linear combinations of composition operators on...

LINEAR COMBINATIONS OF COMPOSITION OPERATORSON THE FOCK-SOBOLEV SPACES

HONG RAE CHO, BOO RIM CHOE, AND HYUNGWOON KOO

ABSTRACT. We study linear combinations of composition operators acting onthe Fock-Sobolev spaces of several variables. We show that such an opera-tor is bounded only when all the composition operators in the combination arebounded individually. In other words, composition operators on the Fock-Sobolevspaces do not possess the same cancelation properties as composition operatorson other well-known function spaces over the unit disk. We also show the ana-logues for compactness and the membership in the Schatten classes. In particu-lar, compactness and the membership in some/all of the Schatten classes turn outto be the same.

1. INTRODUCTION

For a fixed positive integer n, let H(Cn) be the space of all entire functions onthe complex n-space Cn. Given an entire map ϕ : Cn → Cn, we denote by Cϕthe composition operator with inducing function ϕ on H(Cn) defined by

Cϕf := f ◦ ϕ

for f ∈ H(Cn). It is clear that Cϕ maps H(Cn) into itself. The purpose of thispaper is to study boundedness, as well as related facts, of linear combinations ofcomposition operators acting on the Fock-Sobolev spaces described below.

We denote by z · w the Hermitian inner product of z, w ∈ Cn given by

z · w =n∑j=1

zjwj .

Here, zj denotes the j-th component of a typical point z ∈ Cn so that z =(z1, . . . , zn). We also write

|z| = (z · z)1/2

for z ∈ Cn.

Date: June 3, 2011; (Revised) February 6, 2014.2010 Mathematics Subject Classification. Primary 47B33; Secondary 46E20, 32A36.Key words and phrases. Linear combination of composition operators, Fock space, Fock-Sobolev

space.H. Cho was supported by the National Research Foundation of Korea(NRF) grant funded by

the Korea government(MEST) (NRF-2011-0013740) and B. Choe was supported by Basic ScienceResearch Program through the National Research Foundation of Korea(NRF) funded by the Ministryof Education, Science and Technology(2013R1A1A2004736).

1

2 H. CHO, B. CHOE, AND H. KOO

We first recall the well-known Fock space over Cn. Let dG be the Gaussianmeasure on Cn given by

dG(z) =1

πne−|z|

2dV (z), z ∈ Cn

where dV is the ordinary volume measure on Cn. The Fock space, denoted byF 2 = F 2(Cn), is then the space L2(G) ∩ H(Cn). Being considered as a closedsubspace of L2(G), the space F 2(Cn) is a Hilbert space with inner product 〈·, ·〉and norm ‖ · ‖ given by

〈f, g〉 :=

∫Cn

f(z)g(z) dG(z)

and

‖f‖2 :=

∫Cn|f(z)|2 dG(z)

for f, g ∈ F 2.Now, given a nonnegative integer m, we denote by F 2

m = F 2m(Cn) the space of

all f ∈ H(Cn) such that ∑|α|≤m

‖∂αf‖2 <∞. (1.1)

Here, we are using the standard multi-index notation. Namely, given an n-tupleα = (α1, . . . , αn) of nonnegative integers, |α| := α1 + · · · + αn and ∂α :=∂α1

1 · · · ∂αnn where ∂j denotes the partial differentiation with respect to the j-thcomponent. Note that F 2

m is again a Hilbert space, equipped with the inner productnaturally associated with the left-hand side of (1.1). In Section 2 we will pro-vide a more convenient inner product on F 2

m that induces an equivalent norm. Thespace F 2

m is naturally referred to as the Fock-Sobolev space of order m, becauseof the similarity to the way the classical Sobolev spaces are defined. We remarkin passing that F 2

m is continuously embedded in F 2m−1 for each m ≥ 1 by defini-

tion. Of course, those embeddings are all proper. For example, consider the series∑∞j=1(j!j2)−1/2zj1 ∈ F 2 \ F 2

1 . These Fock-Sobolev spaces have been recentlystudied by Cho and Zhu [3] in a context other than composition operators.

The Fock-Sobolev spaces introduced in the preceding paragraph is actually justone example of various kinds of such spaces. In fact the study of the Fock-Sobolevspaces was initiated by the seminal work [14] in 1984 of Meyer, who establisheda deep Sobolev-type inequality with respect to Wiener measures. On the settingof the real n-space Rn, Meyer’s inequality yields the Gaussian Riesz transforminequality, asserting the equivalence of Gaussian norms of a given function and itsGaussian Riesz transform associated with the Ornstein-Uhlenbeck operator. Earlierin 1969 Muckenhoupt [16] had already considered such an inequality on R. Afterthose two landmark papers, various other proofs were subsequently obtained in [6],[7], [18], [19] and [22]. Also, higher-order versions of Meyer’s Gaussian Riesztransform inequality are obtained subsequently in [8], [17] and [22]. As a result,with respect to the Gaussian measure, it follows that a Sobolev order is converted

LINEAR COMBINATIONS OF COMPOSITION OPERATORS 3

into the Ornstein-Uhlenbeck derivative of the corresponding order, and vice versa.More recently, holomorphic Fock-Sobolev spaces have been studied as the imagespaces of the generalized Segal-Bargmann transform on the Sobolev spaces over acompact Lie group (see [10]) or over a compact symmetric space (see [21]). Onthose holomorphic settings, it turns out that a Sobolev order is converted into acertain natural polynomial weight of the corresponding degree, and vice versa.

The theory of composition operators on various function spaces over a classicaldomain such as the unit disk or the unit ball has been quite extensively establishedduring the past several decades; see, for example, [5] and [20]. We mention twoknown facts that are related to our works in this paper. One is the fact that everyholomorphic self-map of the unit disk induces a bounded composition operator onthe Hardy or Bergman space over the unit disk, which is a well-known consequenceof the Littlewood Subordination Principle. The other is the characterization, dueto Moorhouse [15], for compact difference of two composition operators acting onthe weighted Bergman spaces over the unit disk. Roughly speaking, Moorhouse’scharacterization asserts that a difference of two composition operators is compactwhen suitable cancelation occurs at all bad boundary points of inducing functions.

Composition operators turn out to behave quite differently on the Fock space.First, a composition operator is rarely bounded on F 2 (even for n = 1). The char-acterization due to Carswell, MacCluer and Schuster [2] shows that a compositionoperator is bounded on F 2 if and only if its inducing function is an affine trans-formation satisfying certain additional conditions; the case m = 0 of Theorem 1.1below is the precise statement. Next, no cancelation phenomenon exists on theFock space. More precisely, a recent result of Choe, Kou Hei Izuchi and Koo [4]shows that a linear sum of two composition operators is bounded(compact, resp.)on F 2 only if both composition operators are already bounded(compact, resp.) in-dividually.

In connection with the result for linear sums of two composition operators men-tioned in the preceding paragraph, the following problem was raised in [4]:

Problem. Let ϕ1, . . . , ϕN be arbitrarily finitely many entire mapsfrom Cn to Cn. Suppose that a non-degenerate linear combination ofCϕ1 , . . . , CϕN is bounded(compact, resp.) on F 2. Does it then follow thateach Cϕj is bounded(compact, resp.) on F 2?

In this paper we settle this problem in the affirmative. In fact our solution will begiven in a more general setting of the Fock-Sobolev spaces. As a necessary steptowards our solution, we first characterize single composition operators that arebounded on F 2 as in Theorem 1.1 below, which is our first result.

In what follows we use the notation M(n) for the class of all n × n complexmatrices. We will freely identify A ∈ M(n) with the linear transformation A :Cn → Cn. We write ‖A‖ for the operator norm of A.

Theorem 1.1. Let ϕ : Cn → Cn be an entire map. Given a nonnegative integerm, the following statements hold:


(a) Cϕ is bounded on F 2m if and only if ϕ(z) = Az + B for some A ∈ M(n)

with ‖A‖ ≤ 1 and B ∈ Cn such that Aζ · B = 0 whenever ζ ∈ Cn and|Aζ| = |ζ|;

(b) Cϕ is compact on F 2m if and only if ϕ(z) = Az + B for some A ∈ M(n)

with ‖A‖ < 1 and B ∈ Cn.

The above theorem shows that the characterization due to Carswell, MacCluerand Schuster [2] for the case m = 0 remains unchanged for general m. Such aphenomenon is not too surprising in the sense of the remark given in the last sectionof [2]. In fact Carswell, MacCluer and Schuster considered a more general Fock-type space FW := H(Cn) ∩ L2(e−WdV ), where W is a unbounded (eventuallypositive) radial function, and remarked that their method can be applied as long asW is good enough to admit a nice asymptotic behavior for the norm of reproducingkernels. The explicit examples they mentioned are functions of type W (z) = a|z|bwith a and b positive. Our contribution here is that the Fock-Sobolev space oforderm can be realized as FW , withW (z) = |z|2−m log |z|2, whose reproducingkernels turn out to have a nice asymptotic norm-behavior (Proposition 2.7).

Our main result is the following.

Theorem 1.2. Let ϕ1, . . . , ϕN : Cn → Cn be arbitrarily finitely many entiremaps and assume ϕj 6= ϕk whenever j 6= k. Let a1, . . . , aN ∈ C \ {0} and m bea nonnegative integer. Then

∑Nj=1 ajCϕj is bounded(compact, resp.) on F 2

m if andonly if Cϕj is bounded(compact, resp.) on F 2

m for each j = 1, . . . , N .

As an application, we also show that compactness and membership in the Schat-ten classes are the same for operators under consideration and that non-cancelationphenomenon extends to Schatten class operators; see Corollary 5.2.

Constants. In the rest of the paper we use the same letter C, often with sub-scripts attached, to denote various positive constants which may change at eachoccurrence. These constants depend on the fixed dimension n most of the time.Variables (other than n) indicating the dependency of constants C will be oftenspecified in parentheses. We use the notation X . Y or Y & X for nonnega-tive quantities X and Y to mean X ≤ CY for some inessential constant C > 0.Similarly, we use the notation X ≈ Y if both X . Y and Y . X hold.

2. FOCK-SOBOLEV SPACES

There are several ways of defining inner products that induce equivalent normson F 2

m. For example, the norm defined as the square root of (1.1) is induced by theinner product

(f, g) 7→∑|α|≤m

〈∂αf, ∂αg〉. (2.1)

It is well known that normalized (holomorphic) monomials form an orthonormalbasis for F 2. Thus we also see that normalized monomials form an orthonormalbasis for F 2

m with respect to the inner product (2.1). In particular, the set of all poly-nomials on Cn is dense in F 2

m. However, the natural inner product (2.1) has some


disadvantage in the sense that it is not easy to find reproducing kernels explicitly.In this section we find an inner product which admits very convenient reproducingkernels.

Our scheme is to convert the Sobolev order m into the natural weight |z|2m sothat the space F 2

m can be identified with a Fock space with respect to a radially-weighted Gaussian measure. Such convertibility between Sobolev orders and natu-ral weights is now a general phenomenon for various types of Fock-Sobolev spaceson more general settings; see [10] and [21]. While one may probably use ideas andtechniques from [10] or [21] to fit to our concrete setting Cn, we here provide a di-rect proof by means of purely analytic approach for both completeness and reader’sconvenience.

We first fix some notation. In what follows we again use the conventional multi-index notation, i.e., zα := zα1

1 · · · zαnn and α! := α1! · · ·αn! for a multi-indexα = (α1, . . . , αn). For multi-indices α and β we write β ≤ α if βj ≤ αj for eachj = 1, . . . , n. We also write z not only for a typical point in Cn but also for theidentity map on Cn. We make the convention ‖f‖ =∞ when f ∈ H(Cn) \ F 2.

The identity in the next lemma is the standard bosonic commutation relation forthe annihilation operator and the creation operator in quantum field theory; see,for example, [9, Section 14.4.1]. Quite a simple proof is included for reader’sconvenience.

Lemma 2.1. The identity

‖zjf‖2 = ‖f‖2 + ‖∂jf‖2

holds for f ∈ H(Cn) and j = 1, . . . , n.

Proof. Since the polynomials form a dense subset of F 21 , it suffices to prove the

identity for polynomials. Let f =∑

α cαzα be a polynomial. Since ‖zα‖2 = α!

and monomials are mutually orthogonal, we have

‖f‖2 =∑α

|cα|2α!

‖∂jf‖2 =∑α

αj |cα|2α!

‖zjf‖2 =∑α

(αj + 1)|cα|2α!

for each j. This completes the proof. �

Lemma 2.2. Given a nonnegative integer m, there is a constant C = C(m) > 0such that ∑

|α|≤m

‖∂αf‖2 ≤ C∑|α|=m

‖zαf‖2

for f ∈ H(Cn).

Proof. Let f ∈ H(Cn). First, we observe a simple consequence of Lemma 2.1.We have by Lemma 2.1

‖f‖ ≤ ‖zjf‖


for each j. Repeating this inequality, we obtain

‖zβf‖ ≤ ‖zαf‖ (2.2)

whenever β ≤ α. Next, given a multi-index α, we claim that there is a constantC = C(α) > 0 such that

C‖∂αf‖ ≤ ‖zαf‖+∑|β|<|α|

‖zβf‖ (2.3)

for f ∈ H(Cn). This, together with (2.2), implies the lemma.We prove (2.3) by induction on the order |α| of α. The case |α| = 0 is trivial.

Now, let k ≥ 0 and assume that (2.3) holds for multi-indices of order k. By Lemma2.1 we have

‖zβ∂jf‖ = ‖∂j(zβf)− (∂jzβ)f‖ ≤ ‖zjzβf‖+ ‖(∂jzβ)f‖

for all j and multi-indices β. Thus, given α with |α| = k, we have by inductionhypothesis

C1‖∂j∂αf‖ ≤ ‖zα∂jf‖+∑|β|<k

‖zβ∂jf‖

≤ ‖zjzαf‖+ C2

∑|β|≤k

‖zβf‖

for each j where C1 and C2 are positive constants depending only on k (and n).This shows that (2.3) also holds for multi-indices of order k + 1. This completesthe induction and the proof of the lemma. �

We also have the reverse estimate.

Lemma 2.3. Given a nonnegative integer m, there is a constant C = C(m) > 0such that ∑

|α|=m

‖zαf‖2 ≤ C∑|α|≤m

‖∂αf‖2

for f ∈ H(Cn).

Proof. Given a multi-index α 6= 0, we claim that there is a constantC = C(α) > 0such that

‖zαf‖ ≤ C∑β≤α‖∂βf‖ (2.4)

for f ∈ H(Cn). This implies the lemma.We prove (2.4) by induction on the order |α| of α. The case |α| = 0 is trivial.

Now, let k ≥ 0 and assume that (2.3) holds for multi-indices of order k. Given αwith |α| = k, we have by Lemma 2.1 and induction hypothesis

‖zjzαf‖ ≤ ‖zαf‖+ ‖∂j(zαf)‖≤ ‖zαf‖+ ‖∂j(zα)f‖+ ‖zα(∂jf)‖

for each j. By induction hypothesis the first two terms of the above are dominatedby some constant times

∑β≤α ‖∂βf‖. Also, the last term is dominated by some


constant times∑

β≤α ‖∂β∂jf‖ again by induction hypothesis. So, (2.4) also holdsfor multi-indices of order k + 1. This completes the induction and the proof of thelemma. �

Proposition 2.4. Given a nonnegative integerm, there is a constant C = C(m) >0 such that

C−1‖|z|mf‖2 ≤∑|α|≤m

‖∂αf‖2 ≤ C‖|z|mf‖2

for f ∈ H(Cn).

Proof. Since

|z|2m =∑|α|=m

m!

α!|z1|2α1 · · · |zn|2αn =

∑|α|=m

m!

α!|zα|2,

we have

‖|z|mf‖2 =∑|α|=m

m!

α!‖zαf‖2

for f ∈ H(Cn). So, the proposition follows from Lemmas 2.2 and 2.3. �

Let m be a nonnegative integer. Suggested by Proposition 2.4, we denote bydGm the weighted Gaussian measure given by

dGm(z) = ωm|z|2m dG(z), z ∈ Cn

where

ωm = ωm(n) :=(n− 1)!

(m+ n− 1)!(2.5)

is the normalizing constant. Having Proposition 2.4, from now on we identify F 2m

with H(Cn) ∩ L2(Gm). So, the inner product 〈·, ·〉m and the norm ‖ · ‖m on F 2m

are given by

〈f, g〉m =

∫Cn

f(z)g(z) dGm(z)

and

‖f‖2m =

∫Cn|f(z)|2 dGm(z)

for f, g ∈ F 2m.


Given z ∈ Cn and f ∈ F 2m, subharmonicity of |f(z + w)e−w·z|2|z + w|2m (as

a function of w) yields

|f(z)|2|z|2m .∫|w|<1

∣∣f(z + w)e−w·z∣∣2 |z + w|2me−|w|2 dV (w)

≤∫Cn

∣∣f(z + w)e−w·z∣∣2 |z + w|2me−|w|2 dV (w)

= e|z|2

∫Cn|f(u)|2|u|2me−|u|2 dV (u).

Thus we have

|f(z)|2 ≤ (constant)‖f‖2me|z|

2

1 + |z|2m, z ∈ Cn (2.6)

for all f ∈ F 2m. This shows that each point evaluation is a bounded linear functional

on F 2m. So, to each z ∈ Cn corresponds the reproducing kernel Km

z at z such that

f(z) = 〈f,Kmz 〉m

for f ∈ F 2m.

As is well known, we have

Kmz (w) =

∑α

eα(w)eα(z) (2.7)

where {eα} is any orthonormal basis for F 2m. Recall that polynomials form a dense

subset of F 2m. Also, note that monomials are mutually orthogonal. We thus see that

normalized monomials form an orthonormal basis for F 2m. Namely, { zα

‖zα‖m } is anorthonormal basis for F 2

m.We now proceed to compute the reproducing kernel in a closed form. Set

Pm(λ) = Pm,n(λ) :=

∞∑k=n−1+m

λk−m

k!(2.8)

for λ ∈ C. In other words,

Pm(λ) =eλ −Qn−1+m(λ)

λm(2.9)

where Q0 = 0 and Qk is the Taylor polynomial of eλ of order k − 1 for k ≥ 1.Note that Pm is an entire function on C.

In the rest of the paper we are concerned with only one fixed Sobolev order m.So, we simply write

Kz = Kmz

for short.

Theorem 2.5. Let m be a nonnegative integer. Then

Kz(w) =1

ωmP (n−1)m (w · z)

for z, w ∈ Cn. Here, ωm is the constant specified in (2.5).


Proof. An elementary computation yields

‖zα‖2m = ωmα!(n− 1 +m+ |α|)!

(n− 1 + |α|)!for each multi-index α. Thus, setting eα := zα/‖zα‖m, we have

eα(z) =

√(n− 1 + |α|)!

ωmα!(n− 1 +m+ |α|)!zα.

Now, given z, w ∈ Cn, a little manipulation, together with (2.7), yields

Kz(w) =1

ωmh(w · z)

where

h(λ) =

∞∑k=0

(n− 1 + k)!

(n− 1 +m+ k)!k!λk.

Note

h(λ) =dn−1

dλn−1

{ ∞∑k=0

λn−1+k

(n− 1 +m+ k)!

}= P (n−1)

m (λ)

where the second equality comes from (2.8). This completes the proof. �

Note P (n−1)0 (λ) = eλ. So, as a consequence of Theorem 2.5, we see that ew·z

is the reproducing kernel for F 2, which is of course well known. We now turn tothe norm estimate of the reproducing kernels. Note

‖Kz‖2m = Kz(z) =1

ωmP (n−1)m (|z|2), z ∈ Cn (2.10)

by the reproducing property and Theorem 2.5.

Lemma 2.6. Given a nonnegative integer m, the estimate∣∣∣P (n−1)m (t)

∣∣∣ ≈ et

1 + tm

holds for t ≥ 0.

Proof. Fix a nonnegative integer m. It is easily seen from (2.8) that P (n−1)m (λ) is

still a power series with positive coefficients. Thus we have

P (n−1)m (t) ≥ P (n−1)

m (0) =(n− 1)!

(n− 1 +m)!

for t ≥ 0. Therefore ∣∣∣P (n−1)m (t)

∣∣∣ ≈ 1 ≈ et

1 + tm

when t ≥ 0 stays bounded.Now, consider t > 0 large. We note from (2.9)

Pm(t) =et

tm−n+m−2∑k=0

tk−m

k!


and hence

P (n−1)m (t) =

et

tm

1 +n−1∑j=1

cjtj

+n+m−1∑j=n

cjtj

(2.11)

for some real coefficients cj . Therefore we see∣∣∣P (n−1)m (t)

∣∣∣ ≈ et

tm≈ et

1 + tm

for t > 0 sufficiently large. This completes the proof. �

As an immediate consequence of (2.10) and Lemma 2.6, we have the followingoptimal norm estimate for the reproducing kernels.

Proposition 2.7. Given a nonnegative integer m, the estimate

‖Kz‖2m ≈e|z|

2

1 + |z|2m

holds for z ∈ Cn.

We remark that the upper estimate above can be regarded as a generalization ofBargmann’s famous pointwise bound for reproducing kernels (see [1]). Bargmann’sbounds have already been generalized in various directions: one can even replaceCn by a compact Lie group and replace the Gaussian measure by a heat kernelmeasure; see [10] and [21].

3. SINGLE COMPOSITION OPERATORS

In this section we prove Theorem 1.1. In fact the key steps are already done inthe previous section. The rest of the proof is an easy modification of the proof in[2] required for the setting of the Fock-Sobolev spaces. Nevertheless, we include aproof (made as short as possible) for the sake of completeness.

Before proceeding, we pause to mention some preliminary observations, whichare also needed in the next section. Given a bounded linear operator T on F 2

m, wedenote by ‖T‖ the operator norm of T and by |||T ||| the essential norm of T , i.e.,

|||T ||| = inf{‖T − L‖ : L is compact on F 2m}.

So, T is compact if and only if |||T ||| = 0. As is well-known we have ‖T ∗‖ =‖T‖ and |||T ∗||| = |||T ||| where the superscript ∗ means the Hilbert space adjointoperator. Also is well known that a bounded linear operator on a Hilbert space iscompact if and only if it maps weakly convergent sequences to norm convergentones.

In connection with the remark at the end of the preceding paragraph, we notethat weakly convergent sequences in the Fock-Sobolev spaces are characterized asin the next proposition. The complete analogue is well known on various othersettings. We omit the proof which is a standard application of the Uniform Bound-edness Principle and normal family with pointwise growth rate (2.6).


Proposition 3.1. Let m be a nonnegative integer and {fj} be a sequence in F 2m.

Then fj → 0 weakly in F 2m if and only if supj ‖fj‖m < ∞ and fj → 0 uniformly

on every compact set in Cn.

Given a composition operator Cϕ bounded on F 2m, the reproducing property

easily yields C∗ϕKz = Kϕ(z) for z ∈ Cn. Thus we have

supz∈Cn

‖Kϕ(z)‖m‖Kz‖m

≤ ‖C∗ϕ‖ = ‖Cϕ‖. (3.1)

As a consequence of Propositions 2.7 and 3.1, we see that the normalized kernelKz/‖Kz‖m weakly converges to 0 in F 2

m as |z| → ∞. Thus we also have

lim sup|z|→∞

‖Kϕ(z)‖m‖Kz‖m

≤ |||C∗ϕ||| = |||Cϕ|||. (3.2)

We now turn to the proof of Theorem 1.1. We split it into two parts, i.e., thenecessity and the sufficiency. We first prove the necessity.

Proposition 3.2. Let ϕ : Cn → Cn be an entire map. Given a nonnegative integerm, the following statements hold:

(a) If Cϕ is bounded on F 2m, then ϕ(z) = Az + B for some A ∈ M(n) with

‖A‖ ≤ 1 and B ∈ Cn such that Aζ · B = 0 whenever ζ ∈ Cn and|Aζ| = |ζ|.

(b) If, in addition, Cϕ is compact on F 2m, then ‖A‖ < 1.

Proof. We first prove (a). We have by (3.1) and Proposition 2.7

supz∈C

1 + |z|2m

1 + |ϕ(z)|2me|ϕ(z)|2−|z|2 <∞. (3.3)

We claim

lim sup|z|→∞

|ϕ(z)||z|

≤ 1. (3.4)

Suppose not. Then there is a sequence {zj} such that |zj | → ∞ and

limj→∞

|ϕ(zj)||zj |

> 1.

So, setting xj = |ϕ(zj)|2/|zj |2, we have

limj→∞

1 + |zj |2m

1 + |ϕ(zj)|2me|ϕ(zj)|2−|zj |2 = lim

j→∞

e|zj |2(xj−1)

xmj=∞,

which is impossible by (3.3). Now, having (3.4), we have ϕ(z) = Az+B for someA ∈M(n) with ‖A‖ ≤ 1 and B ∈ Cn; see the proof of [2, Theorem 1].

Next, consider ζ ∈ Cn such that |Aζ| = |ζ|. We may assume |ζ| = 1. Wemay further assume Aζ = ζ by a unitary change-of-variable; see the proof of [2,


Theorem 1]. Pick a unimodular number γ so that γ(Aζ · B) = |Aζ · B|. Letw = tγζ. Then |ϕ(w)|2 = t2 + |B|2 + 2t|Aζ ·B|. Thus

|w|2m

|ϕ(w)|2me|ϕ(w)|2−|w|2 =

e|B|2+2t|Aζ·B|

(1 + t−2|B|2 + 2t−1|Aζ ·B|)m

stays bounded as t → ∞ by (3.3), which forces Aζ · B = 0, as desired. Thiscompletes the proof of (a).

We now prove (b). So, assume ϕ(z) = Az +B induces a compact compositionoperator on F 2

m. We have by (3.2) and Proposition 2.7

lim|z|→∞

1 + |z|2m

1 + |ϕ(z)|2me|ϕ(z)|2−|z|2 = 0.

Thus an easy modification of the argument above shows that there is no ζ 6= 0 suchthat |Aζ| = |ζ|. So, ‖A‖ < 1. This completes the proof of (b). �

We now proceed to the proof of the sufficiency. We first recall the notion of thesingular value decomposition. LetA ∈M(n). Let σ1, . . . , σn be the (nonnegative)eigenvalues of (AA∗)1/2 listed in non-increasing order so that σ1 ≥ σ2 ≥ · · · ≥σn ≥ 0. The number σj is usually called the j-th singular value of A. Let ∆ ∈M(n) be the diagonal matrix whose j-th diagonal entry is σj . Then it is knownthat there are unitary matrices U1, U2 ∈M(n) such that

A = U1∆U2; (3.5)

this is called the singular value decomposition of A. A proof can be found in [11].In fact this decomposition extends to the Hilbert space setting; see, for example,[24, pp. 16-17].

The singular value decomposition (3.5) provides us with a useful normalizationas follows. Let B ∈ Cn and put B′ = U∗1B. Then the maps ϕ(z) = Az + B andϕ(z) = ∆z+B′ are related by ϕ = U1◦ϕ◦U2 and henceCϕ = CU2CϕCU1 . SinceCU1 and CU2 are unitary on F 2

m, we see that ‖Cϕ‖ = ‖Cϕ‖ and |||Cϕ||| = |||Cϕ|||.We call ϕ a normalization of ϕ.

Note that the singular values of A will all be less than or equal to 1 if ‖A‖ ≤ 1,and at least one is equal to 1 if ‖A‖ = 1. The following lemma, taken from [2,Lemma1], gives an information on how the hypothesis “Aζ · B = 0 whenever|Aζ| = |ζ|” results in a normalization.

Lemma 3.3. Let A ∈ M(n) with ‖A‖ ≤ 1 and B ∈ Cn. Let ϕ(z) = Az + Bwith a normalization ϕ(z) = ∆z + B′. Let s be the number of singular values ofA whose values are 1. Then the hypothesis “Aζ · B = 0 whenever |Aζ| = |ζ|”implies that “the first s components of B′ are 0”.

We now prove the sufficiency and thereby complete the proof of Theorem 1.1.

Proposition 3.4. LetA ∈M(n) with ‖A‖ ≤ 1 andB ∈ Cn. Put ϕ(z) = Az+B.Given a nonnegative integer m, the following statements hold:

(a) If Aζ · B = 0 whenever ζ ∈ Cn and |Aζ| = |ζ|, then Cϕ is bounded onF 2m.


(b) If ‖A‖ < 1, then Cϕ is compact on F 2m.

Proof. Fix a nonnegative integer m. Let ϕ(z) = ∆z+B′ be a normalization of ϕ.If ∆ is not invertible, then the proofs of (a) and (b) are exactly the same as in [2].So, assume that ∆ is invertible for the rest of the proof and put D = ∆−1. Notethat the j-th diagonal entry of D is 1/σj where σj is the j-th singular value of A.

In order to prove (a), it is sufficient to show the boundedness of Cϕ on F 2m.

Given f ∈ F 2m, we have by a change of variables

‖f ◦ ϕ‖2m =

∫Cn|f(∆z +B′)|2 dGm(z)

= (det ∆)−2

∫Cn|f(w)|2 |D(w −B′)|2m

|w|2me|w|

2−|D(w−B′)|2 dGm(w).

Meanwhile, denoting by s the maximal index such that σs = 1, we have by Lemma3.3

|w|2 − |D(w −B′)|2 =

n∑j=s+1

{|wj |2 −

∣∣∣∣wj −B′jσj

∣∣∣∣2}

=n∑

j=s+1

|B′j |21− σ2j

−

(1

σ2j

− 1

)∣∣∣∣∣wj − B′j1− σ2

j

∣∣∣∣∣2

for all w ∈ Cn. So, the expression |w|2 − |D(w − B′)|2 has maximum ν :=∑nj=s+1 |B′j |2/(1−σ2

j ), asw ranges over Cn. Also, note |D(w−B′)| ≤ ‖D‖(|w|+|B|). Therefore

‖f ◦ ϕ‖2m ≤ (det ∆)−2eν‖D‖2m∫Cn|f(w)|2 (|w|+ |B|)2m

|w|2mdGm(w)

.∫Cn|f(w)|2(1 + |w|2m) dG(w)

. ‖f‖2mwhere the last inequality holds by Proposition 2.4. Thus Cϕ is bounded on F 2

m, asdesired.

Next, we prove (b). Assume ‖A‖ < 1. As above it is enough to prove thecompactness of Cϕ on F 2

m. Let {fj} be a sequence weakly convergent to 0 in F 2m

as j →∞. In order to complete the proof it is sufficient to show ‖fj ◦ ϕ‖m → 0.Changing variables as above, we have

‖fj ◦ ϕ‖2m .∫Cn|fj(w)|2 (|w|+ |B|)2m

|w|2me|w|

2−|D(w−B′)|2 dGm(w) (3.6)

for all j. Note that all the singular values of A are strictly less than 1, because‖A‖ < 1. So, all the diagonal entries of D are strictly greater than 1. Hence,

(|w|+ |B|)2m

|w|2me|w|

2−|D(w−B′)|2 → 0


as |w| → ∞. Now, using this and Proposition 3.1, one can deduce from (3.6) that‖fj ◦ ϕ‖m → 0 as j →∞. This completes the proof. �

4. LINEAR COMBINATIONS OF COMPOSITION OPERATORS

In this section we prove our main result, Theorem 1.2. This will be accom-plished through a sequence of several lemmas.

The starting point of our proof is the analogues of (3.1) and (3.2) for linearcombinations of composition operators. That is, for arbitrarily finitely many entiremaps ϕ1, · · · , ϕN from Cn into itself and a1, . . . , aN ∈ C we have

supz∈Cn

1

‖Kz‖2m

∥∥∥∥∥∥N∑j=1

ajKϕj(z)

∥∥∥∥∥∥2

m

≤

∥∥∥∥∥∥N∑j=1

ajCϕj

∥∥∥∥∥∥2

(4.1)

and

lim sup|z|→∞

1

‖Kz‖2m

∥∥∥∥∥∥N∑j=1

ajKϕj(z)

∥∥∥∥∥∥2

m

≤

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣N∑j=1

ajCϕj

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣2

. (4.2)

Going one step further, under the restriction that all aj are nonzero, we aim toestablish the estimate

N∑j=1

‖Kϕj(z)‖2m .

∥∥∥∥∥∥N∑j=1

ajKϕj(z)

∥∥∥∥∥∥2

m

for all z such that the points ϕ1(z), . . . , ϕN (z) are pairwise separated by somefixed positive distance. Such an estimate is the content of Lemma 4.3 below, whichis the key estimate for our proof of Theorem 1.2. We need the next inequality.

Lemma 4.1. Given a nonnegative integer m, there is a constant C = C(m) > 0such that

|Kz(w)| ≤ Ce−ρ2(z,w)

2 (‖Kz‖2m + ‖Kw‖2m)

for all z, w ∈ Cn. Here, ρ(z, w) = min{|z|, |w|, |z − w|}.

Proof. Fix a nonnegative integer m. Given z, w ∈ Cn, put ρ = ρ(z, w) for sim-

plicity. In case |z| ≤ 1 or |w| ≤ 1, we have 0 ≤ ρ ≤ 1 so that 1/√e ≤ e−

ρ2

2 . Thuswe have

|Kz(w)| = |〈Kz,Kw〉m| ≤ ‖Kz‖m ‖Kw‖m

≤√e

2e−

ρ2

2 (‖Kz‖2m + ‖Kw‖2m).

So, assume |z| ≥ 1 and |w| ≥ 1. If |z · w| ≥ 1, we have by Theorem 2.5 and(2.11)

|Kz(w)| . eRe(z·w) + 1. (4.3)


This estimate remains valid even for |z · w| < 1, because |Kz(w)| then staysbounded by Theorem 2.5. Note

eRe(z·w) = e−12|z−w|2e

12

(|z|2+|w|2) ≤ e−ρ2

2 e|z|2+|w|2 .

We also have e−ρ2

2 e|z|2+|w|2 ≥ 1, because ρ2 ≤ 2(|z|2 + |w|2). We thus have by

(4.3)

|Kz(w)| . e−ρ2

2 e|z|2+|w|2 ≈ e−

ρ2

2 (‖Kz‖2m + ‖Kw‖2m)

where the last equivalence comes from Proposition 2.7, because |z| ≥ 1 and |w| ≥1. The proof is compete. �

The proof of Lemma 4.3 below will be divided into a few subcases. Its proof isquite long and the hardest subcase is thus isolated as in the next lemma.

Lemma 4.2. Letm be a nonnegative integer. Given a1, . . . , aN ∈ C\{0} and pos-itive numbers L,M, ε with L > ε, there are positive constantsC andR (dependingon m,L,M, ε and a1, . . . , aN ) such that∥∥∥∥∥∥

N∑j=1

ajKzj

∥∥∥∥∥∥2

m

≥ CN∑j=1

‖Kzj‖2m

for all z1, . . . , zN ∈ Cn satisfying(i) ε ≤ |zj − zk| ≤ L whenever j 6= k;

(ii) |zj | ≤ |zk|+ M|zk| whenever j 6= k;

(iii) |zj | ≥ R for all j.

Proof. To avoid triviality let N ≥ 2. Fix a1, . . . , aN ∈ C \ {0} and positivenumbers L,M, ε with L > ε. Let z1, . . . , zN ∈ Cn and assume that (i) and (ii) aresatisfied. Put r = minj |zj | and assume r ≥ R where R is a large positive numberto be chosen later. So, we have by (ii)

R ≤ r ≤ |zj | ≤ r +M

r(4.4)

for all j. Note ∥∥∥∥∥∥N∑j=1

ajKzj

∥∥∥∥∥∥m

=

∥∥∥∥∥∥N∑j=1

ajKUzj

∥∥∥∥∥∥m

for all unitary operators U on Cn; this is easily seen by the fact that Kz(w) =KUz(Uw). Thus we may assume z1 = (r, 0, . . . , 0) by (4.4). Choose real numbersxj and yj such that

zj1 = r +xjr

+ iyj (i =√−1) (4.5)

for each j = 1, . . . , N . Note x1 = y1 = 0. Let

wj := (iyj , zj2, . . . , z

jn)

for each j.


If xj ≥ 0, then xj ≤ M by (4.4). If xj ≤ 0, then we have by the secondinequalities of (4.4) and (i)

−2xj ≤(xjr

)2+ |wj |2 = |z1 − zj |2 ≤ L2 (4.6)

so that |xj | ≤ L2/2. We thus have

|xj | ≤M +L2

2(4.7)

for all j. This, together with (i), yields

ε ≤ |zj − zk| ≤ |xj − xk|r

+ |wj − wk| ≤ 2M + L2

r+ |wj − wk|

whenever j 6= k. Accordingly, for any R ≥ 2(2M + L2)/ε, we have by (iii)

|wj − wk| ≥ ε

2(4.8)

whenever j 6= k. Also, note by (4.6)

|wj | ≤ L (4.9)

for all j.We now proceed to the norm estimate of the given linear combination of repro-

ducing kernels. Given j and k, we have by (4.5)

zj · zk = r2 + (xj + iryj) + (xk − iryk) + cjk + wj · wk (4.10)

where

cjk =xjxkr2

+i

r(xkyj − xjyk).

Note by (4.7) and (4.9)

ecjk = 1 +O(cjk) = 1 +O(

1

r

)(4.11)

and, for each integer ` ≥ 0,

1

(zj · zk)`=

1

r2`

{1 +O

(1

r

)}(4.12)

where, and in the rest of the proof, the “big Oh” estimate refers to an estimatewhich is uniform in z1, . . . , zN .

Note by Theorem 2.5∥∥∥∥∥∥N∑j=1

ajKzj

∥∥∥∥∥∥2

m

=∑j,k

aj ak〈Kzj ,Kzk〉m

=1

ωm

∑j,k

ajak P(n−1)m (zk · zj). (4.13)


In addition, each term of the above sum is equal to

ajak

[ezk·zj

(zk · zj)m

(1 +

n−1∑`=1

c`

(zk · zj)`

)+n+m−1∑`=n

c`

(zk · zj)`

](4.14)

where c` are the coefficients determined by (2.11). Meanwhile, setting

bj = bj(xj , wj) := aje

xj+iryj ,

we have by (4.11) and (4.12)

ajakezk·zj

(zk · zj)m=

er2

r2mbjbke

wk·wj{

1 +O(

1

R

)}(4.15)

for all j and k. Also, note

|bj | ≤ |aj |eM+L2/2 (4.16)

for all j by (4.7). Thus bjbkewk·wj = O(1) by (4.7) and (4.9). So, by (4.15) and

(4.12) the quantity (4.14) admits the estimate

er2

r2m

{bjbke

wk·wj +O(

1

R

)}for all j and k. Put kwj (wk) = ew

k·wj . Note that kwj is the reproducing kernel atwj for F 2. Thus, replacing each term of the sum in (4.13) by the above, we obtain

r2me−r2

∥∥∥∥∥∥N∑j=1

ajKzj

∥∥∥∥∥∥2

m

=∑j,k

bjbkkwj (wk) +O

(1

R

)

=

∥∥∥∥∥∥N∑j=1

bjkwj

∥∥∥∥∥∥2

+O(

1

R

). (4.17)

Consider for a moment the norm in (4.17) as a function of independent vari-ables w1, . . . , wN on the compact region determined by (4.8) and (4.9). Sincew1, . . . , wN are all distinct by (4.8), one may easily verify via the reproducingproperty that kw1 , . . . kwN are linearly independent. Thus, since each bj is nevervanishing, we see that the norm ‖

∑Nj=1 bjkwj‖ is never zero. In addition, since

each bj continuously depends on wj , we see that the norm ‖∑N

j=1 bjkwj‖ is a con-tinuous function of w1, . . . , wN . So, the norm ‖

∑Nj=1 bjkwj‖ has a positive lower

bound, say 2C1, on the aforementioned compact region.Now, taking R sufficiently large, we obtain from (4.17)∥∥∥∥∥∥

N∑j=1

ajKzj

∥∥∥∥∥∥2

m

≥ C1er

2

r2m(4.18)


for r ≥ R. Furthermore, since r2 ≤ |zj |2 ≤ r2 + 2M + 1 ≤ 2r2 (with Rsufficiently large), we have

er2

r2m≈ e|z

j |2

|zj |2m≈ ‖Kzj‖2m

for all j. Consequently, we conclude the lemma by (4.18) and the above. The proofis complete. �

We now proceed to the key estimate for our proof.

Lemma 4.3. Let m be a nonnegative integer. Given a1, . . . , aN ∈ C \ {0} andε > 0, there is a constant C = C(m, ε; a1, . . . , aN ) > 0 such that∥∥∥∥∥∥

N∑j=1

ajKzj

∥∥∥∥∥∥2

m

≥ CN∑j=1

‖Kzj‖2m

for all z1, . . . , zN ∈ Cn such that |zj − zk| ≥ ε whenever j 6= k.

Proof. Let ε > 0 be given. Note that forN = 1 it is trivially true. We now proceedby induction N . So assume N ≥ 2 and that the assertion holds up to N − 1. Leta1, . . . , aN ∈ C \ {0} be given. Let z1, . . . , zN ∈ Cn and assume |zj − zk| ≥ εwhenever j 6= k for the rest of the proof.

We first fix several pieces of notation. By the induction hypothesis there is aconstant C1 > 0, independent of z1, . . . , zN , such that∥∥∥∥∥∥

∑j∈I

ajKzj

∥∥∥∥∥∥2

m

≥ C1

∑j∈I‖Kzj‖

2m (4.19)

for any nonempty proper subset I of J := {1, . . . , N}. Using Proposition 2.7, picka constant M1 = M1(m) > 0 such that

1

M1≤ e|z|

2

|z|2m‖Kz‖2m≤M1 (4.20)

for all z ∈ Cn with |z| ≥ 1. Put ν = maxj |aj | and let C4 = C4(m) > 0be the constant provided by Lemma 4.1. Fix L > 0 sufficiently large so thatL > max{ε, 1} and

C2 := 2C1 − C4Nν2e−

L2

2 > 0. (4.21)

Also, pick M > 0 such that

C3 := C ′1 − 2(ν + C ′1

)M1e

−M > 0 (4.22)

where C ′1 =√

C1N . Finally, letR = R(m,L,M, ε; a1, . . . , aN ) > 0 be the number

provided by Lemma 4.2. We may assume R ≥ L.Now we assume |zj | ≥ R for all j and continue the induction process. Suppose

that there is a nonempty J1 ( J with the following property:

|zj − zk| > L whenever j ∈ J1 and k /∈ J1. (4.23)


Note∥∥∥∥∥∥N∑j=1

ajKzj

∥∥∥∥∥∥2

m

=

∥∥∥∥∥∥∑j∈J1

ajKzj

∥∥∥∥∥∥2

m

+

∥∥∥∥∥∥∑k/∈J1

akKzk

∥∥∥∥∥∥2

m

+∑

j∈J1,k /∈J1

ajakKzj (zk)

≥

∥∥∥∥∥∥∑j∈J1

ajKzj

∥∥∥∥∥∥2

m

+

∥∥∥∥∥∥N∑

k/∈J1

akKzk

∥∥∥∥∥∥2

m

− ν2∑

j∈J1,k /∈J1

|Kzj (zk)|.

The first two terms of the above are taken care of by (4.19). In conjunction withthe last term, we have by Lemma 4.1∑

j∈J1,k /∈J1

|Kzj (zk)| ≤ C4e

−L2

2

∑j>k

(‖Kzj‖2m + ‖Kzk‖2m)

= C4(N − 1)e−L2

2

N∑j=1

‖Kzj‖2m.

It follows from these observations and (4.21) that∥∥∥∥∥∥N∑j=1

ajKzj

∥∥∥∥∥∥2

m

≥ C2

N∑j=1

‖Kzj‖2m,

as desired.Next, assume that (4.23) is not satisfied for any nonempty J1 ( J . This means

that we have

|zj − zk| ≤ L for all j and k. (4.24)

By Lemma 4.2 we may further assume that there is a nonempty J2 ( J with thefollowing property:

|zk|+ M

|zk|≤ |zj | whenever j ∈ J2 and k /∈ J2. (4.25)

By (4.19) we have∥∥∥∥∥∥N∑j=1

ajKzj

∥∥∥∥∥∥m

≥

∥∥∥∥∥∥∑j∈J2

ajKzj

∥∥∥∥∥∥m

−∑k/∈J2

|ak|‖Kzk‖m

≥ C ′1∑j∈J2

‖Kzj‖m − ν∑k/∈J2

‖Kzk‖m

= C ′1

N∑j=1

‖Kzj‖m −(ν + C ′1

) ∑k/∈J2

‖Kzk‖m.


Meanwhile, for j ∈ J2 and k /∈ J2, we have by (4.24) |zj | ≤ L+ |zk| ≤ 2|zk| andtherefore by (4.20) and (4.25)

‖Kzj‖2m ≥1

M1

e|zj |2

|zj |2m≥ e2M

4M1

e|zk|2

|zk|2m≥ e2M

4M21

‖Kzk‖2m.

This yields ∑k/∈J2

‖Kzk‖m ≤ 2M1e−M

N∑j=1

‖Kzj‖m.

So, we conclude by these observations and (4.22)∥∥∥∥∥∥N∑j=1

ajKzj

∥∥∥∥∥∥m

≥ C3

N∑j=1

‖Kzj‖m,

which implies the desired inequality.So far, we have proved the asserted inequality when |zj | ≥ R for all j. Now, let

J3 = {j ∈ J : |zj | ≥ R} and assume J3 ( J . We need to take care of the tworemaining cases depending on whether J3 is empty or not. Consider the case whenJ3 is nonempty. We have by (4.19)

∥∥∥∥∥∥N∑j=1

ajKzj

∥∥∥∥∥∥m

≥

∥∥∥∥∥∥∑j∈J3

ajKzj

∥∥∥∥∥∥m

−∑k/∈J3

|ak|‖Kzk‖m

≥ C ′1∑j∈J3

‖Kzj‖m −NM2ν

= C ′1

{1− NM2ν∑

j∈J3 ‖Kzj‖m

}∑j∈J3

‖Kzj‖m

where M2 = max|z|≤R ‖Kz‖m. Note∑

j∈J3 ‖Kzj‖m → ∞ as maxj∈J3 |zj | →∞. We thus obtain ∥∥∥∥∥∥

N∑j=1

ajKzj

∥∥∥∥∥∥m

≥ C ′12

N∑j=1

‖Kzj‖m

(and hence the desired inequality) when some |zj |, j ∈ J3, is bigger than a suf-ficiently large number, say R1 with R1 ≥ R. Finally, consider the case whenJ3 is empty. In this case, further allowing |zj | ≤ R1 for all j to cover all pos-sible remaining cases, we easily obtain the desired inequality, because the norm∥∥∥∑N

j=1 ajKzj

∥∥∥m

has a positive lower bound (as in the proof of Lemma 4.2), while

the sum∑N

j=1 ‖Kzj‖m has a finite upper bound. This completes the proof. �

As an immediate consequence of (4.1), (4.2) and Lemma 4.3, we have the nextlemma.


Lemma 4.4. Let ϕj : Cn → Cn be entire maps for j = 1, . . . , N and assumeϕj 6= ϕk whenever j 6= k. Let a1, . . . , aN ∈ C \ {0}. Given ε > 0, let Sε be theset of all z ∈ Cn such that |ϕj(z)−ϕk(z)| ≥ ε for all j 6= k. If T :=

∑Nj=1 ajCϕj

is bounded on F 2m, then

supz∈Sε

1

‖Kz‖2m

N∑j=1

‖Kϕj(z)‖2m ≤ C‖T‖2

and

lim sup|z|→∞, z∈Sε

1

‖Kz‖2m

N∑j=1

‖Kϕj(z)‖2m ≤ C|||T |||2

for some constant C = C(m, ε; a1, . . . , aN ) > 0.

Using Lemmas 4.3 and 4.4, we now reduce our proof of Theorem 1.2 to the casewhen all the inducing functions are affine transformations.

Lemma 4.5. Let m be a nonnegative integer. Given Aj ∈M(n) and Bj ∈ Cn forj = 1, . . . , N , put ϕj(z) = Ajz + Bj and assume ϕj 6= ϕk whenever j 6= k. Leta1, . . . , aN ∈ C \ {0}. If

∑Nj=1 ajCϕj is bounded(compact, resp.) on F 2

m, thenCϕj is bounded(compact, resp.) on F 2

m for each j = 1, . . . , N .

Proof. Letp := max

1≤j≤N‖Aj‖.

We may assume p > 0. Given a unit vector ζ ∈ Cn, put

Γ(ζ) := {j : |Ajζ| = p}.Fix ζ such that Γ(ζ) 6= ∅. Let ν be the number of equivalent classes induced by theequivalence relation ∼ on Γ(ζ) given by

j ∼ k def⇐⇒ Ajζ = Akζ and Bj = Bk.

We consider two possibilities(i) ν < ]Γ(ζ)

(ii) ν = ]Γ(ζ)

and provide a proof for each case separately.

Case (i): In this case at least one equivalence class, say E, contains more thanone element. Note ζ ∈ ker(Aj − Ak) ( Cn for any distinct j, k ∈ E. Thus theintersection of all possible such kernels is a proper subspace, still containing ζ, ofCn. So, we can pick a unit vector η = ηE ∈ Cn such that η ·ζ = 0 andAjη 6= Akηfor any distinct j, k ∈ E. Fix a nonzero ω ∈ C with the property

either Ajζ 6= Akζ or Bj + ωAjη 6= Bk + ωAkη (4.26)

for any distinct j, k ∈ Γ(ζ). Such a choice of ω is arbitrary when j, k ∈ E, andalso certainly possible otherwise. Put

ψj(z) = ψj,ω(z) := Ajz + (Bj + ωAjη)


for each j = 1, . . . , N .For any distinct j, k ∈ Γ(ζ), note by (4.26) that the slice map λ 7→ ψj(λζ) −

ψk(λζ) is either a nonzero constant or a linear polynomial map. Thus there is someε > 0 such that

|ψj(λζ)− ψk(λζ)| ≥ ε as |λ| → ∞

for any distinct j, k ∈ Γ(ζ). Accordingly, we have by Lemma 4.3 some constantC > 0 such that ∥∥∥∥∥∥

∑j∈Γ(ζ)

ajKψj(λζ)

∥∥∥∥∥∥m

≥ C∑j∈Γ(ζ)

‖Kψj(λζ)‖m (4.27)

for all λ ∈ C with |λ| sufficiently large.Meanwhile, for j ∈ Γ(ζ), we have by Proposition 2.7

‖Kψj(λζ)‖2m ≈

e|λAjζ+Bj+ωAjη|2

|λAjζ +Bj + ωAjη|2m≈ ep

2|λ|2

p2m|λ|2m

as |λ| → ∞. On the other hand, for j /∈ Γ(ζ), we have |Ajζ| < p and thus

‖Kψj(λζ)‖2m .

e(|λAjζ|+|Bj+ωAjη|)2

p2m|λ|2m= o

(ep

2|λ|2

p2m|λ|2m

)as |λ| → ∞. Therefore we obtain from (4.27)∥∥∥∥∥∥

N∑j=1

ajKψj(λζ)

∥∥∥∥∥∥m

≥ C∑j∈Γ(ζ)

‖Kψj(λζ)‖m −∑j /∈Γ(ζ)

|aj |‖Kψj(λζ)‖m

&∑j∈Γ(ζ)

‖Kψj(λζ)‖m

for all |λ| sufficiently large. Equivalently, we have proved∥∥∥∥∥∥N∑j=1

ajKϕj(λζ+ωη)

∥∥∥∥∥∥2

m

&∑j∈Γ(ζ)

‖Kϕj(λζ+ωη)‖2m (4.28)

for all |λ| sufficiently large.Now, suppose that

∑Nj=1 ajCϕj is bounded on F 2

m. Then we have by (4.28) andLemma 4.4

sup|λ|&1

‖Kϕj(λζ+ωη)‖2m‖Kλζ+ωη‖2m

<∞ for each j ∈ Γ(ζ).

Easily modifying the proof of Proposition 3.2(a), we conclude p ≤ 1 and

Ajζ ·Bj + ωAjη = 0

whenever j ∈ Γ(ζ). Moreover, the above remains true for any other (certainlypossible) choices of ω satisfying (4.26). So, we conclude

Ajζ ·Bj = 0


whenever j ∈ Γ(ζ). Thus all operators Cϕ1 , . . . , CϕN are individually bounded onF 2m by Proposition 3.4.Similarly, if

∑Nj=1 ajCϕj is compact on F 2

m, then we have p < 1 and thus con-clude that all operators Cϕ1 , . . . , CϕN are individually compact on F 2

m by Propo-sition 3.4. This completes the proof for Case (i).

Case (ii): In this case each equivalent class must be a singleton. So, givendistinct j, k ∈ Γ(ζ), we have either Ajζ 6= Akζ or Bj 6= Bk. So, pretendingη = 0, one may repeat the proof of Case (i). This completes the proof for Case (ii)and the proof of the lemma. �

We are now ready to complete the proof of Theorem 1.2.

Conclusion of the proof of Theorem 1.2. Since the sufficiency is trivial, we onlyneed to prove the necessity. By Lemma 4.5 we only need to show that, if

∑Nj=1 ajCϕj

is bounded on F 2m, then each ϕj is an affine transformation.

Assume that∑N

j=1 ajCϕj is bounded on F 2m. Put Φjk := ϕj − ϕk for j 6= k.

Since each Φjk is not identically 0, we can pick ε > 0 sufficiently small so thatthe set Sε introduced in Lemma 4.4 is nonempty. We have by Lemma 4.4 andProposition 2.7

supz∈Sε

∑j=1

1 + |z|2m

1 + |ϕj(z)|2me|ϕj(z)|

2−|z|2 <∞. (4.29)

Now, easily modifying the proof of (3.4), we obtain

M := supz∈Sε

1

1 + |z|

N∑j=1

|ϕj(z)| <∞, (4.30)

which yields |Φjk(z)| ≤M(1 + |z|) + ε for all z ∈ Cn and thus that Φjk is eitherconstant or an affine transformation for all j 6= k.

We now show that ϕj are all affine transformations. Pick Ajk ∈ M(n) andBjk ∈ Cn such that Φjk(z) = Ajkz + Bjk for j 6= k; recall either Ajk 6= 0 orBjk 6= 0. Note that the number ε in the preceding paragraph can be chosen assmall as we want. So, we may assume ε ≤ minΦjk(0)6=0 |Φjk(0)| for the rest of theproof. Let

Λ :=⋃

Ajk 6=0

kerAjk.

Note that Λ ⊂ Cn is of measure 0, because it is a finite union of proper subspacesof Cn. Fix an arbitrary unit vector ζ ∈ Cn \ Λ. Note Bjk 6= 0 if Ajkζ = 0,because Ajkζ = 0 only when Ajk = 0. Thus if Ajkζ = 0, then |Φjk(λζ)| =|Bjk| = |Φjk(0)| ≥ ε for all λ ∈ C. Otherwise, |Φjk(λζ)| → ∞ as |λ| → ∞. So,we have λζ ∈ Sε for |λ| sufficiently large, say |λ| ≥ C1 = C1(ζ) > 0, and henceby (4.30)

sup|λ|≥C1

N∑j=1

|ϕj(λζ)| ≤M(1 + |λ|).


Consequently, each component function of the slice map λ 7→ ϕj(λζ), 1 ≤ j ≤ N ,is at most a linear polynomial. Now, since ζ /∈ Λ is arbitrary and Λ is of measure0, we see that such a property of being “at most a linear polynomial” extendsto arbitrary unit vector ζ ∈ Cn. This implies that each ϕj must be an affinetransformation, as asserted. The proof is complete. �

5. AN APPLICATION

As an application of our characterization for compactness, we proceed to thecharacterization of linear combinations that belong to the Schatten classes.

Let us briefly recall the notion of Schatten class operators. A positive compactoperator T on F 2

m is said to belong to the trace class if

tr(T ) :=∑β

〈Teβ, eβ〉m <∞

for some orthonormal basis {eβ} of F 2m. As is well known, the sum above, called

the trace of T, is independent of choice of {eβ}. For 0 < p < ∞ and a gen-eral compact operator T on F 2

m, not necessarily positive, we say T ∈ Sp(F2m),

the Schatten p-class, if (TT ∗)p/2 belongs to the trace class. It is known that theSchatten p-class gets smaller, as p decreases. We refer to [24, Chapter 1] for moreinformation on the Schatten classes. We also recall the trace formula

tr(T ) =

∫Cn〈TKz,Kz〉m dGm(z)

valid for positive operators T ∈ S1(F 2m); one may verify this by the same proof for

the trace formula [24, Theorem 6.4] in the setting of the weighted Bergman spacesover the unit disk.

Proposition 5.1. Let m be a nonnegative integer and ϕ : Cn → Cn be a holo-morphic map. If Cϕ is compact on F 2

m, then Cϕ ∈ Sp(F 2m) for all 0 < p <∞.

Proof. We only need to consider p small. So, let 0 < p ≤ 2. Since 0 < p ≤ 2, wehave by [24, Proposition 1.31]⟨

(CϕC∗ϕ)p/2

Kz

‖Kz‖m,

Kz

‖Kz‖m

⟩m

≤⟨

(CϕC∗ϕ)

Kz

‖Kz‖m,

Kz

‖Kz‖m

⟩p/2m

=

(‖Kϕ(z)‖m‖Kz‖m

)pfor all z ∈ Cn. Therefore we have by the trace formula and Proposition 2.7

tr[(CϕC

∗ϕ)p/2

]≤∫Cn

(‖Kϕ(z)‖m‖Kz‖m

)p‖Kz‖2m dGm(z)

.∫Cn

ep2

(|ϕ(z)|2−|z|2) dV (z).

Now, assuming that Cϕ is compact on F 2m, we see by Theorem 1.1(b) that the

last integral in the displayed expression above is finite. Thus we conclude Cϕ ∈Sp(F

2m) for 0 < p ≤ 2 and hence for all 0 < p <∞. The proof is complete. �


As an easy consequence, we obtain the following corollary.

Corollary 5.2. Let ϕj : Cn → Cn be holomorphic maps for j = 1, . . . , N andassume ϕj 6= ϕk whenever j 6= k. Let a1, . . . , aN ∈ C \ {0}. Given 0 < p < ∞and a nonnegative integer m, the following three statements are equivalent:

(a)N∑j=1

ajCϕj is compact on F 2m;

(b)N∑j=1

ajCϕj ∈ Sp(F 2m);

(c) Cϕj ∈ Sp(F 2m) for each j = 1, . . . , N .

Proof. The implications (c) =⇒ (b) =⇒ (a) are straightforward. Also, the impli-cation (a) =⇒ (c) is an immediate consequence of Theorem 1.2 and Proposition5.1. �

Proposition 5.1 and Corollary 5.2 reveal Fock-space phenomena that are differ-ent from what occurs in many Hilbert spaces of functions on bounded domains.For example, see [12], [13] and [23] for earlier works on the Schatten classes inthe setting of the weighted Bergman spaces over the unit disk.

Acknowledgement. We would like to thank the referee for his/her suggestionsthat improved much the exposition of the paper, especially for drawing our atten-tion to the work of P. -A. Meyer [14] and related works.

REFERENCES

[1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral trans-form, Part I, Commun. Pure Appl. Math. 14 (1961), 187–214.

[2] B. Carswell, B. MacCluer and A. Schuster, Composition operators on the Fock space, ActaSci. Math. (Szeged) 74(2008), 807–828.

[3] H. R. Cho and K. Zhu, Fock-Sobolev spaces and their Carleson measures, J. Funct. Anal.263(2012), 2484–2506.

[4] B. R. Choe, K. Izuchi, and H. Koo, Linear sums of two composition operators on the Fockspace, J. Math. Anal. Appl. 369(2010), 112–119.

[5] C. Cowen and B. MacCluer, Composition operators on spaces of analytic functions, CRCPress, New York, 1995.

[6] R. Gundy, Sur les transformations de Riesz pour le semi-groupe d’Ornstein-Uhlenbeck, C.R. Acad. Sci. Paris S’er. I Math. 303(19)(1986), 967–970.

[7] C. E. Gutierrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal.120(1)(1994), 107–134.

[8] C. E. Gutierrez, C. Segovia, and J. Torrea, On higher Riesz transforms for Gaussian mea-sures, J. Fourier Anal. Appl. 2(6)(1996), 583–596.

[9] B. C. Hall, Quantum Theory for Mathematicians, GTM 267, Springer, New York, 2013.[10] B. C. Hall and W. Lewkeeratiyutkul, Holomorphic Sobolev spaces and the generalized

Segal-Bargmann transform, J. Funct. Anal. 217(2004), 192–220.[11] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990.[12] S. Y. Li, Trace ideal criteria for composition operators on Bergman spaces, Amer. J. Math.

117(1995), 1299–1323.[13] D. Luecking and K. Zhu, Composition operators belonging to the Schatten ideals, Amer.

J. Math. 114(1992), 1127–1145.


[14] P. -A. Meyer, Transformations de Riesz pour les lois gaussiennes, Seminar on probability,XVIII, 179-193, Lecture Notes in Math., 1059, Springer, Berlin, 1984.

[15] J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219(2005),70–92.

[16] B. Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc. 139(1969), 243–260.

[17] S. Perez, Boundedness of Littlewood-Paley g-functions of higher order associated with theOrnstein-Uhlenbeck semigroup, Indiana Univ. Math. J. 50(2)(2001), 1003-1014.

[18] S. Perez and F. Soria, Operators associated with the Ornstein-Uhlenbeck semigroup, J.London Math. Soc. 61(23)(2000), 857-871.

[19] G. Pisier, Riesz transforms: a simpler analytic proof of P. -A. Meyer’s inequality, Seminairede Probabilit‘es, XXII, 485-501, Lecture Notes in Math., 1321, Springer, Berlin, 1988.

[20] J. Shapiro, Composition opertors and classical function theory, Springer, New York, 1993.[21] S. Thangavelu, Holomorphic Sobolev spaces associated to compact symmetric spaces, J.

Funct. Anal. 251(2007), 438–462.[22] W. Urbina, On singular integrals with respect to the Gaussian measure, Ann. Scuola Norm.

Sup. Pisa Cl. Sci. (4) 17(4)(1990), 531–567.[23] K. Zhu, Schatten class composition operators on weighted Bergman spaces of the unit disk,

J. Operator Theory 46(2001), 173–181.[24] K. Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Mono-

graphs, 138. Amer. Math. Soc, Providence, 2007.

DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, PUSAN 609-735, RE-PUBLIC OF KOREA

E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY, SEOUL 136-713, REPUBLIC OF KO-REA


DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY, SEOUL 136-713, REPUBLIC OF KO-REA


linear combinations of composition operators on...

Documents