Composition Operators on the Bloch space of the Unit Ball of a Hilbert Space

Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.


Introduction
Let E be a complex Hilbert space of arbitrary dimension and denote B E its open unit ball. The space B(B E ) of Bloch functions was introduced in [1]. There it was shown that it can be endowed with a (modulo the constant functions) norm that is invariant under the automorphisms of B E ; see section 3 below for the basics. This article studies composition operators acting on B(B E ), i.e., self-maps of B(B E ) defined according to C ϕ (f ) = f • ϕ, for a given analytic map ϕ : B E → B E . As in the finite dimensional case, every composition operator is bounded, actually of norm not greater than 1 for the invariant norm if the symbol vanishes at 0, and also the hyperbolic metric on B E measures the distance between evaluations in the dual space. We also study the compactness of composition operators, providing necessary and sufficient conditions. There are two common requirements for both the necessity and the sufficiency: (1 − z 2 ) Rϕ(z) 1 − ϕ(z) 2 = 0 and lim ϕ(z) →1 (1 − z 2 )| ϕ(z), Rϕ(z) | 1 − ϕ(z) 2 = 0, where Rϕ(z) denotes the radial derivative at z. The fact that for all 0 < δ < 1, ϕ(δB E ) is relatively compact completes a necessary condition, while the additional assumption ϕ(B E ) ∩ δB E being relatively compact, provides a sufficient one. Such compactness requirements are trivially satisfied in the finite dimensional case, thus the two limits above yield an apparently new characterization. Some of our techniques are inspired by J. Dai's paper [4]. However, there are some obstacles to avoid when allowing an infinite number of variables, like the lack of relative compactness of the ball, the number of components of the symbol or the use of the invariant Laplacian. And still a major one: uniform convergence on compact sets does not imply uniform convergence on compact sets of the derivatives; this only happens in the finite dimensional setting (see [3]). Such obstacle causes the lengthy proof of our main result Theorem 4.13. In the final section we present several examples that discuss the relations among the conditions we have found.

Background
Let (e k ) k∈Γ be an orthonormal basis of E that we fix at once. Then every z ∈ E can be written as z = k∈Γ z k e k and we write z = k∈Γ z k e k .
Given an analytic function ϕ : B E → B E we write ϕ(x) = k∈Γ ϕ k (x)e k , ϕ (x) : E → E its derivative at x, and Rϕ(x) = ϕ (x)(x) its radial derivative at x.
We shall denote by ϕ a the Möbius transforms for Hilbert spaces. For each a ∈ B E , ϕ a : where s a = 1 − a 2 , m a : B E → B E is the analytic function m a (x) = a − x 1 − x, a and P a = 1 a 2 a⊗a where u⊗v(x) = x, u v and Q a = Id−P a are the orthogonal projection on the one dimensional subspace generated by a and on its orthogonal complement respectively. Since ϕ a • ϕ a (x) = x one has (ϕ a ) −1 = ϕ a and ϕ a (a) = (ϕ a (0)) −1 .
Actually (see for instance [1, Lemma 3.2]) (2.1) ϕ a (0) = −s 2 a P a − s a Q a , and The pseudo-hyperbolic and hyperbolic metrics on B E are respectively defined by ρ E (x, y) := ϕ x (y) and β E (x, y) : It is known ( [6] p. 99) that where ρ is the pseudo-hyperbolic metric on the open unit disk D in the complex plane given by ρ(z, w) = z−w 1−zw and H ∞ (B E ) denotes the Banach space of bounded analytic functions on B E endowed with the sup-norm.
Since (s + t)/(1 + st) is an increasing function of s and t for 0 ≤ s, t ≤ 1, the sharpened form of the triangle inequality for ρ(z, w) easily yields the same inequality for ρ E (x, y), The following estimate holds (see [1], Lemma 4.1): The open unit ball of H ∞ (B E ) is invariant under post-composition with conformal self-maps of D. By composing f with a conformal self-map of D that maps f (y) to 0, one obtains that Recall that if f : The following result gives an explicit formula to compute the invariant gradient. It is a modification of Lemma 3.5 in [1] in a form that fits our purposes.
Lemma 2.1. Let f : B E → C be an analytic function and x ∈ B E . Then Now we can replace w by ϕ x (0) −1 (w) in the above formula, so In the proof of Lemma 3.5 in [1] it is shown that so the statement follows.
The following version of Schwarz-Pick lemma will be needed in the sequel. The analogue of these results in several variables has been proved in [2].
For background on analytic (or holomorphic) mappings on infinite dimensional complex spaces we refer to [3].

The Bloch space
The classical Bloch space B is the space of analytic functions on the open unit disk, f : D → C, such that the semi-norm f B = sup z∈D (1 − |z| 2 )|f (z)| is bounded; it becomes a Banach space when endowed with the norm f Bloch = |f (0)| + f B . See [10] for general background on the classical Bloch space. The Bloch space of functions defined on the finite dimensional Euclidean ball was introduced by R. Timoney in [8]. See [9] for further information.
The space of Bloch functions is denoted by B(B E ) and it has been studied in [1]. As in the finite dimensional case, the space H ∞ (B E ) is strictly contained in B(B E ) (see [1,Corollary 4.3]) and the following inequality holds for any f ∈ H ∞ (B E ): An equivalent semi-norm for the space of Bloch functions is given by This semi-norm satisfies f • ϕ inv = f inv for any f ∈ B(B E ) and any automorphism ϕ of B E . The space B(B E ) is usually endowed with the norm f Bloch(B E ) = |f (0)| + f inv and then it becomes a Banach space.
Another equivalent semi-norm is given by We refer to [1,Thm 3.8] for all the equivalences of these semi-norms. In particular, we have the following inequalities: The following result extends Theorem 5.5 in [10] to an infinite dimensional Hilbert space E: Theorem 3.1. Let f : B E → C be an analytic function. Then, Proof. First we prove that If f inv = ∞, then we are done. So take f ∈ B(B E ) and x, y ∈ B E . Then, Consider f •ϕ y ∈ B(B E ). By the inequality above, (3.2) and bearing in mind that f •ϕ inv = f inv for any automorphism ϕ, we have that If x = tu and taking limits when x → 0, we have that and we are done.
Proof. From Theorem 3.1, we have for any Also In particular, we observe that the norm topology of B(B E ) is finer than the compact open topology co.
As consequence of Theorem 3.1, we have that Notice that the metric β E (x, y) can be also recovered from the Bloch semi-norm f inv : To check the other inequality follow the same pattern as Theorem 3.9 in [9] and recall [1, Lemma 3.3].

Composition operators
4.1. Boundedness. As it occurs in the finite dimensional case, every composition operator on B(B E ) is bounded.
where last inequality holds because ρ E (x, y) is contractive for analytic maps ϕ : Further, using Corollary 3.2, , and we conclude that C ϕ is bounded.
We provide another proof that relies on magnitudes that will appear further on.
By combining this with Lemma 2.2 we conclude that Thus the boundedness of C ϕ is immediate if we assume ϕ(0) = 0. If ϕ(0) = x = 0, then we consider the mapping ψ = ϕ x • ϕ, for which ψ(0) = 0, and the bounded operator C ψ . Since f • ϕ x inv = f inv it follows, using Corollary 3.2 as well, that C ϕx is continuous. Hence C ϕ = C ψ • C ϕx is continuous.

4.2.
Compactness. Now we proceed to discuss necessary and sufficient conditions for a composition operator on B(B E ) to be compact. We begin with some necessary ones. 4.2.1. Necessary conditions. The following result is a little improvement of a result due to Dai [4] for finitely many variables.
Proof. We use the projection theorem for Hilbert spaces, so for each Now by Montel's theorem (see [3,Theorem 17.21]), there is a subnet (f n(α) ) of (f n ) that converges uniformly on compact subsets of Proof. First we prove (4.1): Indeed, since the set The fact that C * ϕ (δ z ) = δ ϕ(z) allows us to conclude that ϕ(δB E ) is relatively compact by appealing to (3.4). Let (n k ) be an increasing sequence in N and (ξ k ) a sequence in E with ξ k ≤ 1. According to [1,  We have ϕ, ξ k n k rad = sup Let us first show (4.3). We suppose that there exist ε > 0 and a sequence (z k ) ∈ B E such that ϕ(z k ) → 1 and for each k, Let n k be the integer part of , ϕ(z k ) |, which gives a contradiction if (4.5) holds. Thus (4.3) holds.
Remark 4.7. Note that ϕ(z) = z satisfies (4.2) and fails (4.3). Also observe that Proof. For z ∈ B E and w ∈ E we consider the linear functional λ z,w acting on f ∈ B(B E ) ac- is relatively compact because for the function e u (z) = z, u , we have RC ϕ (e u )(z) = Rϕ(z), u = λ ϕ(z),Rϕ(z) (e u ) and hence There are also necessary conditions in terms of the components of the symbol ϕ. Recall that (e k ) k∈Γ is an orthonormal basis of E and ϕ = k∈Γ ϕ k (x)e k . Here, ϕ k = ϕ, e k .  C ϕ k,l : B → B is compact for all k, l ∈ Γ, where ϕ k,l (λ) := ϕ k (λe l ), λ ∈ D. Also, In particular, lim k∈Γ ϕ k B(B E ) = 0. And further, Proof. Let y ∈ E \ {0} and ξ ≤ 1. We write F ξ (x) = F ( x,ξ ), x ∈ B E , for each F ∈ H(D), and f y (λ) = f (λ y y ), λ ∈ D, for each f ∈ H(B E ).
If f ∈ B(B E ) and y ≤ 1 then it is an easy calculation that f y ∈ B and f y B ≤ f B(B E ) . Hence the operator R y : f ∈ B(B E ) → f y ∈ B is continuous. For each y, ξ ∈ B E and F ∈ B we can write So C ϕ y,ξ = R y • C ϕ • E ξ is compact. Then (4.7) follows because ϕ k,l = ϕ e k ,e l . Let us now show (4.8). Given a weakly null net (ξ k ) k∈κ ∈ E with ξ k ≤ 1, we consider f k (z) = log Assume now that (4.8) does not hold. Then there exist ε > 0, and a subnet (n k ) such that for every n k there is z k with Selecting now ξ k = e n k ϕ n k (z k ), we get a weakly null net for which thus (4.10) holds. Then that contradicts (4.11). Finally, we prove (4.9). Let n ∈ Γ and assume that (4.9) does not hold, that is there is ε > 0 and a sequence (z l ) with lim l→∞ |ϕ n (z l )| = 1 and Let F l (λ) = log . We may assume that ϕ n (z l ) converges to some w 0 , |w 0 | = 1. This means that (g l ) coconverges to g 0 (x) = F 0 ( x, e n ) = log 1 1− x,en w 0 where F 0 (λ) = log 1 1−λw 0 . Next, notice that C ϕ (g l )(x) = F l ϕ(x), e n = F l • ϕ n (x).

Compactness criteria.
Lemma 4.10. Let f : B E → C be analytic and x, y ∈ B E . Then and using that P x is self-adjoint, (4.14) Therefore, s 2 x y, ∇f (x) = (s x − 1) By the Cauchy formula we have that Thus equality (4.14) becomes and we conclude by taking y = x.
Remark 4.11. From (4.14) we deduce the following identity that might be of independent interest Lemma 4.12. For every 0 < δ < 1, there exists C δ > 0 such that x, x ∈ δB E and y ≤ 1, y ≤ 1, and f ∈ B(B E ).
Next, using Cauchy formula we have for x, x ∈ δB E , y ≤ 1, y ≤ 1, From this, Theorem 3.1 and the equivalence of the semi-norms, we get that for some constant C > 0 Applying (2.6) we find a constant C δ > 0 depending only on δ such that Theorem 4.13. Let ϕ : B E → B E be analytic. Assume that Proof. We are going to apply Lemma 4.4. Let (f α ) be a bounded net in B(B E ) converging to zero uniformly on compact sets. Recall that Let ε > 0. By (ii) and (iii) there exists δ < 1 such that for ϕ(z) > δ we have and hence using Lemma 2.1, we have Denote A δ = {z ∈ B E : ϕ(z) ≤ δ}. For z ∈ A δ we use formula (4.15) obtained in the proof of Lemma 4.10 to have Rϕ(z) 2 Rϕ(z) , ∇f (ϕ(z)) = Hence for each z ∈ A δ , Bearing in mind (2.12) in Lemma 2.2 and that lim →0 In particular, for each δ < 1 there exists C δ > 0 such that To finish the proof we use that ϕ(A δ ) is relatively compact in B E . So, given ε > 0 there exists a finite family of points {z k : 1 ≤ k ≤ N } ⊂ A δ such that for each z ∈ A δ there exists z k for which ϕ(z) − ϕ(z k ) < ε. Now bearing in mind (2.11) to observe that s 2 y Rϕ(y) ≤ 1, we may use Lemma 4.12 to have for each z, z ∈ A δ ∇f α (ϕ(z)), Hence sup The proof is then complete using (4.19). (1 − z 2 ) Rϕ(z) In particular for ϕ n (z) = 2n j=n z, e j , C ϕ is compact on B(B E ). Proof. Note that sup z <1 ϕ(z) 2 ≤ ( ∞ n=1 sup z <1 |ϕ n (z)| 2 ) < 1. Moreover, ϕ(B E ) is relatively compact since it lies inside the Hilbert cube given by the sequence ( ϕ n ∞ ). Now, apply Corollary 4.15.
Next, we introduce a class of symbols ϕ that allows a characterization of the compactness of C ϕ . We say that the analytic mapping ϕ : In particular any map with bounded radial derivative satisfies (4.20). (1 − z 2 )|Rϕ n (z)| = 0 for all n ∈ N and then ϕ(z) = ∞ n=1 ϕ n (z)e n ∈ B 0 (B E , B E ). Proof. Given ε > 0 there exist N ∈ N and 0 < δ j < 1 for j = 1, · · · , N such that (1 − z 2 ) Rϕ(z) Proof. In case ϕ ∞ < 1, both right hand limits are null and both left hand limits vanish according to the assumption. Since ϕ(z) ≤ z by Lemma 2.2, the limits on the right hand side are not greater than those on the left hand side. Now, in case ϕ ∞ = 1, there is a sequence (z n ) ⊂ B E such that z n → 1 and lim sup z →1 . From the bounded sequence ( ϕ(z n ) ) we get a convergent subsequence that we denote the same. If lim n ϕ(z n ) = 1, we have lim sup ϕ(z) →1 that leads to the equality (i), while if lim n ϕ(z n ) < 1, then lim sup z →1 holds as well. The analogous argument for (ii).
In the following result we replace condition (i) in Theorem 4.13 by the weaker one given by (4.1) and conditions (4.2) and (4.3) by the stronger ones given by taking lim ϕ(z) →1 instead of lim z →1 . Since the proof follows the same arguments as in Theorem 4.13, it will only be sketched. (1 − z 2 ) Rϕ(z) Proof. By Lemma 2.2, we have ϕ(z) ≤ z . The analogous estimate to (4.18) holds for z > δ.
In the remaining case z ≤ δ, also ϕ(z) ≤ δ so the estimates in the proof of Theorem 4.13 hold, that is, if z ≤ δ, then Now the final argument in the proof of Theorem 4.13 relies on the relative compactness of ϕ({ z ≤ δ}).
Let us mention that (4.21) implies that ϕ ∈ B 0 (B E , B E ) and that combining the necessary condition obtained in Theorem 4.5 and Proposition 4.18 we get the following:

Examples
In this section we provide a number of examples to discuss the relations among the various conditions we have found above.
Assume now that ξ n 0 = 1. Selecting z = λξ n 0 we have that ϕ(z) = λe n 0 and lim ϕ(z) →1 This gives (ii). Now (iii) follows using that |ϕ n (z)| ≤ ξ n for each n. Hence ϕ(B E ) is contained in the Hilbert cube given by the sequence ( ξ n ).
To check (iv), choose δ = sup k F k ∞ and use (5.2). Since ϕ(B E ) is contained in the Hilbert cube given by the sequence ( F k ∞ ) it is relatively compact. Thus (v) holds.
Finally to show (vi) we use the estimate for analytic functions F : D → D with F (0) = 0 given by |F (λ)| ≤ F B β(0, λ) to obtain that ϕ(δB E ) is contained in the Hilbert cube given by the sequence ( F k B β(0, δ)). This gives (4.1).
Example 5.3. Let {e k } be an orthonormal sequence in E. Let us consider ϕ(z) = k ϕ k (z)e k where (5.3) ϕ k (z) = z, e k k .
Then ϕ satisfies (4.1) and fails (4.8). In particular C ϕ is non-compact on B(B E ).
Proof. Notice that ϕ(z) ∈ B E for each z ∈ B E because It is clear that Rϕ k (z) = kϕ k (z). To show (4.1) just observe that sup z ≤δ |ϕ k (z)| ≤ δ k . Denote Let z = λe k and estimate A k ≥ sup Then ϕ and ψ satisfy (4.1) but fail (4.7). Hence C ϕ and C ψ are non-compact on B(B E ).