Composition operator on model spaces

0.15 MB·English

by  Yurii I Lyubarskii

COMPOSITION OPERATORS ON MODEL SPACES
YURIII. LYUBARSKIIANDEUGENIA MALINNIKOVA
3
1
Dedicated to Nikolai K. Nikolski on the occasion of his 70th birthday
0
2
n Abstract. Let ϕ : D → D be a holomorphic function, ϑ : D → D be
a an inner function and K (D)=H2(D)⊖ϑH2(D) be the corresponding
ϑ
J model space. Westudy the composition operator C on K and give a
ϕ ϑ
0 necessaryandsufficientconditionforC :K →H2tobecompact. The
ϕ ϑ
3 condition involves an interplay between ϑand theNevanlinnacounting
function of ϕ. For a one-component ϑ a characterization of compact
]
V composition operators Cϕ in terms of the Aleksandrov-Clark measures
of ϕ and thespectrum of ϑ is also given.
C
.
h
t
a
m
1. Introduction
[
LetDbetheunitdiskinthecomplexplane. Givenaholomorphicfunction
3
ϕ: D → D, denote by
v
2 C : f 7→ f ◦ϕ
ϕ
7
1 the composition operator defined on holomorphic functions in D. This op-
5 erator is bounded on the Hardy space H2(D) (see e.g. [15]). One of the
.
5 intensively studied questions is when C is a compact operator on various
ϕ
0
spaces of analytic functions. We refer the reader to the monographs [9,16]
2
1 for the history and basic results on composition operators.
: Loosely speaking C is compact on H2(D) if ϕ(z) does not approach the
v ϕ
i unit circle T too fast as z → T. J. Shapiro [15] quantified this idea by using
X
the Nevanlinna counting function
r
a
N (w) = −log|z|.
ϕ
X
ϕ(z)=w
He proved in particular that C is compact on H2(D) if and only if
ϕ
(1) lim N (w)/(−log|w|) = 0.
ϕ
|w|→1−
2010 Mathematics Subject Classification. Primary 47B33; Secondary 30H10, 30J05,
47A45.
Keywordsandphrases. Compositionoperator,modelspace,Nevanlinnacountingfunc-
tion, 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
2 YURIII.LYUBARSKIIANDEUGENIAMALINNIKOVA
The basic tool in his argument is the Stanton formula
(2) kC fk2 = 2 |f′(z)|2N (z)dA(z)+|f(ϕ(0))|2,
ϕ ϕ
ZD
where A is the normalized area measure. It is obtained from the identity
1
(3) kfk2 =2 |f′(z)|2log dA(z)+|f(0)|2, f ∈ H2(D),
ZD |z|2
by substituting f ◦ϕ in place of f.
Another way to describethe compactness property of C is related to the
ϕ
Aleksandrov-Clark measures of ϕ. These are the positive measures µ on T
α
defined by the relation
α+ϕ(z)
ℜ = P dµ ,
z α
α−ϕ(z) ZT
where P is the Poisson kernel, α ∈ T. We refer the reader to the surveys
z
[13,18] for more details. In [14] D. Sarason showed how C can be treated
ϕ
as an integral operator on the spaces L1(T) and M(T) and proved that C
ϕ
is compact on these spaces if and only if each µ is absolutely continuous.
α
Due to [17], it is further equivalent to C being compact on H2(D) as well
ϕ
as on other Hardy spaces Hp(D), see also [5].
In this article we study the compactness of the operator C : K →
ϕ ϑ
H2(D), where ϑ is an inner function in D and K = H2(D)⊖ϑH2(D) is the
ϑ
corresponding model space. Consider the canonical factorization of ϑ
ξ +z
ϑ(z) = B (z)exp dω(ξ) ,
Λ
(cid:18)ZT ξ −z (cid:19)
where Λ is the zero set of ϑ, B is the corresponding Blachke product, and
Λ
ω is a singular measure on T. Functions in K admit analytic continuation
ϑ
through T\Σ(ϑ), where
Σ(ϑ)= (T∩Clos(Λ))∪supp(ω)
is the spectrum of ϑ (see [12], Lecture 3). Therefore the compactness prop-
erty of C does notsuffer as thevalues of ϕapproach points in T\Σ(ϑ). We
ϕ
quantify this idea below and give a condition that is necessary and sufficient
for the compactness of C : K → H2(D).
ϕ ϑ
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 3
2. Nevanlinna counting function
In this section we give a counterpart of the condition (1) for the operator
C : K → H2(D). The proof follows the ideas of [15]. The main new
ϕ ϑ
ingredientsareestimatesforthereproducingkernelsanditsderivativesgiven
in Lemma 1 below.
Let κ be the reproducing kernel for K ,
w ϑ
1−ϑ(w)ϑ(ζ) 1−|ϑ(w)|2
κ (ζ)= , kκ k2 = ,
w 1−w¯ζ w 1−|w|2
and let κ˜ be its normalized version
w
1−|w|2 1/2 1−ϑ(w)ϑ(ζ)
κ˜ (ζ) = .
w (cid:18)1−|ϑ(w)|2(cid:19) 1−w¯ζ
By
1 (1−|w|2)1/2
(4) k (ζ) = , k˜ (ζ) =
w w
1−w¯ζ 1−w¯ζ
we denote the reproducing kernel for H2 and its normalized version.
Lemma 1. Let {w }⊂ D, |w |→ 1 be such that
n n
(5) |ϑ(w )| <a,
n
for some a ∈ (0,1). Then
w∗
(i) κ˜ −−→ 0 as n → ∞;
wn
(ii) there exist ǫ > 0 , c > 0 and n such that
0
c
(6) |κ′ (ζ)|> , ζ ∈ D (w )
wn (1−|w |2)2 ǫ n
n
holds for any n> n , where D (w) = {ζ;|ζ−w|< ǫ|1−ζ¯w|} is a hyperbolic
0 ǫ
disk with center at w.
Proof. (i) It suffices to show that
(1−|w |2)1/2
n (1−ϑ(w )ϑ(ζ)) −w−→∗ 0 in L2(T) as |w |→ 1.
n n
1−w¯ ζ
n
This in turn follows from the known fact that the normalized reproducing
kernels k˜ for the Hardy space H2(D) tend weakly to 0 as |w | → 1, see
wn n
e.g. [15].
(ii) We start with the following well-known estimate
1−|ϑ(ζ)|2
(7) |ϑ′(ζ)|≤ , ζ ∈ D.
1−|ζ|2
Together with (5) it readily yields
(8) |ϑ(ζ)| < b, ζ ∈ ∪ D (w ),
n ǫ n
for some b < 1 and ǫ > 0.
4 YURIII.LYUBARSKIIANDEUGENIAMALINNIKOVA
We claim now that for sufficiently large n
0
const
(9) |κ′(ζ)| > , ζ ∈ ∪ D (w ).
ζ (1−|ζ|2)2 n>n0 ǫ n
Indeed,
ϑ′(ζ)ϑ(ζ) 1−|ϑ(ζ)|2
κ′(ζ)= − +ζ¯ =A +A .
ζ 1−|ζ|2 (1−|ζ|2)2 1 2
It follows from (8) that |A |> c(1−|ζ|2)−2 for some c > 0, and in order to
2
prove (9) it suffices to show that
|A | < q|A | for some q ∈(0,1),
1 2
when ζ ∈ ∪ D (w ). The relation (7) yields
n>n0 ǫ n
1−|ϑ(ζ)|2 b
|A | ≤ |ϑ(ζ)| < |A |, ζ ∈ ∪ D (w ).
1 (1−|ζ|2)2 |ζ| 2 n ǫ n
Since b < 1 and inf{|ζ| : ζ ∈ ∪ D } → 1 as m → ∞, the required
n>m ǫ(wn)
estimate follows.
The inequality (9) proves (6) for the special case ζ = w . In order to
n
complete the proof consider the function
ϑ′(ζ)ϑ(w) 1−ϑ(ζ)ϑ(w)
g(w,ζ) = κ′ (ζ)= − +w .
w 1−ζ¯w (1−ζ¯w)2
We have |g(w ,ζ)| = |κ′ (ζ)|. On the other hand
n wn
(10) |g(w ,ζ)−g(ζ,ζ)| < |g′(w˜,ζ)||ζ −w |,
n n
for some point w˜ ∈ [ζ,w ], where the derivative is taken with respect to the
n
first variable. A straightforward estimate shows
const
|g′(w˜,ζ)| < , w˜,ζ ∈ D (w ),
(1−|w |)3 ǫ n
n
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). (cid:3)
In what follows we assume for simplicity that ϕ(0) = 0.
Theorem 1. The following statements are equivalent
(C) C :K → H2 is a compact operator.
ϕ ϑ
(N) The Nevanlinna counting function of ϕ satisfies
1−|ϑ(w)|2
(11) N (w) → 0 as |w| ր 1.
ϕ 1−|w|2
Proof (N) ⇒ (C). Since N (w)(1−|(w)|2)−1 and 1−|ϑ(w)|2 are bounded,
ϕ
the condition (N) means that for any a < 1
lim N (w)(1−|w|2)−1 =0.
ϕ
|ϑ(w)|<a,|w|→1−
COMPOSITION OPERATORS ON MODEL SPACES 5
In particular, for any p > 0
(1−|ϑ(w)|)p
N (w) → 0 as |w| ր 1.
ϕ
1−|w|
We use the following inequality, see [8, page 187] and [3],
1−|z|
(12) kfk2 ≥ C |f′(z)|2 dA(z)+|f(0)|2, f ∈ K ,
pZD (1−|ϑ(z)|)p ϑ
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 oper-
ator in some weighted spaces of analytic functions can be found in [9, Ch.
3.2].
Let K(n) = {f ∈ K ;f has zero of order n at the origin}, and let Π(n) :
ϑ ϑ
(n)
K → K be the corresponding orthogonal projection. We will prove that
ϑ ϑ
kC Π(n)k → 0, n → ∞.
ϕ Kϑ→H2
Thus C is compact as it can be approximated by the finite-rank operators
ϕ
C (I −Π(n)).
ϕ
Indeed, given f ∈ K , kfk = 1, denote g = Π(n)f. We have kg k ≤ 1
ϑ n n
and, for each R < 1, ǫ > 0 we can choose n(ǫ,R) independent of f and such
that
|g (w)| < ǫ, |g′ (w)| < ǫ, for all n >n(ǫ,R), and |w| < R.
n n
It follows from (12) that
1−|z|
|g′(z)|2 dA(z) < C,
ZD n (1−|ϑ(z)|)p
with C independent of f, n. Next, by (2) we have
kC Π(n)fk2 = |g′ (z)|2N (z)dA(z) ≤ + ≤
ϕ n ϕ
ZD Z|z|<R ZR<|z|<1
max |g′ (z)|2 N (z)dA(z)+
n ϕ
|z|<R(cid:8) (cid:9)Z|z|<R
(1−|ϑ(z)|)p 1−|z|
max N (z) |g′ (z)|2 dA(z).
R<|z|<1(cid:26) ϕ 1−|z| (cid:27)ZR<|z|<1 n (1−|ϑ(z)|)p
thisislessthanC
ChoosingfirstR suchthatthesecon|dsummandissm{zall, andthennla}rge
enough to provide smallness of the first summand we can make the whole
expression arbitrary small for all f ∈ K , kfk = 1. (cid:3)
ϑ
Proof (C) ⇒ (N). Assume that C is compact but (11) does not hold.
ϕ
Then there exists a sequence {w }⊂ D, |w |→ 1, satisfying
n n
1−|ϑ(w )|2
n
(13) N (w ) > c > 0.
ϕ n 1−|w |2
n
6 YURIII.LYUBARSKIIANDEUGENIAMALINNIKOVA
By the Littlewood subordination principle, which implies that N (w) ≤
ϕ
log 1 , there exists a < 1 such that (5) holds. Applying Lemma 1 (i) and
|w|
the compactness of C , we get kC κ˜ k2 → 0 as n → ∞. On the other
ϕ ϕ wn
hand, (5), Lemma 1 (ii) and the subharmonicity inequality for N (see [15])
ϕ
imply
(14) kC κ˜ k2 ≥ |κ˜′ (ζ)|2N (ζ)dA(ζ) ≥
ϕ wn ZD wn ϕ
c
c |κ′ (ζ)|2(1−|w |2)N (ζ)dA(ζ) ≥ 2 N (ζ)dA(ζ) ≥
1ZD wn n ϕ (1−|wn|2)3 ZDǫ(wn) ϕ
c N (w )
ǫ ϕ n
.
1−|w |2
n
We combine the last estimate with (13) to get a contradiction. (cid:3)
3. Aleksandrov-Clark measures
For α ∈ T let as before µ be the Aleksandrov-Clark measure of ϕ corre-
α
sponding to α and let
dµ = h dm+dσ
α α α
be its decomposition into absolutely continuous and singular parts, where
m is the normalized Lebesgue measure on T. Then
1−|ϕ(ζ)|2
h (ζ)=
α |α−ϕ(ζ)|2
for almost every ζ on T. As above, we assume for simplicity that ϕ(0) = 0,
then kµ k= 1.
a
We give a condition in terms of the Aleksandrov-Clark measures that is
sufficient for the compactness, it is also necessary if ϑ is a one-component
inner function, i.e. the set {z ∈ D : |ϑ(z)| < r} is connected for some r ∈
(0,1).The one-component inner functions were introduced by W. S. Cohn in
[7],seealso[2]foranumberofequivalentcharacterizationsofone-component
inner functions.
Theorem 2. Let ϑ be a one-component inner function. The following state-
ments are equivalent
(C) C :K → H2 is a compact operator.
ϕ ϑ
(S) σ = 0 for all α ∈ Σ(ϑ).
α
Moreover, the implication (S) ⇒ (C) holds for any inner function ϑ.
Theproofmainlyfollowsthepatternasdescribedin[18]section7,see[14]
for the original approach and also [5]. We need the following description of
the spectrum of a one-component inner function.
Lemma A. Let ϑ be a one-component inner function and α ∈ T. The
following statements are equivalent
(a) α ∈ Σ(ϑ); (b) liminf |ϑ(w)| < 1; (c) liminf |ϑ(rα)| < 1.
w→α r→1−
COMPOSITION OPERATORS ON MODEL SPACES 7
The implications (c) ⇒ (a) ⇒ (b) ⇒ (a) are straightforward and hold
for any inner function, see [12], Lecture 3; (a) ⇒ (c) is true when ϑ is
one-component, it follows from [19], Section 5, see also Theorem 1.11 in [2].
Proof (C) ⇒ (S). Fix α ∈ Σ(ϑ) and chose a sequence r → 1 so that
n
|ϑ(αr )| < a < 1. By Lemma 1, we have
n
1−r2 |1−ϑ¯(αr )ϑ(ϕ(ξ))|2
kC κ˜ k2 ≥ n n |dξ| → 0, as n → ∞.
ϕ αrn ZT |1−α¯rnϕ(ξ)|2 1−|ϑ(αrn)|2
Since |ϑ(αr )| < a < 1, this yields
n
1−r2
kC k˜ k2 = n |dξ| → 0, as n → ∞,
ϕ αrn ZT |1−α¯rnϕ(ξ)|2
where k˜ is the normalized reproducing kernel for H2, see (4).
w
The rest of the proof follows literally [5], see also [18], Lemma 7.6. We
give it here for the sake of completeness. We have
1−|r ϕ(ξ)|2 1−|ϕ(ξ)|2
kC k˜ k2 = n |dξ|− r2 |dξ| =:A −B .
ϕ αrn ZT |α−rnϕ(ξ)|2 ZT n|α−rnϕ(ξ)|2 n n
Clearly,
α+r ϕ(0)
n
A = ℜ = 1.
n
(cid:18)α−r ϕ(0)(cid:19)
n
Further, by the monotone convergence theorem
1−|ϕ|2
lim B = = kh k , as n → ∞.
n→∞ n ZT |α−ϕ|2 α 1
Then kσ k = 1−kh k =0. This completes the proof (C) ⇒ (S). (cid:3)
α α 1
We remark that the one-component condition was employed only in the
description of the spectrum, so the following statement holds for any inner
function: If C :K → H2 is a compact operator then σ =0 for all α ∈ T
ϕ ϑ α
such that liminf |ϑ(rα)| < 1.
r→1−
Proof (S) ⇒ (N). We will prove this implication and refer to Theorem 1.
Suppose that (N) is false, then
1−|ϑ(w )|2
N (w ) n > c> 0, for some {w } ⊂D,w → α ∈ T.
ϕ n 1−|w |2 n n
n
Clearly α∈ Σ(ϑ) and (5) holds for some a < 1. Further,
(1−|w |)−1N (w )> c > 0.
n ϕ n 1
We obtain a contradiction in the same way as in [5] see also [18], Theorem
7.5. We have, by a simple version of (14)
N (w )
kC k˜ k2 ≥ C ϕ n > c > 0.
ϕ wn 1−|w |2 2
n
8 YURIII.LYUBARSKIIANDEUGENIAMALINNIKOVA
On the other hand by the Fatou lemma,
1−|ϕ(ξ)|2
limsupkC k˜ k2 =1−liminf |w |2 |dξ| ≤
n→∞ ϕ wn n→∞ ZT n |1−w¯nϕ(ξ)|2
1−|ϕ(ξ)|2
1− |dξ| = kσ k = 0,
ZT |α−ϕ(ξ)|2 α
which leads to a contradiction. (cid:3)
4. Examples and concluding remarks
Inner functions with one point spectra. Consider the Paley-Wiener space
K generated by
ϑ1
z+1
ϑ1(z) = ez−1,
this space can be obtained from the classical Paley-Wiener space of entire
functions by the substitution ζ 7→ z−i. Then ϑ is a one-component inner
z+i 1
function and Σ(ϑ ) = {1}. Theorem 1 and explicit calculation show that
1
C :K → H2 is a compact operator if and only if
ϕ ϑ1
N (w)
ϕ
(15) → 0, as w → 1.
max((1−|w|2),|1−w|2)
Consider now D = {w ∈ D;|w −1/4| < 3/4} and let ϕ be a conformal
mapping ϕ :D → D, φ(0) = 0,ϕ(1) = 1. Clearly (15) does not hold and the
operator C : K → H2 is not compact. Evidently, the Aleksandrov-Clark
ϕ ϑ1
measure µ of ϕ is not absolutely continuous. Below we give an example
1
of a (multi-component) inner function ϑ with Σ(ϑ) = {1} and such that
C : K → H2 is a compact operator. Thus (C) does not imply (S) for
ϕ ϑ
general ϑ.
Take a sequence t ց 0 such that {ζ } = {(1−t3 )1/2eitm} is an inter-
m m m
polating sequence in D. Given a sequence {α } ∈ l1, α ∈ (0,1), denote
m m
Λ= {λ } = {(1−α t3 )1/2eitm}, this is also an interpolating sequence. Let
m m m
now ϑ = B be the Blaschke product corresponding to the sequence Λ. We
Λ
claim that C : K → H2 is a compact operator.
ϕ ϑ
Indeed, kC k˜ k ≤ 1, ζ ∈ D, here k˜ is the normalized reproducing kernel
ϕ ζ ζ
for H2, this follows just from the fact that C is contractive. In particular
ϕ
|dξ|
t3 = kC k˜ k2 ≤ 1.
mZT |1−ζ¯mϕ(ξ)|2 ϕ ζm
Since |1−ζ¯ ϕ(ξ)|2 ≤ c|1−λ¯ ϕ(ξ)|2, ξ ∈T, we have,
m m
|dξ|
kC k˜ k2 ≍ α t3 ≤ Cα .
ϕ λm m mZT |1−λ¯mϕ(ξ)|2 m
On the other hand the system {k˜ } forms a Riesz basis in K (see e.g.
λm ϑ
[12], Lecture VII). Compactness of C : K → H2 is now straightforward,
ϕ ϑ
alternatively it could be deduced from Theorem 1.
COMPOSITION OPERATORS ON MODEL SPACES 9
Concluding remarks. In the classical case of H2(D) the essential norm of
the composition operator was obtained by J. Shapiro
N (w)
kC k2 = limsup ϕ .
ϕ e −log|w|
|w|→1−
For a given one-component ϑ the equivalence of the norms proved in [8] and
similar arguments give
1−|ϑ(w)|2
kC :K → H2k2 ≍ limsupN (w) .
ϕ ϑ e ϕ 1−|w|2
|w|→1−
Let ϕ : D → D be a holomorphic function and ϕ∗ be its radial boundary
values. Defineameasureν onD¯ byν (E) = m((ϕ∗)−1(E))foranyE ⊂D¯,
ϕ ϕ
where m is the Lebesgue measure on T. The composition operator C on
ϕ
H2(D) is isometrically equivalent to the embedding of H2 into L2(D¯,ν ),
ϕ
see [9,11] for details. The connecting between the Nevanlinna counting
function and the measure ν was recently studied in details in [10].
ϕ
Respectively, the compactness of the composition operator on K can
ϑ
be reduced to the question of the compactness of the embedding K ֒→
ϑ
L2(D¯,ν ). It is well-known that the embeddings are easier to study for
ϕ
one-component inner functions ϑ, see [7,8], and subsequent works [19] and
[1,2]. The compactness of the embedding K ֒→ L2(D¯,µ) was studied
ϑ
by J. A. Cima and A. L. Matheson [6] and by A. D. Baranov [4]. The
latterarticlecontainsinparticularnecessaryandsufficientconditionsforthe
compactnessof theembeddingforthecaseofone-componentinnerfunction.
This approach also shows that for one-component ϑ the compactness of the
composition operator C :Kp → Hp does not depend on p ∈(1,∞).
ϕ ϑ
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Department of Mathematical Sciences, Norwegian University of Science
and Technology, NO-7491, Trondheim, Norway
E-mail address: [email protected]
E-mail address: [email protected]