One-component inner functions

We explicitely unveil several classes of inner functions $u$ in $H^\infty$ with the property that there is $\eta\in ]0,1[$ such that the level set $\Omega_u(\eta):=\{z\in\mathbb D: |u(z)|<\eta\}$ is connected. These so-called one-component inner functions play an important role in operator theory.


Introduction
Definition 0.1. An inner function u in H ∞ is said to be a one-component inner function if there is η ∈]0, 1[ such that the level set (also called sublevel set or filled level set) Ω u (η) := {z ∈ D : |u(z)| < η} is connected.
One-component inner functions, the collection of which we denote by I c , were first studied by B. Cohn [10] in connection with embedding theorems and Carlesonmeasures. It was shown in [10, p. 355] for instance that arclength on {z ∈ D : |u(z)| = ε} is such a measure whenever Ω u (η) = {z ∈ D : |u(z)| < η} is connected and η < ε < 1.
A thorough study of the class I c was given by A.B. Aleksandrov [1] who showed the interesting result that u ∈ I c if and only if there is a constant C = C(u) such that for all a ∈ D sup z∈D 1 − u(a)u(z) Many operator-theoretic applications are given in [1,2,7,3]. In our paper here we are interested in explicit examples, which are somewhat lacking in literature. For example, if S is the atomic inner function, which is given by then all level sets Ω S (η), 0 < η < 1 are connected, because these sets coincide with the disks (0.1) D η := z ∈ D : z − L L + 1 < 1 L + 1 , L := log 1 η , which are tangential to the unit circle at p = 1. The scheme of our note here is as follows: in section 1 we prove a general result on level sets which will be the key for our approach to the problem of unveiling classes of one-component inner functions. Then in section 2 we first present with elementary geometric/function theoretic methods several examples and then we use Aleksandrov's criterion to achieve this goal. For instance, we prove that BS, B •S and S •B are in I c whenever B is a finite Blaschke product. Considered are also interpolating Blaschke products. It will further be shown that, under the supremum norm, I c is an open subset of the set of all inner functions and multiplicatively closed. In the final section we give counterexamples.

Level sets
We first begin with a topological property of the class of general level sets. Although statement (1) is "well-known" (the earliest appearance seems to be in [26,Theorem VIII,31]), we could nowhere locate a proof. The argument that the result is a simple and direct consequence of the maximum principle is, in our viewpoint, not tenable. (1) Ω 0 is a simply connected domain; that is, C \ Ω 0 has no bounded components 1 .
Proof. We show that (1) holds for every holomorphic function f in D; that is if Ω 0 is a component of the level set Ω f (η), η > 0, then it is a simply connected domain 2 . Note that each component Ω 0 of the open set Ω f (η) is an open subset of D. We may assume that η is chosen so that {z ∈ D : |f (z)| = η} = ∅.
Suppose, to the contrary, that D is a bounded component of C \ Ω 0 . Note that D is closed in C. Then, necessarily, D is contained in D, because the unique unbounded complementary component of Ω 0 contains {z ∈ C : |z| ≥ 1}. Hence D is a compact subset of D.
In fact, given z 0 ∈ ∂H, let U be a disk centered at z 0 . Then U ∩ Ω 0 = ∅, since otherwise U ∪ H would be a connected set strictly bigger than H and contained in the complement of Ω 0 ; a contradiction to the maximality of H.

Explicit examples of one-component inner functions
Let ρ(z, w) = z − w 1 − zw be the pseudohyperbolic distance of z to w in D and D ρ (z 0 , r) = {z ∈ D : ρ(z, z 0 ) < r} the associated disks, 0 < r < 1. Here is a first class of examples of functions in I c . Although the next Proposition must be known (in view of A.B. Aleksandrov's criterion [1]), see 2.12 below), we include a simple geometric proof for the reader's convenience. Proof. Denote by z 1 , . . . , z N the zeros of B, multiplicities included. Let η ∈ ]0, 1[ be chosen so close to 1 that G : because z ∈ G implies that for some n, Since G is connected, there is a unique component Ω 1 of Ω containing G. In particular, Z(B) ⊆ G ⊆ Ω 1 . If, in view of achieving a contradiction, we suppose that Ω := Ω B (η) is not connected, there is a component Ω 0 of Ω which is disjoint with Ω 1 , and so with G. In particular, Since Ω 0 ⊆ Ω B (η) ⊆ D, and |B| = η on ∂Ω 0 , we deduce from the minimum principle that Ω 0 contains a zero of B; a contradiction.
We now generalize this result to a class of interpolating Blaschke products. Recall that a Blaschke product b with zero set/sequence {z n : n ∈ N} is said to be an interpolating Blaschke product if δ(b) := inf(1 − |z n | 2 )|b (z n )| > 0. If b is an interpolating Blaschke product then, for small ε, the pseudohyperbolic disks D ρ (z n , r) = {z ∈ D : ρ(z, z n ) < ε} are pairwise disjoint. Moreover, by Hoffman's Lemma (see below and also [19]), for any η ∈]0, 1[, b is bounded away from zero on {z ∈ D : ρ(z, Z(b)) ≥ η}. Theorem 2.2 (Hoffman's Lemma). ( [18] p. 86, 106 and [13] p. 404, 310). Let δ, η and be real numbers, called Hoffman constants, satisfying 0 < δ < 1, If B is any interpolating Blaschke product with zeros {z n : n ∈ N} such that (2) The following inclusions hold: A large class of interpolating Blaschke products which are one-component inner functions now is given in the following result. Since G is assumed to be connected, there is a unique component Ω 1 of Ω containing G. In particular, Z(b) ⊆ G ⊆ Ω 1 . Now, if we suppose that Ω is not connected, then there is a component Ω 0 of Ω which is disjoint with Ω 1 , and so with G. In particular, By Lemma 1.1, inf Ω 0 |b| = 0. Thus, there is z 0 ∈ Ω 0 be so that |b(z 0 )| < ε. We deduce from Hoffman's Lemma 2.2 that ρ(z 0 , Z(b)) < η < σ. This is a contradiction to (2.2). We conclude that Ω must be connected. It is clear that the condition ρ(z n , z n+1 ) < σ for every n implies that n D ρ (z n , σ) is connected. For the rest, just note that Corollary 2.4. Let B be a Blaschke product with increasing real zeros x n satisfying Proof. Just use Theorem 2.3 and the fact that by the Vinogradov-Hayman-Newman theorem, B is interpolating if and only if Using a result of Kam-Fook Tse [25], telling us that a sequence (z n ) of points contained in a Stolz angle (or cone) {z ∈ D : |1 − z| < C(1 − |z|)} is interpolating if and only if it is separated (meaning that inf n =m ρ(z n , z m ) > 0), we obtain: Corollary 2.5. Let B be a Blaschke product whose zeros (z n ) are contained in a Stolz angle and are separated. Suppose that ρ(z n , z n+1 ) ≤ η < 1. Then B ∈ I c .
Similarily, using a result by M. Weiss [27,Theorem 3.6] and its refinement in [4,Theorem B], we also obtain the following assertion for sequences that may be tangential at 1 (see also Wortman [28]). Corollary 2.6. Let B be a Blaschke product whose zeros z n = r n e iθn satisfy: r n < r n+1 , θ n+1 < θ n , r n 1, θ n 0, Then B is an interpolating Blaschke product contained in I c . This holds in paticular if the zeros are located on a convex curve in D with endpoint 1 and satisfying (2.3).
Other classes of this type can be deduced from [14]. Here are two explicit examples of interpolating Blaschke products in I c whose zeros are given by iteration of the zero of a hyperbolic, respectively parabolic automorphism of D. These functions appear, for instance, in the context of isometries on the Hardy space H p (see [8]).
. Then ϕ is an hyperbolic automorphism with fixed points ±1. If ϕ j := ϕ • · · · • ϕ j−times , then ϕ j ∈ Aut(D) and vanishes exactly at the point This can readily be seen by using that x j+1 = ϕ −1 (x j ) and we deduce from Corollary 2.4 that the Blaschke product associated with these zeros is in I c .
Hence, by Corollary 2.5, (z n ) is an interpolating sequence (see also [11, p.80]) and the associated Blaschke product b = ∞ n=1 e iθn τ n belongs to I c (here θ n is chosen so that the n-th Blaschke factor is positive at the origin). Thus, by, Corollary 2.6, the Blaschke product associated with these zeros is interpolating and belongs to I c .
Proposition 2.8. Let B be a finite Blaschke product or an interpolating Blaschke product with real zeros clustering at p = 1. Then f := BS ∈ I c .
Proof. i) Let B be a finite Blaschke product. Chose η ∈ ]0, 1[ so close to 1 that the disk D η in (0.1), which coincides with the level set Ω S (η), contains all zeros of B. Now D η = Ω S (η) ⊆ Ω f (η). Now Ω f (η) must be connected, since otherwise there would be a component Ω 0 of Ω f (η) disjoint from the component Ω 1 containing D η . But f is bounded away from zero outside D η ; hence f = BS is bounded away from zero on Ω 0 . This is a contradiction to Lemma 1.1 (2). ii) If B is an interpolating Blaschke product with zeros (z n ), then, by Hoffman's Lemma 2.2, B is bounded away from zero outside R := D ρ (z n , ε) for every ε ∈ ]0, 1[. Now, if the zeros of B are real, and bigger than −σ for some σ ∈]0, 1[, this set R is contained in a cone with cusp at 1 and aperture-angle strictly less than π (see for instance [21]). Hence R is contained in D η for all η close to 1. Thus, as above, we can deduce that Ω BS (η) is connected.
The previous result shows, in particular, that certain non one-component inner functions (for example a thin Blaschke product with positive zeros, see Corollary 3.1), can be multiplied by a one-component inner function into I c . In particular, uv ∈ I c does not imply that u and v belong to I c . The reciprocal, though, is true: that is I c itself is stable under multiplication, as we are going to show below. Proof. Let Ω u (η) and Ω v (η ) be two connected level sets. Due to monotonicity (Lemma 1.2), and the fact that λ∈[λ 0 ,1[ Ω f (λ) = D, we may assume that σ satisfies max{η, η } ≤ σ < 1 and is so close to 1 that 0 ∈ Ω u (σ) ∩ Ω v (σ) = ∅.
Next we look at right-compositions of S with finite Blaschke products. We first deal with the case where B(z) = z 2 .
Example 2.11. The function S(z 2 ) is a one-component inner function.
Proof. Let Ω S (η) be the η-level set of S. Then, as we have already seen, this is a disk tangent to the unit circle at the point 1. We may choose 0 < η < 1 so close to 1 that 0 belongs to Ω S (η). Let U = Ω S (η)\] − ∞, 0]. Then U is a simply connected slitted disk on which exists a holomorphic square root q of z. The image of U under q is a simply connected domain V in the semi-disk {z : |z| < 1, Re z > 0}. Let V * be its reflection along the imginary axis. Then E := V * ∪ V is mapped by z 2 onto the closed disk Ω S (η). Then E \ ∂E coincides with Ω S(z 2 ) (η). Using Aleksandrov's criterion (see below), we can extend this by replacing z 2 with any finite Blaschke product. Recall that the spectrum ρ(Θ) of an inner function Θ is the set of all boundary points ζ for which Θ does not admit a holomorphic extension; or equivalently, for which Cl(Θ, ζ) = D, where Cl(Θ, ζ) = {w ∈ C : ∃(z n ) ∈ D N , lim z n = ζ and lim Θ(z n ) = w} is the cluster set of Θ at ζ (see [13, p. 80]). (1) Θ ∈ I c .
Note that, due to this theorem, Θ ∈ I c necessarily implies that ρ(Θ) has measure zero. Proposition 2.13. Let B be a finite Blaschke product. Then S • B ∈ I c .
Proof. Since S is the universal covering map of D \ {0}, each singular inner function S µ writes as S µ = S • v for some inner function v. Since ρ(S µ ) is finite, v necessarily is a finite Blaschke product. (This can also be seen from [15, Proof of Theorem 2.2]). The assertion now follows from Proposition 2.13.
Note that this result also follows in an elementary way from Proposition 2.9 and the fact that every such S µ is a finite product of powers of the atomic inner function S. We now consider left-compositions with finite Blaschke products. Proof. Let τ (z) = (a − z)/(1 − az). Then ρ(τ • Θ) = ρ(Θ). As above, from which we easily deduce the first and second derivatives. By using the formulas 2.5, we obtain Hence, the assumption Θ ∈ I c now yields (via Aleksandrov's criterion 2.12) that This also follows from Corollary 2.6 by noticing that the a-points of S are located on a disk tangent at 1 and that the pseudohyperbolic distance between two consecutive ones is constant (see [20]). There it is also shown that the Frostman shift (S − a)/(1 − aS) is an interpolating Blaschke product. Proof. This is a combination of Propositions 2.15 and 2.9.  Proof. Let ε ∈ ]0, 1[ be arbitrary close to 1. Choose η > 0 and δ > 0 so close to 1 so that ε < η 2 and η < (1 − √ 1 − δ 2 )/δ. By deleting finitely many zeros, say z 1 , . . . , z N of b, we obtain a tail b N such that (1 − |z n | 2 )|b N (z n )| ≥ δ for every n > N . Hence, by Theorem 2.2, and the disks D(z n , η) are pairwise disjoint. This implies that the level set {z ∈ D : |b N (z)| < ε} is not connected. Now choose r so close to 1 that for every z with r ≤ |z| < 1. We show that the level set {|b| < ε 2 } is not connected. In fact, for some r ≤ |z| < 1 we have |b(z)| < ε 2 , then Hence {z : r < |z| < 1, |b(z)| < ε 2 } ⊆ {|b N (z)| < ε} Since the disks D ρ (z n , η) are pairwise disjoint if n > N , we are done.
Corollary 3.2. No finite product B of thin interpolating Blaschke products belongs to I c .
Using the following theorem in [5], we can exclude a much larger class of Blaschke products from being one-component inner functions: In particular this condition is satisfied by finite products of thin Blaschke products (see [17,Proposition 2.2]) as well as by the class of uniform Frostman Blaschke products sup ξ∈T ∞ n=1 1 − |z n | 2 |ξ − z n | < ∞.
Note that this Frostman condition implies that the associated Blaschke product has radial limits of modulus one everywhere [9, p. 33]. As a byproduct of Theorem 2.3 we therefore obtain Corollary 3.4. If b is a uniform Frostman Blaschke product with zeros (z n ) clustering at a single point, then lim sup ρ (z n , z n+1 ) = 1.
Questions 3.5. To conclude, we would like to ask two questions and present three problems: