce that m(clos(σ(b)) ∩ T)). In the case where b is a non-extreme point of the closed unit ball of H ∞ , we can get a more general result without the hypothesis that the symbol φ is continuous.

Theorem 3.5. Let b be a non-extreme point of the closed unit ball of H ∞ and let φ ∈ H ∞ . Then the operator
is compact if and only if φ = .
Proof. Let a be the unique outer function such that (a, b) is an euclidian pair. Note that since b is nonextreme then kz ∈ H(b), for all z ∈ D (see (2)).
Suppose that Tφ ,b is compact. Notice that for every (λn)n ⊂ D such that |λn| → , the sequence ( k λn ||k λn || b )n converges weakly to in H(b).
Indeed, let f ∈ H(b) such that f and f + ∈ H ∞ . Recall that f + is de ned in Theorem 2.2. Then, using that Tb k λn = b(λn)k λn = Tā b(λn) a(λn) k λn (see (3)), we see that Whence by Theorem 2.2, we have On the other hand, it is known that in the non-extreme case: Using this inequality and the inequality we deduce that [8,Theorem 23.23]). Hence, the sequence ( k λn ||k λn || b )n converges weakly to in H(b). Now by compactness of Tφ ,b it follows that for every (λn)n ⊂ D such that |λn| → , But Tφ ,b k λn = φ(λn)k λn (see (3)). Thus (5)  Which implies that φ = .
The proof of Theorem 3.5 obviously doesn't work in the case when b is an extreme point of the closed unit ball of H ∞ , since in that case, the Cauchy kernels k λ do not belong to H(b) when b(λ) ≠ , λ ∈ D.

. Hypercyclic and frequently hypercyclic operators
Let X be a complex in nite-dimensional separable Banach space. An operator T ∈ L(X) is said to be hypercyclic if there is some vector x ∈ X such that the orbit is dense in X. Such a vector x is said to be hypecyclic for T, and the set of all hypercyclic vectors for T is denoted by HC(T).
Moreover we say that T is frequently hypercyclic, if there exists a vector x ∈ X such that for every nonempty open subset U of X, the set N(x, U) = {n ≥ ; T n (x) ∈ U} of instants when the iterates of x under T visit U has positive lower density, i.e. dens (N(x, U) We refer the reader to the recent book [4] for more information on these topics. Frequent hypercyclicity is a much stronger notion than hypercyclicity, and some operators are hypercyclic without being frequently hypercyclic: an example is the Bergman backward shift [3].
Let us complete this section by recalling two criterions for hypercyclicity and frequent hypercyclicity that we will use to study the hypercyclicity properties of the Toeplitz operator Tφ ,b .
We start with the Godefroy-Shapiro Criterion [11], according to which a bounded operator having a large supply of eigenvectors associated to eigenvalues of modulus strictly larger than 1 and strictly smaller than 1 is hypercyclic. Then it was shown by S. Grivaux that an operator T which has "su ciently many" eigenvectors associated to eigenvalues of modulus is automatically frequently hypercyclic.

Then T is frequently hypercyclic (in particular hypercyclic).
It is also a natural question, given a family of hypercyclic operators to ask if they have a common hypercyclic vector. The following result gives a su cient condition for a family of multiples of an operator to have a dense G δ -set of common hypercyclic vectors. Theorem 4.3 (Shkarin,[17]). Let X be a separable Fréchet space, T ∈ L(X), ≤ a < c ≤ ∞. Assume also that for all α, β ∈ R such that a < α < β < c there exists a dense subset E of X and a map S : E −→ E such that TSx = x, α −n T n x → and β n S n x → for each x ∈ E. Then is a dense G δ -set in X.
We nish by giving an example of a hypercyclic Toeplitz operator.
Rolewicz's result [4] in 1960, says that the operator λS * = T λz : H → H for every λ ∈ C, |λ| > , is hypercyclic, this was shown using Kitai's Criterion (a particular case of the Hypercyclity Criterion) [10]. This result of Rolewicz was generalized by Godefroy-Shapiro [11] in 1991. Following the approach of Godefroy-Shapiro, we generalize Theorem 4.4 to operators Tφ ,b when b is a nonextreme point of the closed unit ball of H ∞ and φ ∈ H ∞ , even we get a better result. And according to this generalization we noticed that in the non-extreme case, for every |λ| > the operator λX b = λTz ,b is frequently hypercyclic (in particular hypercyclic). On the contrary when b is extreme, λX b is never hypercyclic.
As we saw in the previous Corollary, for all |λ| > , λX b is hypercyclic, so this naturally raises the question of nding a common hypercyclic vector for (λX b ) |λ|> . We will apply Shkarin's Theorem 4.3 but we need to introduce another operator on H(b).
It is well known that H(b) is invariant under the unilateral forward shift operator S if and only if b is non-extreme [8,Corollary 20.20]. In that case, the mapping gives a well-de ned operator. Moreover S b is bounded on H(b) with ||S b || = + |a( )| − S * b b (see [8,Section 24.1]) In particular, we see that except in the case when b = (corresponding to H(b) = H ), the operator S b has a norm strictly greater than . It is clear that X b S b = I. For all < α < β < ||S b || − , and for all p ∈ P, we have on one hand α −n X n b p → as n → ∞, since from a certain rank n = deg(p)+ , X n b p = , and on the other hand, ||β n S n b p|| b ≤ (β||S b ||) n ||p|| b → as n → ∞. Hence, using Theorem 4.3, we conclude that G is a dense G δ -set of H(b). In other word, is the set HC(λX b ; < |λ| < ∞) a dense G δ -set of H(b)?
In the case where b is extreme, the operator X b is no longer hypercyclic, which shows a signi cant di erence in the H(b) space theory following that log( − |b|) is integrable or not on T. The proof of this result requires basic facts on the spectrum of hypercyclic operators, which we now brie y recall. Let X be a complex Banach space, and let T ∈ L(X) be hypercyclic. Then σp(T * ) = ∅ and every connected component of the spectrum of T intersects the unit circle (see [4,Page 11]).