ISOMETRIC COMPOSITION OPERATORS ON BMOA
JUSSI LAITILA
Abstract. We characterize and provide examples of the analytic selfmaps of the unit disk, which induce isometric composition operators on
the space BM OA equipped with a Möbius invariant H 2 norm.
1. Introduction
Let H 2 denote the classical analytic Hardy space on the unit disk D and
let ϕ be an analytic self-map of D. We will consider composition operators
Cϕ : f 7→ f ◦ ϕ on the space BM OA, which consists of the analytic functions
f ∈ H 2 whose boundary functions have bounded mean oscillation on ∂D.
There are a number of equivalent norms on the space BM OA; see [1], [6],
or [7], for example. In this paper we set
kf kBM OA = |f (0)| + kf k∗ ,
where
kf k∗ = sup kf ◦ σa − f (a)k2 .
a∈D
H2
Then f ∈
belongs to BM OA if and only if kf kBM OA < ∞. Above σa is
the conformal automorphism σa (z) = (a − z)/(1 − az) for a, z ∈ D.
It is well known that all composition operators C ϕ are bounded on the
spaces H 2 and BM OA and, if ϕ vanishes at the origin, then C ϕ is a contraction on both of these spaces. Since H 2 is a Hilbert space, it has plenty
of isometries. However, it follows from a result of Ryff [13, Thm. 3] that a
composition operator Cϕ is an isometry of H 2 only if it is induced by an
inner function ϕ which vanishes at the origin. Another argument was given
in [12] (see [10] for further references).
In this paper we study the question of when a composition operator is an
isometry in the norm k · kBM OA . According to a result of Shapiro [14], if
ϕ is an inner function which vanishes at the origin, then it induces such an
operator. Another proof of this fact was given by Kobayashi ([8], [9]) and
an extension to several variables appears in [2]. However, due to counterexamples of Kobayashi the converse of this result fails.
Our goal here is to complement these results by characterizing all isometric
composition operators on the space BM OA. We point out that the general
question of which linear operators are isometries of BM OA still seems to be
open.
2000 Mathematics Subject Classification. Primary 47B33; Secondary 30D50, 30C80.
Key words and phrases. Composition operator, isometry, bounded mean oscillation.
The author was supported by the Academy of Finland, projects 210970 and 118422.
1
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JUSSI LAITILA
The monographs [15] and [4] are standard references for the theory of
composition operators on classical function spaces.
2. Background
Recall that the Hardy space H 2 consists of the analytic functions on D =
{z ∈ C : |z| < 1} whose power series have square summable coefficients. For
every f ∈ H 2 , the radial limit f (ζ) = limr→1 f (rζ) exists almost everywhere
on ∂D and a standard norm on H 2 is given by
1/2
Z
2
.
|f (ζ)| dm(ζ)
kf k2 =
∂D
Here dm is the Lebesgue measure on ∂D normalized so that m(∂D) = 1. We
refer to [5] and [6] for the basic properties of the classical Hardy spaces.
For any analytic self-map ϕ of the unit disk, the corresponding composition operator Cϕ : f 7→ f ◦ ϕ is bounded on H 2 and
(2.1)
1
1 + |ϕ(0)|
≤ kCϕ : H 2 → H 2 k22 ≤
,
2
1 − |ϕ(0)|
1 − |ϕ(0)|
see [4, p. 123]. In particular, Cϕ is a contraction if and only if ϕ(0) = 0.
Bearing this in mind, it follows from [13, Thm. 3] or [12, Thm. 1] that a
composition operator Cϕ is an isometry of H 2 if and only if ϕ is an inner
function and ϕ(0) = 0. (Recall that ϕ is called inner function if |ϕ(ζ)| = 1
for almost every ζ ∈ ∂D.)
We next consider the operators Cϕ on the space BM OA. For this aim fix
an analytic self-map ϕ of the unit disk. It will be useful to associate with ϕ
the family (ϕa )a∈D of analytic maps on D, where
ϕa = σϕ(a) ◦ ϕ ◦ σa ,
for a ∈ D. Then, for each a ∈ D, the map ϕ a takes the unit disk into itself
and ϕa (0) = 0. Since the automorphisms σϕ(a) are self-invertible, we have
ϕ ◦ σa = σϕ(a) ◦ ϕa , so the right-hand inequality in (2.1) gives
kf ◦ ϕ ◦ σa − f (ϕ(a))k2 = kf ◦ σϕ(a) ◦ ϕa − f (ϕ(a))k2
≤ kf ◦ σϕ(a) − f (ϕ(a))k2 ,
for f ∈ BM OA. This yields the well-known estimate
(2.2)
kCϕ f k∗ ≤ kf k∗ ,
for all f ∈ BM OA; see [16, p. 572], for example. As a consequence, every
composition operator is bounded on BM OA and, if ϕ(0) = 0, then C ϕ is a
contraction.
Let us next consider the question of when C ϕ is an isometry of BM OA.
We start with a basic observation.
Proposition 2.1. A composition operator C ϕ is an isometry of BM OA if
and only if ϕ(0) = 0 and kCϕ f k∗ = kf k∗ for all f ∈ BM OA.
ISOMETRIC COMPOSITION OPERATORS ON BMOA
3
Proof. If ϕ(0) = 0 and kCϕ f k∗ = kf k∗ for all f ∈ BM OA, then Cϕ clearly
is an isometry. On the other hand, if C ϕ is an isometry, then
|g(0)| = kgkBM OA − kgk∗ = kCϕ gkBM OA − kgk∗
= |g(ϕ(0))| + kCϕ gk∗ − kgk∗ ≤ |g(ϕ(0))|,
for all g ∈ BM OA, by (2.2). By choosing g = σ ϕ(0) , we get |ϕ(0)| =
|σϕ(0) (0)| ≤ |σϕ(0) (ϕ(0))| = 0. Hence ϕ(0) = 0 and kCϕ f k∗ = kCϕ f kBM OA −
|f (ϕ(0))| = kf kBM OA − |f (0)| = kf k∗ for all f ∈ BM OA.
Hence the problem of characterizing the isometric composition operators
reduces to the problem of characterizing the operators C ϕ which are isometries with respect to the seminorm k · k ∗ . Examples of such operators are
given by the following known result, which implies that if C ϕ is an isometry
of H 2 , then it is an isometry of BM OA.
Proposition 2.2. If ϕ : D → D is an inner function, then kC ϕ f k∗ = kf k∗
for all f ∈ BM OA.
This result follows from a corresponding result due to Shapiro [14, Thm.
4.1] for the Garsia seminorm on BM OA. See also [8, Thm. 1] or [9] for
another proof. To motivate our main result, we sketch a somewhat different
proof, which is based on the H 2 result. First note that, given f ∈ BM OA,
the map a 7→ kf ◦ σa − f (a)k2 is continuous on D. One way to see this is to
write kf ◦ σa − f (a)k22 = hf (a) − |f (a)|2 , where hf is a harmonic function on
D. Indeed, since in the Hilbert space H 2 the orthogonal projection of f ◦ σa
to the constants is f (a), we have
kf ◦ σa − f (a)k22 = kf ◦ σa k22 − |f (a)|2 .
(2.3)
Moreover, a change of variables yields
Z
Z
kf ◦ σa k22 =
|f (σa (ζ))|2 dm(ζ) =
(2.4)
∂D
|a|2 )/|ζ
|f (ζ)|2 Pa (ζ) dm(ζ),
∂D
a|2
where Pa (ζ) = (1 −
−
is the Poisson kernel. It is a classical fact
that, as a function of a, the Poisson integral defines a harmonic map on D.
If ϕ is now an inner function and a ∈ D, then ϕ a = σϕ(a) ◦ ϕ ◦ σa also is
inner and ϕa (0) = 0. Hence the map f 7→ f ◦ ϕa is an isometry of H 2 , so
kf ◦ ϕ ◦ σa − f (ϕ(a))k2 = kf ◦ σϕ(a) ◦ ϕa − f (ϕ(a))k2
= kf ◦ σϕ(a) − f (ϕ(a))k2 ,
for all f ∈ BM OA. By a theorem of Frostman (see [6, p. 80]), the image of
the inner function ϕ is dense in D. Hence, by continuity,
kf ◦ ϕk∗ = sup kf ◦ σϕ(a) − f (ϕ(a))k2 = sup kf ◦ σb − f (b)k2 = kf k∗ ,
a∈D
b∈ϕ(D)
for f ∈ BM OA. This proves the proposition.
In [9], Kobayashi showed that the converse of Proposition 2.2 does not
hold, that is, there are non-inner functions that induce isometric composition
operators on BM OA. We discuss some further examples in Section 4.
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JUSSI LAITILA
3. The main result
The following theorem is the main result of this paper.
Theorem 3.1. The following conditions are equivalent.
(1) kCϕ f k∗ = kf k∗ for all f ∈ BM OA.
(2) kσw ◦ ϕk∗ = 1 for all w ∈ D.
(3) The map ϕ satisfies the following property:
(S) for every w ∈ D, there is a sequence (a n ) in D such that ϕ(an ) →
w and kϕan k2 → 1, as n → ∞,
where ϕan = σϕ(an ) ◦ ϕ ◦ σan for n ∈ N.
Proof. For the implication (1) to (2), we only need to recall that kσ w k∗ = 1
for all w ∈ D. Indeed, given a ∈ D, a simple calculation shows that σ w ◦ σa =
λσb , where b = σa (w) and |λ| = 1. Then σw (a) = λb and
p
kσw k∗ = sup kσw ◦ σa − σw (a)k2 = sup kσb − bk2 = sup 1 − |b|2 = 1.
a∈D
b∈D
b∈D
We next prove that (2) implies (3). Let w ∈ D. Then there is a sequence
(an ) in D such that
lim kσw ◦ ϕ ◦ σan − σw (ϕ(an ))k2 = kσw ◦ ϕk∗ = 1.
n→1
By (2.3),
|σw (ϕ(an ))|2 = kσw ◦ ϕ ◦ σan k22 − kσw ◦ ϕ ◦ σan − σw (ϕ(an ))k22
≤ 1 − kσw ◦ ϕ ◦ σan − σw (ϕ(an ))k22 → 0,
so |σw (ϕ(an ))| → 0 and ϕ(an ) → w, as n → ∞. Since σw ◦ σϕ(an ) = λn σbn ,
where bn = σϕ(an ) (w) and |λn | = 1, we have
(σw ◦ ϕ ◦ σan )(z) − σw (ϕ(an )) = (σw ◦ σϕ(an ) ◦ ϕan )(z) − σw (ϕ(an ))
= λn ((σbn ◦ ϕan )(z) − bn )
(3.1)
= λn
(|bn |2 − 1)ϕan (z)
,
1 − bn ϕan (z)
for z ∈ D. Therefore,
kσw ◦ ϕ ◦ σan − σw (ϕ(an ))k2 ≤ (1 + |bn |)kϕan k2 .
Since |bn | = |σw (ϕ(an ))| → 0, as n → ∞, we get limn→∞ kϕan k2 = 1, so
that ϕ satisfies the property (S).
We finally prove that (3) implies (1). Let f ∈ BM OA and ε > 0 be
arbitrary. Then there is w ∈ D such that
(3.2)
kf k∗ ≤ kf ◦ σw − f (w)k2 + ε
and, by property (S), there is a sequence of points a n in D such that ϕ(an ) →
w and kϕan k2 → 1, as n → ∞.
We next utilize the non-univalent change of variables formula
Z
kg ◦ ψ − g(ψ(0))k22 = 2 |g 0 (z)|2 N (ψ, z) dA(z),
(3.3)
D
ISOMETRIC COMPOSITION OPERATORS ON BMOA
5
which holds for all g ∈ H 2 and analytic maps ψ : D → D; see [15, p. 179] or
[4, p. 35]. Here dA is the normalized Lebesgue measure on D and
X
N (ψ, z) =
− log |w|
(z ∈ D \ {0})
w∈ψ −1 (z)
is the Nevanlinna counting function. In particular, by choosing g(z) = z and
ψ(z) = ϕan (z) in (3.3), we get
Z
kϕan k22 = 2 N (ϕan , z) dA(z).
D
Since 2
R
D (− log |z|) dA(z)
(3.4)
2
Z
D
=4
R1
0
(−r log r) dr = 1, it follows that
Kn (z) dA(z) = 1 − kϕan k22 → 0,
as n → ∞, where
Kn (z) = − log |z| − N (ϕan , z)
(z ∈ D \ {0}).
By a fundamental inequality of Littlewood (see [15, p. 187] or [4, p. 33]),
the counting function satisfies N (ϕ an , z) ≤ − log |z| for z ∈ D \ {0}. Hence
Kn (z) ≥ 0 for z 6= 0 and (3.4) implies that K n → 0 a.e. on D, as n → ∞.
By choosing g(z) = (f ◦ σw )(z) and ψ(z) = z in (3.3), we get
Z
2 |(f ◦ σw )0 (z)|2 (− log |z|) dA(z) = kf ◦ σw − f (w)k22
D
and, by choosing ψ(z) = ϕan (z), we get
Z
2 |(f ◦ σw )0 (z)|2 N (ϕan , z) dA(z) = kf ◦ σw ◦ ϕan − f (w)k22 .
D
Since
Z
Z
0
2
2 |(f ◦ σw ) (z)| Kn (z) dA(z) ≤ 2 |(f ◦ σw )0 (z)|2 (− log |z|) dA(z)
D
D
≤ kf k∗ < ∞,
the dominated convergence theorem yields
kf ◦ σw − f (w)k2 − kf ◦ σw ◦ ϕan − f (w)k2
Z
= 2 |(f ◦ σw )0 (z)|2 Kn (z) dA(z) → 0,
D
as n → ∞. Combining this with (3.2) yields
(3.5)
kf k∗ ≤ lim kf ◦ σw ◦ ϕan − f (w)k2 + ε.
n→∞
We claim that
(3.6)
lim kf ◦ σw ◦ ϕan − f (w)k2 ≤ kCϕ f k∗ .
n→∞
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JUSSI LAITILA
By (2.1),
kf ◦ σw ◦ ϕan − f (w)k2 ≤ kf ◦ σw ◦ ϕan − f ◦ ϕ ◦ σan k2
+ kf ◦ ϕ ◦ σan − f (ϕ(an ))k2
+ |f (ϕ(an )) − f (w)|
≤ kf ◦ σw − f ◦ σϕ(an ) k2
+ kCϕ f k∗ + |f (ϕ(an )) − f (w)|,
for n ∈ N, where |f (ϕ(an )) − f (w)| → 0 as n → ∞. Hence we need to check
that
(3.7)
lim sup kf ◦ σw − f ◦ σϕ(an ) k2 = 0.
n→∞
Let δ > 0 be arbitrary. Since the analytic polynomials on D form a dense
subset of H 2 , we can apply (2.1) to find a polynomial p and a number N
such that
k(f − p) ◦ σw − (f − p) ◦ σϕ(an ) k2 ≤ (M (w) + M (ϕ(an )))kf − pk2 ≤ δ,
for all n ≥ N , where M (z)2 = (1 + |z|)/(1 − |z|). Moreover, an elementary
calculation shows that
4|w − ϕ(an )|
,
kσw − σϕ(an ) k2 ≤ sup |σw (z) − σϕ(an ) (z)| ≤
(1 − |w|)(1 − |ϕ(an )|)
z∈D
so that kp ◦ σw − p ◦ σϕ(an ) k2 → 0, as n → ∞. Hence
lim sup kf ◦ σw − f ◦ σϕ(an ) k2 ≤ lim sup k(f − p) ◦ σw − (f − p) ◦ σϕ(an ) k2
n→∞
n→∞
+ lim sup kp ◦ σw − p ◦ σϕ(an ) k2 ≤ δ.
n→∞
Since δ was arbitrary, this yields (3.7) and (3.6).
By combining (3.5) with (3.6) and letting ε → 0, we get kf k ∗ ≤ kCϕ f k∗ .
Since we also have kCϕ f k∗ ≤ kf k∗ , by (2.2), and f ∈ BM OA was arbitrary,
we obtain (1).
The classical Bloch space B consists of the analytic functions f : D → C
for which
kf kB = |f (0)| + sup(1 − |z|2 )|f 0 (z)| < ∞.
z∈D
Isometric composition operators on B have been studied in the recent articles
[3] and [11], where different characterizations have been obtained for maps
ϕ which induce isometric composition operators on B. By [3], these are
precisely the maps ϕ for which ϕ(0) = 0 and kϕk B = 1. By [11], this is
equivalent to the requirements that ϕ(0) = 0 and ϕ either is a rotation of
the unit disk or satisfies the following property:
(M) for every w ∈ D, there is a sequence (a n ) in D such that |an | → 1,
ϕ(an ) → w, and |ϕ∗ (an )| → 1, as n → ∞.
Here ϕ∗ (z) = (1−|z|2 )ϕ0 (z)/(1−|ϕ(z)|2 ) is the hyperbolic derivative of ϕ. Either of these characterizations easily imply that an isometric composition operator on B is an isometry of BM OA. For example, property (M) implies (S),
since by the Cauchy integral formula, we have |ϕ ∗ (an )| = |ϕ0an (0)| ≤ kϕan k2 .
ISOMETRIC COMPOSITION OPERATORS ON BMOA
7
Also a rotation of the unit disk clearly satisfies (S). By Proposition 2.1 and
Theorem 3.1, we arrive at the following result.
Corollary 3.2. If Cϕ is an isometry of B, then it is an isometry of BM OA.
Remark 3.3. Theorem 3.1 can be used for deducing a short proof of Proposition 2.2. In fact, if ϕ is inner, then by Frostman’s theorem, for each w ∈ D,
there is a sequence (an ) such that ϕ(an ) → w. Moreover, for each n, the
map ϕan is inner, so
Z
2
(n ∈ N).
(1 − |ϕan |2 ) dm = 0
1 − kϕan k2 =
∂D
Hence ϕ satisfies the property (S), so kC ϕ f k∗ = kf k∗ for all f ∈ BM OA.
4. Composition operators induced by non-inner functions
When does a non-inner function induce an isometric composition operator
on BM OA? It turns out that if ϕ is not an inner function and it satisfies the
property (S), then it also satisfies the following slightly stronger property:
(S’) for every w ∈ D, there is a sequence (a n ) in D such that |an | → 1,
ϕ(an ) → w, and kϕan k2 → 1, as n → ∞.
Indeed, assume that ϕ satisfies (S), but it does not satisfy (S’). Let w ∈ D
and let (an ) be a sequence in rD for some r < 1 such that ϕ(a n ) → w and
kϕan k2 → 1, as n → ∞, Then there are a point a0 ∈ D and a subsequence,
still denoted by (an ), such that an → a0 ∈ D, as n → ∞. Hence σw (ϕ(a0 )) =
0 and (3.1) shows that
(1 − |σϕ(an ) (w)|)kϕan k2 ≤ kσw ◦ ϕ ◦ σan − σw (ϕ(an ))k2 ≤ 1.
Since the left-hand side approaches 1, as n → ∞, and the map a 7→ kσ w ◦
ϕ ◦ σa − σw (ϕ(a))k2 is continuous on D, we get
kσw ◦ ϕ ◦ σa0 k2 = lim kσw ◦ ϕ ◦ σan − σw (ϕ(an ))k2 = 1.
n→∞
This is possible only if σw ◦ ϕ ◦ σa0 , and therefore ϕ, is inner.
On the other hand, an inner function satisfies (S’) only if it is not a
unimodular constant times a finite Blaschke product.
Corollary 4.1. A composition operator C ϕ satisfies kCϕ f k∗ = kf k∗ for all
f ∈ BM OA if and only if either ϕ satisfies the property (S’) or ϕ = λB,
where λ ∈ C, |λ| = 1, and B is a finite Blaschke product.
We next give some concrete examples of non-inner functions which induce
isometric composition operators on BM OA. Such examples have earlier
been established by Kobayashi ([8], [9]). In [9] such a map was constructed
as a composition of an inner function and a map from D onto its subset. The
following example is a slight improvement of [8, Thm. 2], where it was shown
that for a class of non-inner functions ϕ, the equality kC ϕ gk∗ = kgk∗ holds
for all g in the disk algebra A. But since σ w ∈ A for all w ∈ D, Theorem
3.1 yields that kCϕ f k∗ = kf k∗ for all f ∈ BM OA. Our approach is simpler,
since we only need to check that the maps ϕ satisfy the property (S).
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JUSSI LAITILA
Example 4.2. Let λ be an inner function with singularity at 1. Let ψ : D → D
be an analytic map, not an inner function, which extends countinuously to D
and which satisfies ψ(1) = 1. Then the map ϕ = ψλ is not an inner function,
but it satisfies the property (S).
For example, the map
z(z + 1)
ϕ(z) =
exp
2
z+1
z−1
is of the above form. Moreover, since ϕ(0) = 0, the corresponding operator
Cϕ is an isometry of BM OA, by Proposition 2.1 and Theorem 3.1.
Proof of Example 4.2. Since λ has a singularity at 1, there is a sequence (a n )
such that an → 1 and λ(an ) → w, by Frostman’s theorem; see [6, p. 80].
Thus ϕ(an ) → w. We claim that kϕan k2 → 1. Since λ is an inner function,
also σϕ(an ) ◦ λ ◦ σan is inner for all n, so kσϕ(an ) ◦ λ ◦ σan k2 = 1. By setting
Fn (ζ) = |(σϕ(an ) ◦ λ)(ζ)|2 − |(σϕ(an ) ◦ ϕ)(ζ)|2 for ζ ∈ ∂D, we get
Z
Z
2
2
2
Fn Pan dm,
|σϕ(an ) ◦ λ ◦ σan | − |ϕan | dm =
1 − kϕan k2 =
∂D
∂D
where Pan is the Poisson kernel; see (2.4). Hence it suffices to show that
the integral on the right-hand side tends to 0, as n → ∞. Let ε > 0 be
arbitrary. For almost every ζ ∈ ∂D, we have |λ(ζ) − ϕ(ζ)| = |1 − ψ(ζ)|, so
an elementary calculation gives
Fn (ζ) ≤ 2|σϕ(an ) (λ(ζ)) − σϕ(an ) (ϕ(ζ))| ≤ 2
1 + |ϕ(an )|
|1 − ψ(ζ)|.
1 − |ϕ(an )|
Hence the continuity of ψ on ∂D yields t > 0 and a number N 1 such that
Fn (ζ) < ε for almost every ζ in E := {ζ ∈ ∂D : |ζ − 1| ≤ t} and all n ≥ N 1 .
On the other hand, since PanR converges to 0 uniformly on ∂D \ E, as n → ∞,
there is N2 ≥ N1 such that ∂D\E Fn Pan dm < ε, for n ≥ N2 . Hence
Z
Z
2
Pan dm ≤ 2ε,
Fn Pan dm ≤ ε + ε
1 − kϕan k2 =
E
∂D
for all n ≥ N2 . Since ε > 0 was arbitrary, this completes the proof.
Final remark. For 1 ≤ p < ∞, the quantity
kf k∗,p = sup kf ◦ σa − f (a)kp ,
a∈D
defines an equivalent seminorm on BM OA; see [1], [6], or [7]. Here k · k p is
the norm of the Hardy space H p . Hence kf kBM OAp = |f (0)| + kf k∗,p is an
equivalent norm on BM OA for all 1 ≤ p < ∞. By results of Kobayashi ([8],
[9]), it is known that Proposition 2.2 and examples similar to Example 4.2
hold for the norm k · kBM OAp for all 1 ≤ p < ∞. It remains unclear to the
author whether Theorem 3.1 could be in some way extended to p 6= 2 (with
suitable modifications).
Acknowledgement
I thank Hans-Olav Tylli for his careful reading of and comments on a
preliminary version of this paper.
ISOMETRIC COMPOSITION OPERATORS ON BMOA
9
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Department of Mathematics and Statistics, P.O. Box. 68 (Gustaf Hällströminkatu 2b), FIN-00014 University of Helsinki, Finland
E-mail address: [email protected]