De Branges-Rovnyak spaces and local Dirichlet spaces of higher order

We discuss de Branges-Rovnyak spaces $\mathcal H(b)$ generated by nonextreme and rational functions $b$ and local Dirichlet spaces of order $m$ introduced in [6]. In [6] the authors characterized nonextreme $b$ for which the operator $Y=S|_{\mathcal H(b)}$, the restriction of the shift operator $S$ on $H^2$ to $\mathcal H(b)$, is a strict $2m$-isometry and proved that such spaces $\mathcal H (b)$ are equal to local Dirichlet spaces of order $m$. Here we give a characterization of local Dirichlet spaces of order $m$ in terms of the $m$-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for $b$ in the case when $\mathcal H(b)$ coincides with local Dirichlet space of order $m$ with equality of norms. Finally, we prove a property of wandering vectors of $Y$ analogous to the property of wandering vectors of the restriction of $S$ to harmonically weighted Dirichlet spaces obtained by D. Sarason in [11].


Introduction
Let D denote the open unit disc in the complex plane C, and let T = ∂D.For ϕ ∈ L ∞ (T) the Toeplitz operator on the Hardy space H 2 of the disc D is defined by T ϕ f = P (ϕf ), where P is the orthogonal projection of L 2 (T) onto H 2 .In particular, denote S = T z .
For a function b in the closed unit ball of H ∞ , the de Branges-Rovnyak space H(b) is the image of H 2 under the operator (I − T b T b ) 1/2 with the corresponding range norm • b (see the books [9], [3] and references given there).It is known that H(b) is a Hilbert space with reproducing kernel In the case b is an inner function, H(b) = H 2 ⊖ bH 2 is the so called model space.
The theory of H(b) spaces divides into two cases, according to whether b is or is not an extreme point of the closed unit ball of H ∞ .An important property of H(b) spaces is that they are S * -invariant and in the case when b is nonextreme they are also S-invariant.Moreover, for nonextreme b the operator Y = S |H(b) is bounded and it expands the norm.
Here we will concentrate on the case when the function b is not an extreme point of the unit ball of H ∞ (equivalently, log(1 − |b| 2 ) is integrable on T).Then there exists an outer function a ∈ H ∞ for which |a| 2 + |b| 2 = 1 a.e. on T.Moreover, if we suppose a(0) > 0, then a is uniquely determined, and we say that (b, a) is a pair.If (b, a) is a pair, then the quotient ϕ = b a is in the Smirnov class N + .Conversely, every nonzero function ϕ ∈ N + has a unique representation (the so called canonical representation) ϕ = b a , where (b, a) is a pair.In this case T ϕ is an unbounded operator on H 2 with the domain D(T ϕ ) = {f ∈ H 2 : ϕf ∈ H 2 } = aH 2 .Thus T ϕ is densely defined and closed.Consequently, its adjoint T * ϕ = T ϕ is densely defined and closed.Moreover, D(T ϕ ) = H(b) and for f ∈ H(b), (1) For more details on de Branges-Rovnyak spaces connection with unbounded Toeplitz operators see [12].
For a finite measure µ on T let P µ denote the Poisson integral of µ given by The associated harmonically weighted Dirichlet space D µ consists of functions f analytic in D for which where A denotes the normalized area measure on D. Spaces D(µ) were introduced by S. Richter in [7], where it was proved that certain two-isometries on a Hilbert space can be represented as multiplication by z on a space D(µ).The results on D(µ) stated below were also obtained in [7].
If µ is a finite measure on T such that µ(T) > 0, then D(µ) ⊂ H 2 and D(µ) is a Hilbert space with the norm • D(µ) given by given by ( 2) can be expressed as Moreover, if λ ∈ T is such that D λ (f ) < ∞, then the nontangential limit f (λ) exists.
The following local Douglas formula for D λ (f ) was proved by Richter and Sundberg [8]: if f ∈ H 2 and f (λ) exists, then If f (λ) does not exist, then we set D λ (f ) = ∞.The space D(δ λ ) = D λ is called the local Dirichlet space at λ.It has also been proved in [8] that In [6] S. Luo, C. Gu and S. Richter characterized nonextreme b for which Y is a strict 2m-isometry, m ∈ N. It turns out that the corresponding spaces H(b) are equal to the so called local Dirichlet spaces of order m, D m λ , λ ∈ T, defined as follows: λ if and only if for each j = 0, 1, . . ., m − 1 the function f (j) has a nontangential limit at λ and where In the space D m λ the norm is given by ( 4) In the next section we obtain the following characterization of the space In [10] D. Sarason proved that if for λ ∈ T, then the space D λ coincides with H(b λ ) with equality of norms.It follows from the paper [1] that H(b λ ) = D λ with equality of norms only for b λ given by ( 5).
In [6] the authors also described a relation between local Dirichlet space of higher order D m λ and H(b).In particular, they obtained a necessary and sufficient condition for H(b) = D m λ with equality of norms.In Section 3, we derive explicit formulas for such functions b analogous to (5).
A nonzero vector in a Hilbert space is called a wandering vector of a given operator if it is orthogonal to its orbit under the positive powers of the operator.In [11] the author described the wandering vector of the shift operator S µ = S |D(µ) on the harmonically weighted Dirichlet space D(µ) associated with a finitely atomic measure µ = n i=1 µ i δ λi where λ 1 , . . ., λ n , are distinct points on T and µ 1 , . . ., µ n are positive numbers.One of his results states that the outer part of the wandering vector of S µ lies in the model space generated by a certain finite Blaschke product.In the last section we consider the spaces H(b) when b is nonextreme and rational, and show that in this case the outer part of a wandering vector of the operator Y = S |H(b) has similar property.

Local Dirichlet space of order m
In the proof of Theorem 1 we will need the following technical lemma.
Lemma 1.For a positive integer m and z, w ∈ D, Proof.We proceed by induction.For m = 1 the equality is obvious.
Assume (6) holds true for an m.We will use the following Leibniz formula By the induction hypothesis, Proof of Theorem 1.We will use Sarason's idea applied in the proof of Proposition 1 in [10].Without loss of generality we may assume that λ = 1.Let A m = T (z−1) m be the Toeplitz operator with the symbol (z − 1) m on H 2 .Let M(A m ) be the range of A m equipped with the Hilbert structure that makes A m a coisometry from H 2 onto M(A m ).Since the kernel function for the space H 2 is (1 − wz) −1 , for g ∈ M(A m ) we get where the last inner product is in the space M(A m ).This implies that , where the last inner product is taken in the range space M (m) (A m ) of the operator of differentiation of order m.By Lemma 1, Observe that by (3), . We now note that the reproducing kernel for the space Thus the space {(z − 1) m h : h ∈ A 2 (ρ m )} has the same (up to a constant) kernel as the space {f (m) : then for each j = 0, 1, . . ., m − 1 the function f (j) has a nontangential limit at λ.
Proof.We can clearly assume that λ = 1.It follows from the proof of Theorem 1 that operator T m given by is an isometry of H 2 onto A 2 (ρ m ).If f satisfies condition (7), then there exists g ∈ H 2 such that f (m) = ((z − 1) m g) (m) .This means that f (z) = (z − 1) m g + p, where p is a polynomial of degree < m, and the reasoning used in the proof of Lemma 9.1 in [6] proves the claim.

De Branges-Rovnyak spaces H(b) and local Dirichlet spaces of finite order
We first cite one of the main results contained in [6] using notation from the Introduction.Recall that a Hilbert space operator T is an m-isometry if It is easy to check that every m-isometry is a k-isometry for every k ≥ m.An m-isometry that is not an (m − 1)-isometry is called a strict m-isometry.
Theorem ( [6]).Let b be a non-extreme point of the unit ball of H ∞ with b(0) = 0, and let m ∈ N. Then Y is not a strict (2m + 1)-isometry, and the following are equivalent (i) Y is a strict 2m-isometry, (ii) (b, a) is a rational pair such that a has a single zero of multiplicity m at a point λ ∈ T, (iii) there is a λ ∈ T and a polynomial p of degree < m with p(λ) = 0 such that Proof.We first show that for f ∈ Hol(D) -the space of functions holomorphic on D, It has been derived in [5] that for f ∈ Hol(D), we get where the last equality follows from the fact that for f ∈ Hol(D), Note now that polynomials are a dense subset of both H(b λ ) and D m λ , and by ( 1), ( 4) and ( 9), for each polynomial f , λ with equality of norms.In the following proposition we find the explicit formulas for b λ for which H(b λ ) = D m λ with equality of norms.Proposition 1.Let for λ ∈ T and m ∈ N the function ϕ m λ ∈ N + be defined by (8).
, where z k are preimages of the m-th roots of 1 for odd m and m-th roots of −1 for even m under the Koebe function k Proof.Assume that ϕ m λ = b λ a λ where (b λ , a λ ) is a pair.Then for |z| = 1 By Fejér-Riesz theorem there is a unique polynomial r of degree m without zeros in D, such that r(0) > 0 and Let (b, a) be a rational pair and let λ 1 , λ 2 , . . .λ m be all the zeros of a on T, listed according to multiplicity.It has been proved in [2] that for every such rational pair (11) H where P n denotes the set of polynomials of degree at most n.This means that the space H(b) is in fact determined by the zeros of a on T.Moreover, in the case when λ 1 = λ 2 = . . .= λ m = λ, by (3), H(b) = D m λ (as sets).For fixed λ 1 , λ 2 , . . .λ m ∈ T and a polynomial p such that p(λ j ) = 0, j = 1, 2, . . ., m, consider the function ϕ ∈ N + defined by (12) ϕ(z) = p(z) where (b, a) is a canonical pair.It is worth noting that the pair (b, a) is rational.Moreover, by the aforementioned result from [2], for each polynomial p, the space H(b) is described by (11) and the corresponding norms • b given by (1) are equivalent.For a positive integer m set The next proposition slightly extends the result contained in [6, Theorem 9.4].
Proposition 2. Let ϕ = b/a be defined by (12) with the polynomial p of degree ≤ m, and let q be a polynomial of degree ≤ m such that q(λ j ) = 0 for j = 1, . . ., m.Then if and only if, up to a multiplicative unimodular constant, q(z) = p(z) = z m p 1 z .Proof.Assume that q = p.Then, by Proposition 6.5 in [12], Assume now that (13) holds with a polynomial q.Since we get and so Consequently, if ϕ * (z) = q(z) We now prove the following Theorem 2. Let (b, a) be a rational pair such that the corresponding Smirnov function ϕ is given by (12) with p(z) = α 0 + α 1 z + • • • + α m z m such that p(λ j ) = 0, j = 1, 2 . . .m. Then there is a finite Blaschke product B m of degree m, such that the outer part of a wandering vector of Y lies in H(zB m ).

( 1 −
If the three conditions hold, then there are polynomials p and r of degree ≤ m such that b = p r , a = (z−λ) m r , and p(z) = z m p 1 z for z ∈ D, and |r(z)| 2 = |p(z)| 2 + |z − λ| 2m for all z ∈ T. Furthermore, H(b) = D m λ with equivalence of norms.In particular, H(b) = D m λ with equality of norms if and only if b(z) = z m r(z) , where r is a polynomial of degree m that has no zeros in D and such that |r(z)| 2 = 1 + |z − λ| 2m for all z ∈ T. Now we give another proof of the sufficient condition for H(b) = D m λ with equality of norms.The following proposition is actually contained in the above mentioned result of [6].Proposition ([6]).If for λ ∈ T the function ϕ m λ ∈ N + is defined by λz) m , z ∈ D, and its canonical representation is ϕ m λ = b λ a λ , then H(b λ ) = D m λ with equality of norms.

m j=1 1
− λ j z = b * (z) a * (z) , then H(b) = H(b * ) with equality of norms.By [3, Vol. 2, Cor.27.12], b = c * • b * for some unimodular constant c * .Since |a| = |a * | on T and the modulus of an outer function on T determines that function up to a multiplicative unimodular constant, we obtain that q = c • p for some c ∈ T.