Composition Operators on the Fock Space - Mathematics - San ...

<strong>Composition</strong> <strong>Operators</strong> on the Fock Space
Brent Carswell Barbara D. MacCluer Alex Schuster
Abstract
We determine the holomorphic mappings of C n that induce bounded composition
operators on the Fock space in C n . Furthermore, we determine which
of these composition operators are compact, and we compute the operator
norm of all bounded composition operators in this setting. We also consider
extensions of these results in various generalizations of the Fock space.
1 Introduction
<strong>Composition</strong> operators on spaces of analytic functions have been studied in
many settings. Much has been written about the properties of these operators
on the Hardy, Bergman, and Bloch spaces on the unit disk in the complex
plane, or unit ball in Cn (see, for example [3], [10] or [7]). Our purpose here is
to study composition operators on the Fock space in Cn . We will determine
which composition operators are bounded, and which are compact. We will
also compute the norm of every bounded composition operator in this setting,
an interesting result since finding the norm of such an operator on the Hardy
and Bergman spaces is still largely an open problem (see [1], [2] or [4] for
recent norm results in the Hardy space setting). We remark that while we
will be doing our analysis in the several variable setting, we will make every
effort to interpret our results and methods in the case n = 1 in order to
clarify our ideas.
The Fock space F 2 n is the Hilbert space of all holomorphic functions on
Cn with inner product
〈f, g〉 ≡ 1
(2π) n
n
1
−
f(z)g(z)e 2 |z|2
dν(z).
1

Here ν denotes Lebesgue measure on C n . To simplify notation we will often
use F instead of F 2 n, and we will denote by f the norm of f.
The reproducing kernel functions for the Fock space are given by
kw(z) = e 〈z,w〉/2 ,
where 〈z, w〉 = n
1 zjwj. Note that the substitution f = kw into the reproducing
formula 〈f, kw〉 = f(w), which holds for all f ∈ F and w ∈ C n , leads
to the identity kw = exp(|w| 2 /4).
For a given holomorphic mapping ϕ : C n → C n , the composition operator
Cϕ : F → F is given by Cϕ(f) = f ◦ϕ. Our first main result (see Theorem 1)
will show that if the operator Cϕ is bounded, then ϕ must be of the form
ϕ(z) = Az + B, where A is an n × n matrix and B is an n × 1 vector.
Furthermore, it will follow that A ≤ 1 for bounded Cϕ, and that B will be
restricted by the condition that 〈Aζ, B〉 = 0 for any ζ in C n with |Aζ| = |ζ|.
Theorem 1 will also show that if Cϕ is compact, then A < 1 with no
restriction on B. Our second result will be Theorem 2, which gives the
converse to Theorem 1. In our third main result, Theorem 4, we determine
the operator norm of Cϕ (where ϕ(z) = Az + B is chosen so that Cϕ is
bounded) in terms of A and B. Finally, in the last section, we will outline
some ideas about how to extend our results to various generalizations of the
Fock space.
We thank Christopher Hammond for several helpful remarks on the proof
of Theorem 4.
2 Main results
Our first result concerns the boundedness and compactness of composition
operators on F.
Theorem 1. Suppose ϕ : C n → C n is a holomorphic mapping.
(a) If Cϕ is bounded on F, then ϕ(z) = Az +B, where A is an n×n matrix
and B is an n × 1 vector. Furthermore, A ≤ 1, and if |Aζ| = |ζ| for
some ζ in C n , then 〈Aζ, B〉 = 0.
(b) If Cϕ is compact on F, then ϕ(z) = Az + B, where A < 1.
2

Note that when n = 1, Theorem 1(a) simply says that if Cϕ is bounded
on F, then ϕ(z) = az + b, where |a| ≤ 1, and if |a| = 1, then b = 0.
In the proof of Theorem 1 we will use the fact that every entire function
on Cn has a multi-index power series expansion of the form
α c(α)zα , where
α denotes a multi-index (α1, α2, · · · , αn) of non-negative integers, and z α =
z α1
1 z α2
2 · · · z αn
n . We use the standard notation |α| = α1 + · · · + αn and α! =
α1! · · · αn!. The quantity |α| is referred to as the total order of the multi-index
α.
Proof of Theorem 1. If Cϕ is bounded on F, then
sup
w∈n C∗ ϕ(kw)
kw
= sup
w∈n kϕ(w)
kw
= sup
w∈n
1 2 2
exp |ϕ(w)| − |w|
4
< ∞, (1)
where the first equality follows from the easily verified property C ∗ ϕ(kw) =
kϕ(w). From (1) it follows that
lim sup
|w|→∞
|ϕ(w)|
|w|
≤ 1. (2)
For each ζ ∈ ∂Bn and 1 ≤ j ≤ n, we define the analytic function ϕ j
ζ by the
equation ϕ j
ζ (λ) = ϕj(λζ) for λ ∈ C, where ϕj is the jth coordinate function
of ϕ. By (2) we must have
lim sup
|λ|→∞
|ϕ j
ζ (λ)|
|λ|
≤ 1.
If ϕj(z) =
α c(α)zα has homogeneous expansion ∞ s=0 Fs(z) (defined by
Fs(z) =
|α|=s c(α)zα ), then since ϕ j
ζ (λ) = λsFs(ζ), we must have Fs(ζ) =
0 for all s ≥ 2 and ζ ∈ ∂Bn; that is Fs ≡ 0 for s ≥ 2 and each coordinate
function ϕj is linear. This proves that ϕ(z) = Az+B as desired. If |Aζ| > |ζ|
for some ζ of norm 1, then setting w = tζ, t > 0 in (2) and letting t → ∞ we
obtain a contradiction. Thus we must have A ≤ 1.
Next we show that if |Aζ| = |ζ|, then 〈Aζ, B〉 = 0. As a special case
of this, suppose Aζ = λζ where λ is a complex number of modulus 1. If
〈Aζ, B〉 = 0, we may choose ρ ∈ C, |ρ| = 1 so that ρ〈Aζ, B〉 > 0. Considering
w = tρζ as t → ∞, we obtain a contradiction to (1). Now suppose Aζ = η,
where |ζ| = |η| = 1. Let U be a unitary map of Cn such that Uη = ζ. Then
for τ(z) ≡ ϕ ◦ U(z) = A ◦ U(z) + B, we have Cτ bounded on F and, since
3

A(U(η)) = η, by the special case just considered, we have 〈AUη, B〉 = 0 or
〈Aζ, B〉 = 0 as desired. This completes the proof of (a).
To prove (b) we need only show that if ϕ(z) = Az + B induces a compact
composition operator on F, then A < 1. Since the normalized reproducing
kernel functions kw/kw tend to 0 weakly as |w| → ∞, we have
1
Cϕe ≥ lim sup exp
|w|→∞ 4 (|ϕ(w)|2 − |w| 2
) ,
where for an operator T , T e = inf{T −Q : Q is compact} is the essential
norm of T . (c.f. Proposition 3.13 in [3] for the proof of a similar fact in
other spaces). Now suppose that A = 1 and that ζ = 0 satisfies |Aζ| = |ζ|.
Let w = λζ, where λ ∈ C and λ〈Aζ, B〉 ≥ 0. Letting |λ| → ∞, we see that
Cϕe = 0, and so A is not compact. Thus A < 1.
The converse to Theorem 1 holds as well.
Theorem 2. Suppose ϕ(z) = Az + B, where A is an n × n matrix with
A ≤ 1, and B is an n × 1 vector.
(a) If 〈Aζ, B〉 = 0 whenever |Aζ| = |ζ|, then Cϕ is bounded on F.
(b) If A < 1, then Cϕ is compact on F.
Before we give the proof of this theorem, we need several preliminary
results that will be of use throughout the rest of this section. First we recall
the notion of the singular value decomposition of an n × n matrix.
Theorem 3. If A is an n × n matrix of rank k, then A can be written as
A = V ΣW , where V, W are n × n unitary matrices, and Σ is a diagonal
matrix (σij) with σ11 ≥ σ22 ≥ · · · ≥ σkk > σk+1,k+1 = · · · = σnn = 0. The
σii are the non-negative square roots of the eigenvalues of AA ∗ ; if we require
that they be listed in decreasing order, then Σ is uniquely determined from A.
A proof can be found in [6]. For brevity of notation we will write σi for
σii, the i th diagonal entry of Σ.
Note that if A ≤ 1, then the σi will all be less than or equal to 1, and
at least one will equal 1 if A = 1. We make two notational definitions that
will be in force for the rest of this section. Set
j = max{r : σr = 1}, (3)
4

(where j = 0 in the case that no σr = 1), and
k = max{r : σr > 0}, (4)
so that k = rank A.
The singular value decomposition will allow us to perform a useful normalization
in proving Theorem 2. This normalization will also play a role
later in this section when we compute the norm of Cϕ on F. We remark
that in the one variable setting this normalization is not needed, though its
presence there does no harm.
Proposition 1. Suppose ϕ(z) = Az +B, where A and B are as described in
(a) of Theorem 1. Let ψ(z) = Σz+B ′ , where the singular value decomposition
of A is V ΣW , and B ′ = V ∗ B. Then on F, Cϕ = CW CψCV , where CW and
CV are the unitary operators given by CW (f) = f ◦ W and CV (f) = f ◦ V .
Proof. We have
CW CψCV (f)(z) = (f ◦ V ◦ ψ ◦ W )(z) = f(V ΣW (z) + V B ′ )
= f(Az + B) = Cϕ(f)(z)
as desired, where we have used the relationship V B ′ = B.
Corollary 1. The operator Cϕ is bounded (resp., compact) on F if and only
if Cψ is bounded (resp., compact) on F. Moreover, the norm of Cϕ is equal
to the norm of Cψ.
If ϕ(z) = Az + B and ψ(z) = Σz + B ′ , where Σ and B ′ are as given in
Proposition 1, we call ψ a normalization of ϕ. The following lemma gives
the reformulation of the hypothesis “〈Aζ, B〉 = 0 whenever |Aζ| = |ζ|” from
Theorem 2 as applied to the normalization ψ.
Lemma 1. Suppose that ϕ(z) = Az + B satisfies the hypothesis of (a) of
Theorem 2, and suppose ψ = Σz + B ′ is a normalization of ϕ. Then the first
j coordinates of B ′ are 0.
Proof. Let A have singular value decomposition A = V ΣW as in Theorem 3.
Since |Aζ| = |ζ| implies 〈Aζ, B〉 = 0, we have
0 = 〈ΣW ζ, V ∗ B〉 = 〈ΣW ζ, B ′ 〉 (5)
5

whenever |Aζ| = |ζ|. If ζ is chosen so that W ζ = (0, · · · , 1, · · · , 0) t (where the
1 appears in the m th position, 1 ≤ m ≤ j), then |Aζ| = |ζ| and ΣW ζ = W ζ,
so that (5) implies that the m th coordinate of B ′ is 0.
Proof of Theorem 2. By Corollary 1 and the remarks following it, to
prove (a) we need only show that if ψ(z) = Σz + B ′ , where Σ is as described
in Proposition 1 and the first j coordinates of B ′ are 0 (where j is defined
by (3)), then Cψ is bounded on F. If Σ is invertible, this will follow from a
change of variables argument. We have
|f ◦ ψ(z)| 2 1
−
e 2 |z|2
dν(z) = c |f(w)| 2 exp
n
= c ′
n
n
− 1
−1 ′
Σ (w − B ) 2
| dν
2
|f(w)| 2
exp − 1
2 (|Σ−1w| 2 − |w| 2 − 2Re〈Σ −1 w, Σ −1 B ′ 〉)
1
−
e 2 |w|2
dν
where c = (det Σ) −2 and c ′ = c exp(−|Σ −1 B ′ | 2 /2). It suffices to show that
|Σ −1 w| 2 − |w| 2 − 2Re〈Σ −1 w, Σ −1 B ′ 〉 (6)
is bounded away from −∞ as w ranges over C n . By Lemma 1 we see that
the expression in (6) attains its minimum at points w ∈ C n satisfying
|wm| = |b′ m|
1 − σ 2 m
and wmb ′ m ≥ 0
for j + 1 ≤ m ≤ n. This gives the desired result.
If Σ is not invertible, then we have σm = 0 for k < m ≤ n, where k is
as defined in (4). Let Σ1 = Σ and Σ2 be the diagonal matrix with Σ1 = Σ2
except in the last n − k positions along the diagonal, where Σ1 has entries
equal to 0 and Σ2 has entries equal to 1
2
. By the argument just finished for
the invertible case, Cψ2 is bounded on F, where ψ2(z) = Σ2z + B ′ . Thus if
f = c(α)z α is in F, so is f ◦ ψ2. Notice we can write f ◦ ψ2 as g + h, where
g is
α
c(α)z α1
1 · · · z αj
j (σj+1zj+1 + b ′ j+1) αj+1 · · · (σkzk + b ′ k) αk (b ′ k+1) αk+1 · · · (b ′ n) αn
and h = f ◦ ψ2 − g. The terms of h involve at least one of the variables
zk+1, · · · , zn, but the terms of g do not. The orthogonality of the monomials
6

in F then implies that f ◦ ψ2 ≥ g. But g = f ◦ ψ, and so f ◦ ψ ∈ F. By
the closed graph theorem we conclude that Cψ is bounded on F.
Finally we prove (b). Again first suppose that Σ is invertible. To show
Cψ is compact, it suffices to show that a sequence {fn} that is bounded in F
and converges to 0 uniformly on compact subsets of C n has its image under
Cψ converge to 0 in norm. Changing variables as above, we have
fn ◦ ψ 2
= c
n |fn(w)| 2 1
−
e 2 |Σ−1w−Σ−1B ′ | 2
e 1
2 |w|2
1
−
e 2 |w|2
dν(w).
Since σi < 1 for all 1 ≤ i ≤ n, a calculation shows that
1
−
e 2 |Σ−1w−Σ−1B ′ | 2
e 1
2 |w|2
is as small as desired off a compact subset of C n . Since fn converges to
0 uniformly on compact sets, and fn is bounded, this guarantees that
fn ◦ ψ → 0. In the case Σ is not invertible, we again compare Cψ to Cψ2,
as defined above. Then fn ◦ ψ → 0, since fn ◦ ψ2 → 0 and fn ◦ ψ2 ≥
fn ◦ ψ.
We turn now to the issue of norm calculations.
Theorem 4. Suppose ϕ(z) = Az + B, where either A < 1 and B is
arbitrary, or A = 1 and 〈Aζ, B〉 = 0 whenever |Aζ| = |ζ|. Then on F we
have
1
Cϕ = exp |w0|
4
2 − |Aw0| 2 + |B| 2
, (7)
where w0 is any solution to (I − A ∗ A)w = A ∗ B.
Before turning to the proof of Theorem 4, we make several remarks. First
note that Theorem 4 is trivial in the case that A is unitary, so henceforth we
will assume A is not unitary.
Next note that in one variable, the equation (1 − aa)w = ab has the
unique solution w = ab/(1 − |a| 2 ) if |a| < 1, so that
1 |b|
Cϕ = exp
4
2
1 − |a| 2
.
7

In several variables it is natural to ask whether the equation (I−A ∗ A)w =
A ∗ B always has a solution under our hypothesis on ϕ, and if so, whether the
solution uniquely determines the expression in (7). Indeed, since A ∗ B is
orthogonal to the kernel of I − A ∗ A, a solution will always exist. To verify
that the expression given in Equation 7 is constant on the solution set of
(I − A ∗ A)w = A ∗ B, note that if
(I − A ∗ A)w0 = A ∗ B = (I − A ∗ A)w1
then w0 − w1 ∈ ker (I − A ∗ A), and thus 〈w0 − w1, A ∗ B〉 = 0, so that
|w0| 2 −|Aw0| 2 = 〈w0, w0−A ∗ Aw0〉 = 〈w0, A ∗ B〉 = 〈w1, A ∗ B〉 = |w1| 2 −|Aw1| 2 .
In order to prove Theorem 4, we will use a normalization of ϕ as we
did in the proof of Theorem 2. Indeed, let ψ = Σz + B ′ where Σ =
diag {σi} with σ1 = · · · = σj = 1 and σj+1, · · · , σn < 1, and with B ′ =
(0, · · · , 0, b ′ j+1, · · · , b ′ n) t . We first observe that if (I − A ∗ A)w0 = A ∗ B and
(I − Σ ∗ Σ)w = Σ ∗ B ′ , then
|w0| 2 − |Aw0| 2 + |B| 2 = |w| 2 − |Σw| 2 + |B ′ | 2 .
This, together with the fact that Cϕ = Cψ, implies that in order to prove
Theorem 4 it suffices to prove it for the normalization ψ(z) = Σz + B ′ .
We next observe that solutions to (I − Σ ∗ Σ)w = Σ ∗ B ′ are easily seen to
be
wm =
Furthermore, for such w we have
σmb ′ m/(1 − σ 2 m) for m ≥ j + 1
arbitrary for m ≤ j.
|w| 2 − |Σw| 2 + |B ′ | 2 =
n
m=j+1
|b ′ m| 2
1 − σ2 .
m
This implies that in order to prove Theorem 4 we need only show that
′
1 |b j+1|
Cψ = exp
4
2
1 − σ2 j+1
+ · · · + |b′ n| 2
1 − σ2
,
n
(8)
which we will turn to shortly.
The proof of Theorem 4 will require a representation for the adjoint of
Cϕ, which we obtain in the following result.
8

Lemma 2. If ϕ(z) = Az + B where A, B satisfy the hypothesis of Theorem
1(a), so that Cϕ is bounded on F, then C∗ ϕ = MkBCτ, where τ(z) = A∗z and
is multiplication by the kernel function kB.
MkB
Proof. The proof follows by checking that C ∗ ϕ(kw) = exp(〈z, Aw + B〉/2) =
MkB Cτ(kw) for all w ∈ C n . Since the span of the kernel functions is dense in
F, the result follows.
The next result gives a lower bound for Cψ, where ψ is a normalization
of ϕ.
Lemma 3. Suppose ϕ(z) = Az + B satisfies the hypothesis of Theorem 4
and let ψ be a normalization. We have
Cψ ≥ sup
w∈n C∗ ψ (kw)
′
1 |b j+1|
= exp
kw 4
2
1 − σ2 + · · · +
j+1
|b′ n| 2
1 − σ2
.
n
Proof. We have
C∗ 2
ψ (kw)
kw2 2
kψ(w) 1 2 2
= = exp |ψ(w)| − |w|
kw2 2
.
Since Σ = diag(σi) with σi = 1 for i ≤ j, where j defined is as in Equation 3,
it is easy to check that |ψ(w)| 2 − |w| 2 attains its maximum at points w ∈ C n
which satisfy, for j + 1 ≤ m ≤ n,
and
|wm| = σm|b ′ m|
1 − σ 2 m
arg wm chosen so that wmb ′ m ≥ 0.
Hence the maximum value of |ψ(w)| 2 − |w| 2 is then
|B ′ | 2 +
This gives the desired result.
n
m=j+1
σ 2 m|b ′ m| 2
1 − σ 2 m
=
n
m=j+1
|b ′ m| 2
1 − σ2 .
m
For these normalized maps ψ we will compute Cψ by identifying a
reducing subspace M for Cψ and computing Cψ|M and Cψ|N, where N
is the orthogonal complement of M. The next lemma identifies this subspace
M.
9

Lemma 4. Let ψ(z) = Σz + B ′ be a normalization of ϕ, where ϕ satisfies
the hypothesis of Theorem 4. Define j, possibly 0, by Equation (3). Let
M = {f ∈ F : f depends on zj+1, zj+2, · · · , zn only}.
Then M is a reducing subspace for Cψ and the restriction of Cψ to M is
compact.
Proof. If f is in M, then the power series expansion of f about 0 has
the form c(α)zα where the sum is over multi-indices α = (α1, · · · , αn)
satisfying α1 = · · · = αj = 0. That Cψ(M) ⊂ M is then immediate from the
form of Σ.
To see that C∗ ψ (M) ⊂ M, let f ∈ M and use Lemma 2 to write
C ∗ ψ(f)(z) = kB ′(z)f(Σz) = exp(〈z, B′ 〉/2)f(Σz).
Since the first j coordinates of B ′ are 0, this is a function in M. Thus M is
a reducing subspace for Cψ.
To see that Cψ restricted to M is compact, consider the map ˜ ψ obtained
from ψ by replacing any 1 ′ s along the diagonal of Σ by 1/2 ′ s. By Theorem
2(b), C ˜ ψ is compact on F. Since Cψ and C ˜ ψ agree on the subspace M,
and the restriction of a compact operator to a closed subspace is compact,
the result follows.
Corollary 2. Let ψ and M be as in Lemma 4. Then
1
Cψ|M ≥ exp
4
n |b ′ m| 2
1 − σ2
.
m
Proof. When w ∈ C N satisfies
and
m=j+1
wm = 0 for 1 ≤ m ≤ j
|wm| = σm|b ′ m|
1 − σ 2 m
for j + 1 ≤ m ≤ n
wmb ′ m ≥ 0
10

then kw ∈ M, and by the calculations in the proof of Lemma 3 we have
Cψ|M ≥ C∗ ψ (kw)
n 1 |b
= exp
kw 4
′ m| 2
1 − σ2
.
m
m=j+1
In the next lemma we compute the norm of Cψ restricted to M. We
remark that in one variable this lemma provides the proof of Theorem 4,
since then M = F if |a| < 1.
Lemma 5. Let T be the restriction of Cψ to M, for ψ and M as given in
Lemma 4. Then the norm of T is given by
′
1 |b j+1|
T = exp
4
2
1 − σ2 + · · · +
j+1
|b′ n| 2
1 − σ2
.
n
Proof. Since T T ∗ is a positive, compact, self-adjoint operator, T T ∗ =
T 2 is an eigenvalue for T T ∗ and there exists F ∈ M such that (T T ∗ )F =
T 2 F . By Lemma 2 we have
kB ′(Σz + B′ )F (Σ(Σz + B ′ )) = T 2 F (z). (9)
Evaluate both sides of this equation at any point z0 with (I −Σ ∗ Σ)z0 = Σ ∗ B ′ .
If F (z0) = 0 we obtain immediately that T 2 = kB ′(Σz0 + B ′ ). Since the
condition (I − Σ ∗ Σ)z0 = Σ ∗ B ′ implies that (1 − σ 2 i )(z0)i = σib ′ i for 1 ≤ i ≤ n
and b ′ i = 0 for 1 ≤ i ≤ j we obtain
T 2 = exp
1
2
n
m=j+1
|b ′ m| 2
1 − σ 2 m
as desired.
Thus it just remains to handle the case that F (z0) = 0. For motivation
we first show how to handle this case in the one variable setting, with ψ(z) =
σz + b ′ . If F has a zero of order k at z0, then consider the equation
F (a(az + b))
kb(ψ(z))
F (z)
= Cψ 2
and take limits as z → z0. Using L’Hopitals’ rule we get
exp
|σ| 2k = Cψ 2 .
|b ′ | 2
2(1 − |σ| 2 )
11

This cannot be correct, since by Lemma 3
Cψ ≥ exp
|b ′ | 2
4(1 − |σ| 2 )
and |σ| < 1.
For F a function of several complex variables, we say F has a zero of
order k at z = z0 if in the power series expansion for F about z0, F (z) =
α c(α)(z − z0) α , there is a multi-index α of total order k such that c(α) =
0, but for all multi-indices of order less than k, c(α) = 0. Write z0 =
(z 1 0, z 2 0, · · · , z n 0 ) and consider z ′ s of the form (z 1 0 + γζ1, z 2 0 + γζ2, · · · , z n 0 + γζn)
where (ζ1, ζ2, · · · , ζn) is a fixed point of ∂Bn and γ varies over the complex
plane. Substitute such z ′ s into the equation
kB ′(Σz + B′ ) F (Σ(Σz + B′ ))
F (z)
= T 2
and let γ → 0. L’Hopital’s rule (differentiating k times with respect to γ)
yields
kB ′(Σz0 + B ′
|α|=k
)
c(α)σ2α1 1 · · · σ2αn n ζ α1
1 · · · ζαn n
= T 2
(10)
|α|=k c(α)ζα1 1 · · · ζαn n
for any choice of (ζ1, · · · , ζn) for which the denominator in this expression
is not 0. The quotient on the left side of Equation (10) must be some positive
constant, say µ, since T 2 and kB ′(Σz0 + B ′ ) are positive. Thinking of
ζ1, · · · , ζn as variables for the moment, this says the homogeneous polynomi-
als
and
|α|=k
c(α)σ 2α1
1
µ
|α|=k
· · · σ 2αn
n ζ α1
1 · · · ζ αn
n
c(α)ζ α1
1 · · · ζ αn
n
must agree and hence have the same coefficients. Thus
c(α)σ 2α1
1
· · · σ 2αn
n
= µc(α)
for every multi-index α of total order k. Since for some such α, c(α) = 0,
= µ; this forces µ to be at most 1. But µ < 1
we must have σ 2α1
1
· · · σ 2αn
n
12

yields a contradiction to Corollary 2, so we are left with µ = 1. Hence
kB ′(Σz0 + B ′ ) = T 2 , as desired.
The final matter before we complete the proof of Theorem 4 in the several
variable case is to estimate the norm of Cψ restricted to the orthogonal complement
of our reducing subspace M. We address this issue next. Without
loss of generality we may assume 1 ≤ j < n, where n is the dimension and j
is defined by Equation (3).
It is a straightforward calculation (see, e.g., Section 1.4.9 of [9], where
a slightly different normalization is used) to see that for a multi-index α,
z α 2 = 2 |α| α!. In particular
z k1
1 · · · z kj
j zkj+1 j+1 · · · zkn n 2 = 2 k1+···+kjk1! · · · kj!z kj+1
j+1 · · · zkn n 2 .
Lemma 6. If f ∈ F depends on zj+1, · · · , zn only then
z k1
1 · · · z kj
j f2 = 2 k1+···+kj k1! · · · kj!f 2
for any non-negative integers k1, · · · , kj.
Proof. Write f = c(α)z α where the sum is over multi-indices α with
α1 = α2 = · · · = αj = 0. Then
z k1
1 · · · z kj
j f2 =
α=(0,··· ,0,αj+1··· ,αn)
= |c(α)| 2 z k1
1 · · · z kj
j zα 2
z k1
1 · · · z kj
j c(α)zα 2
= 2 k1+···+kj k1! · · · kj! |c(α)| 2 2 |α| αj+1! · · · αn!
= 2 k1+···+kj k1! · · · kj!f 2
We next obtain an estimate on the norm of Cψ restricted to the orthogonal
complement of the subspace M. To do this we will take a function g in M ⊥
and write Cψ(g) as a sum of terms each of which is a function in Cψ multiplied
by a monomial of the form z α1
1 · · · z αj
j . The desired estimate will then follow
by orthogonality and Lemma 6.
Proposition 2. Suppose ψ(z) = Σz+B ′ and M are as described in Lemma 4.
Let g ∈ M ⊥ . Then g ◦ ψ 2 ≤ T 2 g 2 , where T denotes the restriction of
Cψ to M.
13

Proof. Write g as a sum of pairwise orthogonal functions of the form zα′ gα ′
where α ′ = (α1, · · · , αj, 0 ′ ) with α1, · · · , αj not all zero and gα ′ is a function
depending on zj+1, · · · , zn only. This is accomplished by writing g is
its power series expansion about 0 and grouping together all terms of the
form c(α)z α1
1 · · · z αj
j (z′′ ) β , where α1, · · · , αj are fixed and not all zero and z ′′
denotes (zj+1, · · · , zn). Note that
〈z α′
gα ′, z ˜ α ′
gα ˜′〉 = 0
if α ′ = ˜α ′ . Using Lemma 6 this orthogonality implies
g 2 = z α′
gα ′2 = 2 |α′ | α1! · · · αj!gα ′2
where α ′ = (α1, · · · , αj, 0 ′ ). We have g ◦ ψ = zα′ gα ′ ◦ ψ and, by the
form of ψ, the terms of this sum are pairwise orthogonal as α ′ ranges over
multi-indices of the form α ′ = (α1, · · · , αj, 0 ′ ). Thus
g ◦ ψ 2 = z α′
gα ′ ◦ ψ2
= 2 |α′ |
α1! · · · αj! gα ′ ◦ ψ2
≤ 2 |α′ | α1! · · · αj! T 2 gα ′2 = T 2 g 2
where the inequality holds since gα ′ is in M.
Finally, we combine the above results to obtain Theorem 4.
Proof of Theorem 4. As previously discussed (see Equation (8), it
suffices to show
Cψ = exp
′
1 |b j+1|
4
2
1 − σ2 + · · · +
j+1
|b′ n| 2
1 − σ2
n
(11)
where ψ is a normalization of ϕ. By Lemma 5 we know that the expression
on the right hand side of Equation (11) is the norm of the restriction of Cψ to
the reducing subspace M, and by Proposition 2 the norm of the restriction
of Cψ to the orthogonal complement of M is bounded above by this same
expression. This gives the result.
14

3 Concluding remarks
In this final section we consider the possibility of extending the results of the
previous section to various generalizations of F. For simplicity we restrict
to the case n = 1. First consider the, in general, non-Hilbert Fock space F p
defined, for 0 < p < ∞, to be the space of all entire functions f on C for
which the norm
f p p ≡ 1
|f(z)|
2π C
p 1
−
e 2 |z|2
dA(z) < ∞
where dA denotes Lebesgue measure in C.
The boundedness and compactness of Cϕ can be determined by the Carleson
properties of a certain measure associated with ϕ. In particular, recall
that a finite measure µ on C is a (p, q)-Carleson measure if the space F p is
a subset of L q (µ). By the closed graph theorem, this is equivalent to the
existence of a positive constant C such that
fL q (µ) ≤ Cfp
(12)
for all f ∈ F p . The inequality (12) is equivalent to the boundedness of the
inclusion map i : F p → Lq (µ). If the map i is compact, we say that µ is a
vanishing (p, q)-Carleson measure. (A linear map is compact if the image of
the closed unit ball has compact closure.)
For a given entire function ϕ, the weighted pullback measure µϕ on C is
given by
µϕ(E) =
ϕ −1 (E)
1
−
e 2 |z|2
dA(z)
for every Borel subset E of C. Since one can easily see that Cϕ(f)q =
fL q (µϕ) for every f in F p , the operator Cϕ : F p → F q is bounded (resp.,
compact) if and only if the pullback measure µϕ is (p, q)-Carleson (resp.,
vanishing (p, q)-Carleson).
The next two results give a characterization of the Carleson and the
vanishing Carleson measures. We will not give the proofs of these results since
they are slight improvements of results contained in a paper of Ortega [8].
Here ∆(ζ, r) denotes the Euclidean disk with center ζ and radius r.
Theorem 5. Let 0 < p ≤ q < ∞. The following statements are equivalent:
(i) The measure µ is a (p, q)-Carleson measure.
15

(ii) For every r > 0 there is a constant C for which
∆(ζ,r)
e αq|z|2
p dµ(z) ≤ C (13)
for all ζ ∈ C.
(iii) There exists r > 0 and a constant C for which (13) holds for all
ζ ∈ C.
For vanishing Carleson measures, we have the following.
Theorem 6. Let 0 < p ≤ q < ∞. The following statements are equivalent:
(i) The measure µ is a vanishing (p, q)-Carleson measure.
(ii) For every r > 0
∆(ζ,r)
as |ζ| → ∞.
(iii) There exists r > 0 for which (14) holds.
e αq|z|2
p dµ(z) → 0 (14)
From the previous results, we can easily obtain Proposition 3 by replacing
the measure µ with the measure µϕ.
Proposition 3. Consider the operator Cϕ : F p → F q for 0 < p ≤ q < ∞.
(a) If Cϕ is bounded, then ϕ(z) = az + b. Moreover,
q
|a| = 1, then b = 0.
p
q
(b) If Cϕ is compact, then |a| < 1.
p
Conversely, suppose that ϕ(z) = az + b.
q
(c) If p |a| = 1 and b = 0, then Cϕ is bounded.
q
(d) If p |a| < 1, then Cϕ is compact.
q
|a| ≤ 1, and if
p
It is natural to ask whether one can compute the operator norm Cϕ :
F p → F q for 0 < p ≤ q < ∞ when at least one of p and q is different than
2. When p = q an obvious conjecture for this norm is
exp
1
2q
|b| 2
1 − |a| 2
16

when |a| < 1. When p < q a lower bound for the norm is
1
2π
1/p−1/q
exp
1
2q
(q/p)|b| 2
1 − (q/p)|a| 2
These results follow an argument similar to that in Lemma 3, using the point
evaluation functionals on F p and F q .
Returning to a Hilbert space setting, one might also consider generalizations
to the Fock space obtained by defining FW to consist of those entire
functions on C satisfying
|f(z)| 2 exp(−W (z))dA(z) < ∞
C
for some positive weight function W (z) = W (|z|) satisfying appropriate hypotheses.
To the extent that one can obtain good asymptotic information
about kw, as |w| → ∞, (where, as usual, kw denotes the reproducing kernel
function for evaluation at w in FW ) the methods of Theorem 1 will apply.
For example, if W (z) = |z| 2c , c > 0, then a theorem of Folland and Rochberg
(see, e.g. Theorem 4.6 in [?]) shows that
.
kw 2 = exp(|w| 2c )|w| 2c−2 c 2 (1 + O(c 2 |w| 2c ) −γ )
for some γ > 0, as |w| → ∞. From this it follows, as in the proof of
Theorem 1, that if Cϕ is bounded on FW , then
lim sup
|z|→∞
|ϕ(z)|
|z|
and hence that ϕ must be linear.
The analogue of Theorem 2 also holds for W (z) = exp(−|z| 2c ), c > 0 with
a similar proof, involving a change of variable argument, and elementary
estimates on
|w| 2c (1 − |a| −2c |1 − b
w |2c )
for |w| large.
It seems unlikely that a precise calculation of Cϕ can be made in this
setting, however it would be reasonable to expect that
CϕFW
≤ 1
C
= sup
w∈C
∗ ϕ(kw)
kw
might continue to hold when W (z) = exp(−|z| 2c ), c > 0.
17

References
[1] M. Appel, P. Bourdon, and J. Thrall, “Norms of composition operators
on the Hardy space”, Experiment. Math., 5 (1996), 111-117.
[2] P. Bourdon and D. Retsek, “Reproducing kernels and norms of composition
operators”, Acta Sci. Math. (Szeged), 67, (2001), 387-394.
[3] C. Cowen and B. MacCluer, <strong>Composition</strong> <strong>Operators</strong> on Spaces of Analytic
Functions, CRC Press, Boca Raton, FL, 1995.
[4] C. Hammond, On the norm of a composition operator with linear fractional
symbol, preprint.
[5] F. Holland and R. Rochberg, “Bergman kernels and Hankel forms on
generalized Fock spaces”, Contemp. Math. 232, (1999), 189-200.
[6] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press,
Cambridge, 1990.
[7] F. Jafari, B. MacCluer, C. Cowen and D. Porter, eds. Studies on <strong>Composition</strong>
<strong>Operators</strong>, Contemp. Math. 213, Amer. Math. Soc., 1998.
[8] J. Ortega-Cerdà, “Sampling Measures,” Publ. Mat. 42 (1998), no. 2,
559-566.
[9] W. Rudin, Function Theory in the Unit Ball of C n , Springer-Verlag,
New York, 1980.
[10] J. Shapiro, <strong>Composition</strong> <strong>Operators</strong> and Classical Function Theory,
Springer, New York, 1993.
The University of Michigan
Department of Mathematics
East Hall, 525 East University Ave
Ann Arbor, MI 48109-1109
Email: carswell@umich.edu
Department of Mathematics
PO Box 400137 Kerchof Hall
University of Virginia
18

Charlottesville, VA 22904-4137
Email: bdm3f@weyl.math.virginia.edu
San Francisco State University
Department of Mathematics
San Francisco, CA 94132
Email: schuster@sfsu.edu
19