Composition operator on model spaces

arXiv:1205.5172v3 [math.CV] 30 Jan 2013

COMPOSITION OPERATORS ON MODEL SPACES YURII I. LYUBARSKII AND EUGENIA MALINNIKOVA Dedicated to Nikolai K. Nikolski on the occasion of his 70th birthday Abstract. Let ϕ : D → D be a holomorphic function, ϑ : D → D be an inner function and Kϑ (D) = H 2 (D) ⊖ ϑH 2 (D) be the corresponding model space. We study the composition operator Cϕ on Kϑ and give a necessary and sufficient condition for Cϕ : Kϑ → H 2 to be compact. The condition involves an interplay between ϑ and the Nevanlinna counting function of ϕ. For a one-component ϑ a characterization of compact composition operators Cϕ in terms of the Aleksandrov-Clark measures of ϕ and the spectrum of ϑ is also given.

1. Introduction Let D be the unit disk in the complex plane. Given a holomorphic function ϕ : D → D, denote by Cϕ : f 7→ f ◦ ϕ the composition operator defined on holomorphic functions in D. This operator is bounded on the Hardy space H 2 (D) (see e.g. [15]). One of the intensively studied questions is when Cϕ is a compact operator on various spaces of analytic functions. We refer the reader to the monographs [9, 16] for the history and basic results on composition operators. Loosely speaking Cϕ is compact on H 2 (D) if ϕ(z) does not approach the unit circle T too fast as z → T. J. Shapiro [15] quantified this idea by using the Nevanlinna counting function X Nϕ (w) = − log |z|. ϕ(z)=w

He proved in particular that Cϕ is compact on H 2 (D) if and only if (1)

lim Nϕ (w)/(− log |w|) = 0.

|w|→1−

2010 Mathematics Subject Classification. Primary 47B33; Secondary 30H10, 30J05, 47A45. Key words and phrases. Composition operator, model space, Nevanlinna counting function, Aleksandrov-Clark measure, one-component inner function. The authors were partly supported by the Research Council of Norway grants 213638/F20 and 213440/BG.. 1

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YURII I. LYUBARSKII AND EUGENIA MALINNIKOVA

The basic tool in his argument is the Stanton formula Z 2 (2) kCϕ f k = 2 |f ′ (z)|2 Nϕ (z)dA(z) + |f (ϕ(0))|2 , D

where A is the normalized area measure. It is obtained from the identity Z 1 2 (3) kf k = 2 |f ′ (z)|2 log 2 dA(z) + |f (0)|2 , f ∈ H 2 (D), |z| D by substituting f ◦ ϕ in place of f . Another way to describe the compactness property of Cϕ is related to the Aleksandrov-Clark measures of ϕ. These are the positive measures µα on T defined by the relation Z α + ϕ(z) Pz dµα , = ℜ α − ϕ(z) T where Pz is the Poisson kernel, α ∈ T. We refer the reader to the surveys [13, 18] for more details. In [14] D. Sarason showed how Cϕ can be treated as an integral operator on the spaces L1 (T) and M (T) and proved that Cϕ is compact on these spaces if and only if each µα is absolutely continuous. Due to [17], it is further equivalent to Cϕ being compact on H 2 (D) as well as on other Hardy spaces H p (D), see also [5]. In this article we study the compactness of the operator Cϕ : Kϑ → H 2 (D), where ϑ is an inner function in D and Kϑ = H 2 (D) ⊖ ϑH 2 (D) is the corresponding model space. Consider the canonical factorization of ϑ  Z ξ +z dω(ξ) , ϑ(z) = BΛ (z) exp T ξ −z where Λ is the zero set of ϑ, BΛ is the corresponding Blachke product, and ω is a singular measure on T. Functions in Kϑ admit analytic continuation through T \ Σ(ϑ), where Σ(ϑ) = (T ∩ Clos(Λ)) ∪ supp(ω) is the spectrum of ϑ (see [12], Lecture 3). Therefore the compactness property of Cϕ does not suffer as the values of ϕ approach points in T \ Σ(ϑ). We quantify this idea below and give a condition that is necessary and sufficient for the compactness of Cϕ : Kϑ → H 2 (D). Acknowledgments. This work was started when the authors visited the Mathematics Department of the University of California, Berkley. It is our pleasure to thank the Department for the hospitality and Donald Sarason for useful discussions. The authors will also thank Anton Baranov for his comments on the preliminary version of this note and for showing us the inequality in [8] that is crucial for the proof of Theorem 1.

COMPOSITION OPERATORS ON MODEL SPACES

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2. Nevanlinna counting function In this section we give a counterpart of the condition (1) for the operator Cϕ : Kϑ → H 2 (D). The proof follows the ideas of [15]. The main new ingredients are estimates for the reproducing kernels and its derivatives given in Lemma 1 below. Let κw be the reproducing kernel for Kϑ , κw (ζ) =

1 − ϑ(w)ϑ(ζ) , 1 − wζ ¯

kκw k2 =

1 − |ϑ(w)|2 , 1 − |w|2

and let κ ˜ w be its normalized version 1/2  1 − ϑ(w)ϑ(ζ) 1 − |w|2 . κ ˜w (ζ) = 1 − |ϑ(w)|2 1 − wζ ¯ By (4)

kw (ζ) =

1 , 1 − wζ ¯

(1 − |w|2 )1/2 k˜w (ζ) = 1 − wζ ¯

we denote the reproducing kernel for H 2 and its normalized version. Lemma 1. Let {wn } ⊂ D, |wn | → 1 be such that (5)

|ϑ(wn )| < a,

for some a ∈ (0, 1). Then w∗ (i) κ ˜wn −−→ 0 as n → ∞; (ii) there exist ǫ > 0 , c > 0 and n0 such that c (6) |κ′wn (ζ)| > , ζ ∈ Dǫ (wn ) (1 − |wn |2 )2 ¯ holds for any n > n0 , where Dǫ (w) = {ζ; |ζ − w| < ǫ|1 − ζw|} is a hyperbolic disk with center at w. Proof. (i) It suffices to show that (1 − |wn |2 )1/2 w∗ (1 − ϑ(wn )ϑ(ζ)) −−→ 0 in L2 (T) as |wn | → 1. 1−w ¯n ζ This in turn follows from the known fact that the normalized reproducing kernels k˜wn for the Hardy space H 2 (D) tend weakly to 0 as |wn | → 1, see e.g. [15]. (ii) We start with the following well-known estimate (7)

|ϑ′ (ζ)| ≤

1 − |ϑ(ζ)|2 , 1 − |ζ|2

ζ ∈ D.

Together with (5) it readily yields (8)

|ϑ(ζ)| < b,

for some b < 1 and ǫ > 0.

ζ ∈ ∪n Dǫ (wn ),

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YURII I. LYUBARSKII AND EUGENIA MALINNIKOVA

We claim now that for sufficiently large n0 const , ζ ∈ ∪n>n0 Dǫ (wn ). (9) |κ′ζ (ζ)| > (1 − |ζ|2 )2 Indeed, κ′ζ (ζ) = −

ϑ′ (ζ)ϑ(ζ) ¯ 1 − |ϑ(ζ)|2 +ζ = A1 + A2 . 1 − |ζ|2 (1 − |ζ|2 )2

It follows from (8) that |A2 | > c(1 − |ζ|2 )−2 for some c > 0, and in order to prove (9) it suffices to show that |A1 | < q|A2 |

for some q ∈ (0, 1),

when ζ ∈ ∪n>n0 Dǫ (wn ). The relation (7) yields |A1 | ≤ |ϑ(ζ)|

b 1 − |ϑ(ζ)|2 < |A2 |, 2 2 (1 − |ζ| ) |ζ|

ζ ∈ ∪n Dǫ (wn ).

Since b < 1 and inf{|ζ| : ζ ∈ ∪n>m Dǫ(wn ) } → 1 as m → ∞, the required estimate follows. The inequality (9) proves (6) for the special case ζ = wn . In order to complete the proof consider the function g(w, ζ) = κ′w (ζ) = −

1 − ϑ(ζ)ϑ(w) ϑ′ (ζ)ϑ(w) +w ¯ ¯ 2 . 1 − ζw (1 − ζw)

We have |g(wn , ζ)| = |κ′wn (ζ)|. On the other hand (10)

|g(wn , ζ) − g(ζ, ζ)| < |g ′ (w, ˜ ζ)||ζ − wn |,

for some point w ˜ ∈ [ζ, wn ], where the derivative is taken with respect to the first variable. A straightforward estimate shows const |g′ (w, ˜ ζ)| < , w, ˜ ζ ∈ Dǫ (wn ), (1 − |wn |)3 the constant being independent of n. Now, given any η > 0 we can choose ǫ′ < ǫ such that the right-hand side in (10) does not exceed η(1 − |ζ|2 )−2  when ζ ∈ Dǫ′ (wn ). Taking η sufficiently small we obtain (6). In what follows we assume for simplicity that ϕ(0) = 0. Theorem 1. The following statements are equivalent (C) Cϕ : Kϑ → H 2 is a compact operator. (N) The Nevanlinna counting function of ϕ satisfies (11)

Nϕ (w)

1 − |ϑ(w)|2 → 0 as |w| ր 1. 1 − |w|2

Proof (N) ⇒ (C). Since Nϕ (w)(1 − |(w)|2 )−1 and 1 − |ϑ(w)|2 are bounded, the condition (N) means that for any a < 1 lim

|ϑ(w)| 0 (1 − |ϑ(w)|)p → 0 as |w| ր 1. Nϕ (w) 1 − |w| We use the following inequality, see [8, page 187] and [3], Z 1 − |z| 2 dA(z) + |f (0)|2 , f ∈ Kϑ , |f ′ (z)|2 (12) kf k ≥ Cp (1 − |ϑ(z)|)p D which is valid for some p ∈ (0, 1). In our setting this formula replaces (3). We follow the argument of J. Shapiro; a similar argument for compactness of the composition operator in some weighted spaces of analytic functions can be found in [9, Ch. 3.2]. (n) Let Kϑ = {f ∈ Kϑ ; f has zero of order n at the origin}, and let Π(n) : (n) Kϑ → Kϑ be the corresponding orthogonal projection. We will prove that kCϕ Π(n) kKϑ →H 2 → 0,

n → ∞.

Thus Cϕ is compact as it can be approximated by the finite-rank operators Cϕ (I − Π(n) ). Indeed, given f ∈ Kϑ , kf k = 1, denote gn = Π(n) f . We have kgn k ≤ 1 and, for each R < 1, ǫ > 0 we can choose n(ǫ, R) independent of f and such that |gn (w)| < ǫ, |gn′ (w)| < ǫ, for all n > n(ǫ, R), and |w| < R. It follows from (12) that Z |gn′ (z)|2 D

1 − |z| dA(z) < C, (1 − |ϑ(z)|)p

with C independent of f , n. Next, by (2) we have Z Z Z ′ 2 (n) 2 ≤ + |gn (z)| Nϕ (z)dA(z) ≤ kCϕ Π f k = R