Interpolating sequences for weighted spaces of analytic functions on the unit ball of a Hilbert space

We show that an interpolating sequence for the weighted Banach space of analytic functions on the unit ball of a Hilbert space is hyperbolically separated. In the case of the so-called standard weights, a sufficient condition for a sequence to be linear interpolating is given in terms of Carleson type measures. Other conditions to be linearly interpolating are provided as well. Our results apply to the space of Bloch functions of such unit ball.


Introduction and preliminaries
Throughout the paper E stands for a complex Hilbert space of arbitrary dimension and B E = {x ∈ E : x < 1} for its open unit ball. Let υ : B E → (0, ∞) be a weight, that is, a continuous positive function. The weighted space of analytic functions is a Banach space when endowed with the · υ norm. Here we are mostly interested in the standard weights υ α (x) = (1 − x 2 ) α , α ≥ 0. When α = 0 we get the infinite dimensional generalization H ∞ (B E ) of the Hardy algebra H ∞ of the unit disc. Nevertheless, some of our results hold true for more general weights.
An analytic function f : B E → C is said to belong to the Bloch space . Both suprema define equivalent Banach space norms -modulo the constant functions-in B(B E ). We will use both the norms We shall deal also with the following generalization: for α > 0, it is said that f ∈ B α (B E ) whenever sup x∈B E (1 − x 2 ) α |R f (x)| < ∞ and we write The space of Bloch functions on the unit ball of a Hilbert space was introduced and studied in [3] and its generalization B α (B E ) has been considered in [16,28]. If E = C, we get the classical Bloch-type spaces denoted by B α . We shall use the The aim of this article is to study interpolating sequences for these spaces. We will say that (w n ) ⊂ B E is interpolating for H ∞ υ (B E ) if the mapping A sequence (w n ) ⊂ B E \{0} is said to be interpolating (for the radial derivative) for the Bloch space B α (B E ), if the mapping is onto. Notice that 0 cannot be included into the interpolating sequence since R f (0) = 0.
In case S has a linear right inverse, we say that (w n ) is a linear interpolating sequence.
For a given interpolating sequence (w n ), any constant M > 0 such that whenever α ∈ ∞ there exists f with S( f ) = α and f ≤ M α ∞ is called an interpolation constant. Such constants exist by the Open Mapping Theorem.
There are in the mathematical literature several meanings for the expression "interpolating sequence" for B that do not take into account the derivative of the function. See [5,27] for details on different types of interpolation for B. For further information on interpolating sequences on spaces of analytic functions we refer to [25].
After the celebrated result by Carleson which characterizes interpolating sequences for the space H ∞ (see [14]) also interpolating sequences for the spaces H ∞ υ α were characterized by Seip [24]. In connection with Hankel operators, Attele [1] studied in connection with Hankel operators interpolating sequences for the derivatives in the Bloch space B. Such sequences were used by Madigan and Matheson [17] as a tool for the study of compactness of some composition operators. The study of interpolating sequences for Banach spaces of analytic functions on B n , the open unit ball of the Euclidean space C n , was initiated by Berndtsson [2] for H ∞ (B n ) and by Massaneda [18] for H ∞ υ α (B n ) with α > 0. They found some necessary and sufficient conditions for a sequence to be interpolating in those spaces. Interpolating sequences in the n-ball for the so-called fractional derivatives were investigated in [7]. The existence of interpolating sequences for a given Banach function space shows an abundance of elements in the space and it has turned to be very useful in the study of weighted composition operators. The study of interpolating sequences for H ∞ (B E ) was initiated in [11,12] (see also [21]).
In this paper we prove that being hyperbolically separated is also a necessary condition for a sequence to be interpolating in  [1,17]) in the unit disc stating that a sufficiently separated sequence is interpolating. In Sect. 7 we present a number of results concerning B(B E ) that are extension of the classical ones.
For background on analytic functions on subsets of Hilbert or more general Banach spaces, we refer to [19]. We will use, frequently without further notice, the following consequence of Montel's theorem ( [19,Proposition 9.16]).

Lemma 1.1 If F m ∈ H (B E ), the series m F m is pointwise convergent and the series has uniformly bounded partial sums on compact subsets of B E , then it defines an analytic function.
A crucial tool in the study of analytic functions on the unit ball of a Hilbert space is the homogeneity of the ball. Specifically, the existence of the Möbius transforms ϕ a : u v and Q a = I d − P a are the orthogonal projection on the one dimensional subspace generated by a and on its orthogonal complement respectively. We recall that ϕ a • ϕ a (x) = x and ϕ a (a) = 0.
The pseudo-hyperbolic metric on B E is defined by ρ E (x, y) := ϕ x (y) . In case E = C, we write ρ instead of ρ C . The pseudo-hyperbolic disc given by {y ∈ B E : ρ E (x, y) < R} is denoted by D(x, R). We recall some facts to be used later (see [15] p. 99) Let x ∈ B E and R ∈ (0, 1). Then We shall write B n for the unit ball of C n and denote by ν n the normalized measure in B n for n ≥ 1. It is well known that for n = 1 the pseudo-hyperbolic disc D(z, R) becomes an Euclidean disc while in the case n ≥ 2 it is not an Euclidean disc (unless z = 0) but an ellipsoid (see [23, pages 29, 30]) and the value ν n (D(z, R)) is A sequence (w n ) ⊂ B E is said to be hyperbolically r −separated for r > 0 if ρ E (w n , w m ) > r for n = m. We say that the sequence is hyperbolically separated if it is hyperbolically r −separated for some r > 0. We will say that the sequence (w n ) satisfies the Carleson's condition if m =n ρ E (w n , w m ) ≥ δ for some δ and for all m ∈ N.

Carleson measures on the unit ball B E
For any ξ ∈ E, ξ = 1 and 0 < h < 1 we shall denote by S(ξ, h) the Carleson window given by We write S(ξ, h) = B E for h ≥ 1.
Furthermore, for α > β one has Now for k ≥ 1, y ∈ E k and selecting h = 2 k−1 (1 − x 2 ) in the above estimate, we obtain that This completes the proof and give the stated estimates.
Recall that for x ∈ B E , the evaluation map δ x is given by δ In particular η γ, The following characterization is a direct consequence of Lemma 2.2 together with (1.2).

Lemma 2.4
Let (w n ) ∞ n=1 ⊂ B E and α > β > 0. Then for each γ > 0 the measure η γ,(w n ) is a β-Carleson measure if and only if there exists C > 0 such that In particular, the following are equivalent: We consider the following notation introduced in [18] for B n . For p, q > 0 and
Let us now relate the notion of "hyperbolic separation" with β-Carleson measures. Combining Remark 2.6 and [18, Lemma 1.5] it was shown that any hyperbolically separated sequence (z j ) ⊂ B n satisfies that η β,(z j ) is a β-Carleson measure for β > n. We shall give an alternative proof of such a result using the following well known estimate (see [29,Lemma 2.24 for any 0 < r < 1, 0 < p < ∞ and any holomorphic function f in B n .

Proposition 2.7
Let n ∈ N and let (z j ) ⊂ B n be a hyperbolically separated sequence, then η β,(z j ) is a β-Carleson measure for any β > n.
Proof Assume that ρ C n (z j , z k ) ≥ 2R for j = k. Hence D(z j , R) are pairwise disjoint sets in B n . Let us show that (z j ) satisfies (iii) in Lemma 2.4 for any β > n and any α > β. Let z ∈ B n . Applying (2.5) to the function Now use the well-known fact (see [23,Page 18]) that for c > 0 and t > −1

Remark 2.8
For the unit ball of infinite dimensional Hilbert spaces the fact that a sequence is hyperbolically separated does not imply that Invoking now Proposition 2.7, Lemma 2.4 and the fact ρ( x , y ) ≤ ρ E (x, y) for any x, y ∈ B E we obtain the following result.

Examples of interpolating sequences
Let us start by pointing out that in the case of infinite dimensional Hilbert spaces we can find interpolating sequences for H ∞ v (B E ) and in B α (B E ) whose interpolating functions are polynomials.

Proposition 3.1 Let (e n ) ⊂ E be an orthonormal sequence and let u be a bounded radial weight on D. Then for each
Specifically, for any bounded sequence (α k ) ⊂ C and any degree d ≥ 2, there is a polynomial P d ∈ P( d E) such that S(P d ) = (α n ), where P( d E) stands for the set of d-homogeneous polynomials defined on B E and S is the map defined in (1.1).

Proof Consider the d-homogeneous polynomial
Clearly Proposition 3.2 Let (e n ) ⊂ E be an orthonormal sequence and α > 0. Then for each Proof We argue similarly to Proposition 3.1 using now Clearly Q d is well-defined since 2 ⊂ d and Let us give another procedure to generate interpolating sequences where the interpolating functions are explicitly given.
Proof Let (α n ) be a bounded sequence and define The convergence of the series is guaranteed by Bessel's inequality, while the analyticity of f follows from Lemma 1.
This shows that f ∈ H ∞ v α (B E ) and that it interpolates (α n ) at the points (w n ). Now we turn to the case of B α (B E ). We define for each bounded sequence (α n ) the functions To verify that An analogous argument works as well for g 1 , by considering f 1 (t) = log(1 − λt).
Since |z 2 n x, e n 2 | ≤ ||x|| 2 < 1, we get that Therefore by Bessel's inequality, and the analyticity of g α follows again using Lemma 1.1.
Recall that if H (x) = h( x, ξ ) for a given holomorphic function h in the unit disc (1 −z 2 n x, e n 2 ) α .

Now we obtain
Finally since (1 − w n 2 ) α Rg α (w n ) = α n for all n ∈ N, the proof is complete.
Also we can rely in the results in one variable to produce examples. Recall that sequences (z n ) ⊂ D\{0} which are interpolating for any of the spaces we are dealing with in the unit disc satisfy that inf n |z n | > 0. Notice that (z n ) is interpolating for B α if and only if the mapping ϒ : Assume now that (z n ) is an interpolating sequence for B α . Let (α n ) n ∈ ∞ and find ϕ ∈ B α with (1 − |z n | 2 ) α z n ϕ (z n ) = α n for all n ∈ N. Denote f (x) = ϕ( x, ξ ). ϕ ( x, ξ ) x, ξ one has that f ∈ B α (B E ) and that it interpolates (α n ) since The analogue to Proposition 3.4 for finite dimensional Hilbert spaces corresponds to the following procedure. Let L : C n → E be an isometric linear embedding and let P : E → C n , be the orthogonal projection onto L(C n ). Then for any f ∈ H (B E ), and any sequence (z m ) ⊂ C n , one has So we get the following result by also taking into account for g ∈ H (B n ) its compo-

Necessary conditions
In all the known cases (see [1,2,18,24]) a necessary condition for a sequence (z n ) to be interpolating for certain spaces defined in the unit disc D or the unit ball B n is to be hyperbolically separated. To extend this result to the case H ∞ υ (B E ) we shall need a couple of lemmas.
The first one is a Schwarz lemma type consequence of inequality (2.1) in [6] applied to the function w ∈ B E → f (r w).
The second one and its proof are a suitable version of Lemma 14 in [9].
The standard weights υ α , for α ≥ 0, satisfy the assumption with F(r ) = Proof Consider g = f • ϕ x and observe that g is an analytic bounded function on r B E . Indeed, from (4.1) we have Hence sup z ≤r |g(z)| ≤ f υ υ(x) F(r ). Now applying Lemma 4.1 to g, we conclude that for ϕ x (y) ≤ r /2 we have For the final statement, notice that

Theorem 4.3
Let υ be a weight satisfying (4.1) for some F. Any interpolating sequence no subsequence of ( w n ) can converge to 0. Hence inf n { w n : w n = 0} > 0.

Corollary 4.4 Any interpolating sequence for H
Proof Recall (see [8, page 48]) that for any analytic self map γ of the unit disc, Then if x = 0, we apply this inequality to γ (z) = ϕ(z x x ), ϕ(x) to obtain The estimate is obvious for x = 0.

If (w n ) is an interpolating sequence for H ∞ υ (B E ), then ϕ(w n ) is also an interpolating sequence. The analogous statement holds for H
The standard weights υ α satisfy (4.4).

Proof Notice that for
. In order to check that υ α satisfies (4.4), just use Lemma 4.6. The remaining statement follows after realizing that now f

Sufficient conditions
We begin with a result concerning linear interpolation.

Proposition 5.1 Let (w n ) ⊂ B E be a linear interpolating sequence for H
Proof Since (w n ) is a linear interpolating sequence for H ∞ v α (B E ), the corresponding mapping S has a linear right inverse T .
where (e m ) m is the sequence of canonical unit vectors in ∞ . Then we have that the sequence v α (w n )F m (w n ) n = e m and v α ( It is a well defined and bounded operator since To obtain values in H ∞ υ β (B E ) 0 , we can choose the functions h w n (y) = y,w n ||w n || 2 g w n (y) instead of g w n , since w n > r for some r > 0 if all w n = 0.
it is also a linear interpolating sequence (see [20] or [13]). Therefore, we obtain from Proposition 5.1 the following corollary.

Corollary 5.2 If (w n ) ⊂ B E is an interpolating sequence for H ∞ (B E ), then it is linear interpolating for H
In [11,21] it was proved that a sufficient condition for a sequence (w n ) to be interpolating for H ∞ (B E ) is to satisfy the Carleson's condition. Hence, we obtain From Corollary 5.3, we deduce, bearing in mind [12] or ρ( x , y ) ≤ ρ E (x, y), that a sequence (w n ) ⊂ B E which grows exponentially to the unit sphere, that is, In addition, a sequence (w n ) ⊂ B E such that lim n→∞ w n = 1 has a subsequence which is linear interpolating for H ∞ υ α (B E ). In the case E = C all interpolating sequences in H ∞ v p (D) were completely characterized by Seip [24]. Concerning interpolating sequences for H ∞ v α (B n ), let us recall the following results due to Massaneda.
When replacing B n by the open unit ball B E of an infinite dimensional Hilbert space E, the assumptions in both statements above cannot be anymore fulfilled.

Actually Proposition 3.1 shows that interpolating sequences in H
We now shall see that some of the previously known results can be extended to the infinite dimensional case under the additional assumption of η γ,(w n ) being a γ -Carleson measure for some γ.
We now give other sufficient conditions for a sequence to be interpolating in this setting that is inspired by the above results and [17, Proposition 1] Proof For any β + α ≥ > β and from Lemma 2.4 we obtain This guarantees that the series defining (α n ) (x) is uniformly convergent in r B E for any 0 < r < 1 and hence it defines an analytic function on B E . Moreover or equivalently, since /2 ≤ p, Hence is well defined and bounded. We aim to prove that I d − S • < 1, thus S • will be invertible, hence S has • (S • ) −1 as right linear inverse.
Consider now the composition operator, We look at the mth component of (I d − S • ) (α n ) . Further, From this it follows that From (2.2) and picking < β + α/2 we obtain Since inf β< <β+α/2 we conclude that I d − S • < 1 and therefore (i) is proved. For (ii), we get from (5.2) that for all β < ≤ β+α. By choosing = β+α, the assumption gives that I d−S• < 1 as wanted.

Interpolating sequences for radial derivatives in Bloch-type spaces
Our previous results apply to the Bloch-type spaces by means of the following theorem.
is an onto isometric isomorphism.
Proof Notice that for every m-homogeneous polynomial P : E → C we have R P(z) = m P(z). Next for any analytic function f : The integral exists since the integrand function is a continuous one and G(0) = 0. Since for every given x ∈ B E and chosen 1 < λ such that λx < 1, the series turns to be convergent, it follows that Hence G is analytic and for its radial derivative RG(x) = ∞ m=1 Q m (x) = g(x).
Hence the analogous results to Corollary 4.4, 5.2 and Theorem 5.7 hold for the Bloch-type spaces. Let us state those results in this setting.
We are interested in producing sequences which are interpolating for B but not for H ∞ . Recall that a Blaschke sequence (z n ) ⊂ D is a sequence which satisfies ∞ k=1 (1 − |z k |) < ∞. It is well-known that if a sequence satisfies the Carleson's condition, then it is a Blaschke sequence. On the other hand, Proposition 2.7 for the case E = C yields that ∞ n=1 (1−|z n | 2 ) 2 δ z n is 2-Carleson for any hyperbolically separated sequence (z n ). We will show the existence of a sequence which is hyperbolically R−separated for R close enough to 1 which is not a Blaschke sequence. We will adapt the example in [10] to give an example of such a sequence. Proposition 6.4 Consider k an even number, k ≥ 2 and circles C n centered at 0 and radius r n = 1 − 1 k n for any n ≥ 1. In each circle C n , we take z n, j = r n e 2πi j k n−1 for any 0 ≤ j < k n−1 . For any k ≥ 2, the sequence created in this way is hyperbolically R k −separated for some R k so close to 1 as we want. Moreover, the sequence is not a Blaschke sequence.
Proof Take z, w in the sequence. Then there are two possibilites. If z and w are in different circles C n and C m , m > n, then where last inequality is clear since function x−1 x+1 is increasing for x ≥ 0 and k m−n ≥ k. Now suppose that |z| = |w| = r n . Since ρ is invariant for automorphisms, in particular, for rotations, we consider, without loss of generality, that z = r n and w = r n e 2πi j k n−1 for some 1 ≤ j ≤ k n−1 2 since we take by symmetry the semicircle {z ∈ C n : 0 ≤ arg z ≤ π }. Then, Bearing in mind that sin α ≥ 2 π α for any 0 ≤ α ≤ π 2 and j ≥ 1, we have that Hence, for any two terms of the sequence z, w we have that This expression tends clearly to 1 when k → ∞ as we wanted. Let (z n, j ) be the sequence that we have constructed. It is clear that it is not a Blaschke sequence since The proof is now complete. Proof The first statement is true by Proposition 6.4. The other one follows just by considering the sequence (z n ξ) where (z n ) is the sequence defined in Proposition 6.4 and ξ ∈ E such that ξ = 1. If (z n ξ) were interpolating for H ∞ (B E ), then (z n ) would be interpolating for H ∞ and it is well-known that an interpolating sequence for H ∞ satisfies the Carleson's condition and, in particular, it is a Blaschke sequence. Proof Firstly we remark that f is bounded on any ball of radius σ < 1, because according to Theorem 3.1 in [4],

Some Banach space properties of the Bloch space
Fix 0 < s < 1 and pick 1 > σ > s and apply Lemma 4.1 to get Finally for r > r 0 , By letting ε → 0, we deduce that d ( f , B 0 | is a routine verification. , we conclude that there is a subnet { f α j } such that f α j → f ∈ H (B E ) with respect to the τ 0 -topology. For x ∈ B E choose s > 0 such that ||x|| + s < 1. Then for any y ∈ E with ||y|| = 1, we have by the Cauchy integral formula that From this we obtain that Proof Taking into account that f r R(B E ) ≤ f R(B E ) , the result follows immediately from Lemma 7.1.

Remark 7.6
Therefore since E is reflexive, neither B(B E ) nor B 0 (B E ) can have the Dunford-Pettis property. This in sharp contrast to the classical case (E = C) where B is isomorphic to ∞ .
We close this section with an application of the above results to interpolating sequences. DefineT : ∞ → B(B E ) according toT (α n ) = n α n f n . Now using (7.1) and recalling that the series also converges for the τ 0 -topology, we get that Thus (S •T ) = I d ∞ , as wanted.