Interpolating sequences for H∞(BH)

Abstract We prove that under the extended Carleson’s condition, a sequence (xn) ⊂ BH is linear interpolating for H∞(BH) for an infinite dimensional Hilbert space H. In particular, we construct the interpolating functions for each sequence and find a bound for the constant of interpolation.


Introduction
Let A be a space of bounded functions defined on X. A sequence (x n ) in X is called interpolating for A if for any sequence (a n ) ∈ ∞ , there exists f ∈ A such that f (x n ) = a n for all n ∈ N. We consider the linear and continuous R : A → ∞ defined by R(f ) = (f (x n )). The sequence (x n ) is interpolating for A if and only if there exists a map T : ∞ → A such that R • T = Id ∞ . If T is linear, the sequence (x n ) is said to be linear interpolating for A. For any α = (α j ) ∈ ∞ , let M α = inf { f ∞ : f (x j ) = α j , j ∈ N, f ∈ A} . The constant of interpolation for (x n ) is defined by M = sup {M α : α ∈ ∞ , α ∞ ≤ 1} .
It is a classical result in function theory that a sequence (z n ) in the open unit disc D ⊂ C is interpolating for H ∞ , the space of analytic bounded functions on D, if and only if Carleson's condition holds, i. e.: There is δ > 0 such that k =j ρ(z k , z j ) ≥ δ for any j ∈ N, (1.1) where ρ(z k , z j ) denotes the pseudohyperbolic distance for points z k , z j ∈ D, given by Recall the Schwarz-Pick lemma: ρ(f (z), f (w)) ≤ ρ(z, w) for any z, w ∈ D and f ∈ H ∞ , f ≤ 1. If ψ is an automorphism of D, then ρ(ψ(z), ψ(w)) = ρ(z, w).
If we deal with complex Banach spaces E, we denote by B E its open unit ball. A function f : B E → C is said to be analytic if it is Fréchet differentiable. Denote by H(B E ) the space of analytic functions f : B E → C and by H ∞ (B E ) the space {f : B E → C : f is analytic and bounded }, which becomes a uniform Banach algebra when endowed with the sup-norm f = sup{|f (x)| : x ∈ B E } and it is, obviously, the analogue of the space H ∞ for an arbitrary Banach space.
Sufficient conditions for a sequence to be interpolating for H ∞ (B E ) where given by the authors in [GM]. Bearing in mind the Davie-Gamelin extension of f ∈ H ∞ (B E ) tof ∈ H ∞ (B E * * ), the authors proved that a sufficient condition for a sequence (x n ) ⊂ B E * * to be linear interpolating for H ∞ (B E ) is that the sequence of norms ( x n ) is interpolating for H ∞ . Examples of sequences which satisfy this condition are, for instance, those which grow exponentially to the unit sphere, which we call the Hayman-Newman condition: 1− x k+1 < c(1− x k ) for some 0 < c < 1 for any k ∈ N. Interpolating sequences on H ∞ (B E ) have been very useful to study the spectra of composition operators on spaces of analytic functions (see [GGL], [GLR] and [GM2]).
During all the manuscript, we will deal with a complex Hilbert spaceH. The notion of pseudohyperbolic distance can be carried over to H ∞ (B H ) by considering for any x, y ∈ B H , where ρ(z, w) is the pseudohyperbolic distance in D. B. Berndtsson [B] showed that a sequence (x n ) in the open unit Euclidean ball B n of C n is interpolating for H ∞ (B n ) if the following extended Carleson's condition holds: There is δ > 0 such that As P. Galindo, T. Gamelin and M. Lindström pointed out in [GGL], the result given by Berndtsson can be extended to the case of an infinite dimensional complex Hilbert space H by interpolating on finite subsets of the sequence with uniform bounds and applying a normal families argument.
The aim of this paper is to adapt the proof given by Berndtsson to the infinite dimensional case and prove that under the extended Carleson's condition 1.3, a sequence (x n ) ⊂ B H is linear interpolating.
In particular, we will construct the interpolating functions for each sequence and will find a bound for the constant of interpolation.
For our purpose, we will study the automorphisms on B H and will adapt some results given by B. Berndtsson (see [B]) to the infinite dimensional case.

Background
The results of this section and further information about the automorphisms of B H and the pseudohyperbolic distance on B H can be found in [GR].
Automorphisms on B H . Recall that the set of automorphisms on D is denoted by Aut(D). It is well-known that this set is given by all the mappings f : D → D which are the composition of rotations with Möbius transformations m a : D −→ D given by m a (z) = a − z 1 −āz for any a ∈ D. (2.1) The analogues of Möbius transformations on H are ϕ a : P a : H −→ H is the orthogonal projection along the one-dimensional subspace spanned by a, that is, and Q a : H −→ H, is the orthogonal complement, Q a = Id−P a . Recall that P a and Q a are self-adjoint operators since they are orthogonal projections, so P a (x), y = x, P a (y) and Q a (x), y = x, Q a (y) for any x, y ∈ H. The set of automorphisms on B H is given by all the mappings ϕ : Remarks on the pseudohyperbolic distance. It is clear by the definition that for any x, y ∈ B H , so making some calculations we obtain (2.6)

Results
First, we recall Proposition 2.1 in [GLM]: We will call (F n ) a sequence of Beurling functions for (x n ). Under conditions of Proposition 3.1, we have that T : In particular, the constant M is an upper bound for the constant of interpolation.
The following calculations are straightforward and can be found in [GM].
Lemma 3.2. We have the following statements: The following three lemmas are just calculus: The following result will be needed to simplify the proof of Theorem 3.11. The proof is just an exercise.
Lemma 3.5. Let (a n ) ⊂ [0, 1) such that lim n a n = 1. Then, (a n ) can be reordered into a non-decreasing sequence (b n ) such that lim n b n = 1. Now we provide a lemma which includes some calculations related to the automorphisms ϕ x .
Lemma 3.6. Let x, y ∈ B H and ϕ −y : H −→ H the corresponding automorphism defined as in 2.2. Then, we have that Proof. Since for any x ∈ B H we have ϕ y (x) = (s y Q y + P y ) (m y (x)), and bearing in mind that P and Q are orthogonal, we obtain that 3) just making some calculations. Since we have that P a + Q a = Id H for any a ∈ H, we obtain that The complement of the orthogonal projection is given by x, y y, y y, z − z, y y, y y = x, z − 1 y 2 x, y y, z − 1 y 2 x, y y, z + 1 y 2 x, y y, z = x, z − 1 y 2 x, y y, z = x, z y, y − x, y y, z y 2 .
Adding and subtracting 1 and arranging terms, we obtain that the numerator equals to (1 − x, y )(1 − y, z ) − (1 − x, z )(1 − y, y ). Therefore, dividing by the denominator and making calculations, we obtain and the lemma is proved.
Considering z = x, we obtain that formulas 2.5 and 2.6 are the same expression for the pseudohyperbolic distance for x, y ∈ B H .
We will also need some technical lemmas. For the first one, we will need Proposition 5.1.2 in [R], which is stated as follows, Lemma 3.7. Let a, b, c be points in the unit ball of a finite dimensional Hilbert space. Then, Then, we obtain the following lemma, Lemma 3.8. Let H be an infinite dimensional complex Hilbert space and x 1 , x 2 , x 3 ∈ B H . Then, Proof. Let x 1 , x 2 , x 3 ∈ B H and set the space H 1 = span{x 1 , x 2 , x 3 }.
We have that H 1 is itself a Hilbert space and we can consider an orthonormal basis {e 1 , e 2 , e 3 } of H 1 . Consider for j = 1, 2, 3 the vectors y j = (y 1 j , y 2 j , y 3 j ) given by the components of x j in that basis. It is clear that these vectors are in the unit Euclidean ball of C 3 and x j , x k = y j , y k , so we apply Lemma 3.7 to deduce By the arithmetic-geometric means inequality, we have |1− To prove the other result, notice that We have that 1 − | x 1 , x 3 | = |1 − e iα x 1 , x 3 | and 1 − | x 2 , x 3 | = |1 − e iβ x 2 , x 3 | for some α, β ∈ [0, 2π). Then, applying the inequality above, we have that Then, we can prove the following lemma, which extends Lemma 6 in [B] to the infinite dimensional case. We give the proof for the sake of completeness.
Lemma 3.9. Let H be a Hilbert space and x k , and we consider two cases depending on so we are done.
We will also need the following lemma, Then, we have that

5)
and for any j ∈ N, Proof. Taking squares and logarithms in 3.4 we obtain By (3.1), we have that 1−ρ H (x k , x j ) 2 ≤ − log ρ H (x k , x j ) 2 for any k = j, so bearing in mind (2.6), we obtain In consequence, 1 + x j 1 − x j and the lemma is proved. Now we are ready to prove the result for complex Hilbert spaces. In addition, we will provide an upper estimate for the constant of interpolation depending only on δ.
Theorem 3.11. Let H be a Hilbert space and (x n ) a sequence in B H . Suppose that there exists δ > 0 such that (x n ) satisfies the generalized Carleson condition (1.3) for δ. Then, there exists a sequence of Beurling functions (F n ) for (x n ). In particular, the sequence (x n ) is interpolating for H ∞ (B H ) and the constant of interpolation is bounded by 128(1 + 2 log 1 δ ) eδ .
Proof. Define, for any k, j ∈ N, k = j, the analytic function g k,j : . First we check that the infinite product converges uniformly on rB H = {x ∈ B H : x ≤ r} for fixed 0 < r < 1. Let x ∈ rB H . We have, by Lemma 3.6, that so for any j ∈ N, the series k =j |1 − g k,j (x)| is uniformly convergent on rB H by Lemma 3.10. In particular, the infinite product k =j g k,j (x) converges uniformly on compact sets, so B j ∈ H(B H ). In addition, notice that for . It is clear that B j (x k ) = 0 for k = j since ϕ x k (x k ) = 0 and, according to 2.5, we have that Consider the functions q j , A j ∈ H(B H ) for any k ∈ N defined by The function q j is clearly analytic and bounded. By Lemma 3.5, we will consider that the sequence ( x n ) is non-decreasing, so {k : x k ≥ x j } = {k : k ≥ j}. Notice also that for 0 < r < 1 and x ∈ rB H we have that so by Lemma 3.10, the series converges uniformly on rB H and hence and using formula 3.2, we have Consider C δ = 1/(1 + 2 log 1/δ) and for any j ∈ N, the analytic function F j : B H −→ C defined by It is clear that F j (x j ) = 1 and F j (x k ) = 0 for any k = j. We claim that there exists M > 0 such that ∞ j=1 |F j (x)| ≤ M for any x ∈ B H . Indeed, by (3.2) that .
In particular, for x = x j , we obtain .
Using formula (2.6) we obtain that Moreover, to estimate e A j (x) from below we use Lemma 3.9 and we obtain that We define for any k ∈ N, Using that |B j (x j )| ≥ δ, |B j (x)| ≤ 1, the bound for e A j (x j ) and 3.7, we obtain Since 0 ≤ b k (x) ≤ 1, we consider u = b j (x) and t = C δ k≥j |q k (x)| > 0 and apply Lemma 3.3 to conclude that where h(t) = min{1, 256/e 2 t 2 }. Hence, summing on j, we obtain and applying Lemma 3.4 , we obtain that Hence , by Proposition 3.1, we conclude that (x n ) is linear interpolating.
Given (x n ) ⊂ B H satisfying the extended Carleson's condition and any (α n ) ∈ ∞ , the function f (x) = ∞ j=1 α j F j (x), where F j is defined as in Theorem 3.11, is well-defined and interpolates the values α n in the points x n for any n ∈ N.
Notice also that the function f (δ) = 1 δC 2 δ = 1 + 2 log 1/δ δ is non-increasing for 0 < δ ≤ 1. Since lim δ→1 f (δ) = 1, un upper bound for the constant of interpolation is close to 128 e if we deal with sequences satisfying the extended Carleson's condition with δ close to 1. Can the number 128 e be decreased?