Composition Operators on Hilbert Spaces of Entire Functions of Several Variablestle20/Data/DKL_IEOT.pdf · 2017-06-15 · Composition Operators on Hilbert Spaces of Entire Functions

Composition Operators on Hilbert Spaces ofEntire Functions of Several Variables

Minh Luan Doan, Le Hai Khoi and Trieu Le

Abstract. We study composition operators acting on Hilbert spaces ofentire functions in several variables. Depending on the defining weightsequence of the space, different criteria for boundedness and compact-ness are developed. Our work extends several known results on Fockspaces and other spaces of entire functions.

Mathematics Subject Classification (2010). Primary 47B33; Secondary32A15.

Keywords. Hilbert function spaces, reproducing kernels, composition op-erators, boundedness, compactness.

1. Introduction

Let H be a Hilbert space of functions on a certain set X. For a self-mappingϕ of X, the composition operator Cϕ is defined as Cϕf = f ◦ ϕ for f ∈ H.Researchers have been interested in the interaction between the function-theoretic properties of ϕ and the operator-theoretic properties of Cϕ. Morespecifically, the study of boundedness and compactness of composition oper-ators on spaces of analytic functions has been an attractive topic in operatortheory. The books [7, 16] are excellent sources for composition operators onthe Hardy spaces, Bergman spaces and other classical spaces over the unitdisk and the unit ball.

More than a decade ago, Carswell, MacCluer and Schuster [2] studiedcomposition operators on the Fock (also known as Segal–Bargmann) spacesF2ω of entire functions over CN . They provided a complete description of the

boundedness and compactness together with a formula for the norm of Cϕ.

The first-named and second-named authors were supported in part by MOE’s AcRF Tier

1 grant M4011166.110 (RG24/13). The second-named author was also supported in partby MOE’s AcRF Tier 1 grant M4011724.110 (RG128/16). The third-named author was

supported in part by the University of Toledo’s Summer Research Awards and FellowshipsProgram.

2 M. L. Doan, L. H. Khoi and T. Le

A few years later, Guo and Izuchi [10] investigated composition operatorson more general Fock-type spaces over the complex plane. They obtainedseveral results on the boundedness, compactness, spectra, cyclicity and in-variant subspaces of composition operators. In [15], several operator-theoreticand spectral properties of bounded and unbounded composition operators onHilbert spaces whose reproducing kernels are power series with non-negativecoefficients are considered. In [6] and more recently [5], boundedness andcompactness of linear combinations of composition operators are studied onthe Fock and Fock–Sobolev spaces.

Another interesting class of Hilbert spaces of entire functions over thecomplex plane, E2

γ , was first introduced and studied in [4], where the authorsconsidered the problem of cyclicity of translation operators. In [3], some re-sults on the boundedness and compactness of composition operators on thesespaces are obtained. The class E2

γ is quite different from the class of Fock

spaces F2ω since translation operators are always bounded on E2

γ but they

are not bounded on F2ω. On the other hand, there are some similarities be-

tween these two classes: in one variable, a bounded composition operator Cϕmust be necessarily induced by a symbol of the form ϕ(z) = az + b, where|a| ≤ 1. In recent papers [8, 9], the first two authors generalize the class E2

γ toa greater family of spaces on which translation operators are still bounded,and study the boundedness and compactness of composition operators.

All the aforementioned Hilbert function spaces belong to the wide classof weighted Hardy spaces H(β) with appropriate choices of the weight se-quence β. The study of composition operators on spaces H(β) of analyticfunctions on the unit disk and unit ball is discussed in [7]. In the presentpaper, we are interested in those spaces H(β) whose elements are entire func-tions over CN . This class of spaces includes the Fock and Fock-type spacesas well as E2

γ and its generalizations. We characterize entire mappings ϕ thatinduce bounded and compact composition operator Cϕ. It is interesting thatsuch characterization depends closely on the behavior of the weight sequence.We shall also see a difference between the theory of one variable and that ofseveral variables. We recover earlier results obtained by various authors in[2, 4, 8, 9, 10].

The structure of the paper is as follows. Section 2 provides the back-ground and previously known results on composition operators Cϕ on Fockspaces. In Section 3, using the concept of weighted Hardy spaces of formalhomogeneous expansion as our principal tool, we construct Hilbert spaces ofentire functions H(β) for a given weight sequence β. In Section 4, a necessarycondition for the boundedness of Cϕ on spaces H(β) is given (Theorem 4.6);some sufficient conditions for bounded Cϕ are shown (Theorems 4.8 and 4.13).Several lemmas in Sections 3 and 4 are closely related to those in [15], whichconsiders general reproducing kernel Hilbert spaces, so we provide either arefinement or an alternative proof for these results in our particular functionspaces. Sections 5 and 6 develop characterizations (necessary and sufficientconditions) of the boundedness of composition operators on H(β) for specific

Composition Operators 3

classes of weight sequences β, namely, class C1 (Theorem 5.4) and class C2

of Fock-like spaces (Theorem 6.7). These sections, with various examplesprovided, are the most important findings of our research. Section 7 presentssome results on the compactness of composition operators. Finally, in Section8, we give some concluding remarks and the corresponding open questions.

2. Preliminaries

In this section, we recall some important facts and results that will be usedin the paper.

Fix a positive integer N throughout the paper. We write CN for theCartesian product of N copies of the complex plane C. Elements in CNare tuples z = (z1, . . . , zN ), where zj ∈ C for all j ∈ {1, . . . , N}. Withw = (w1, . . . , wN ) ∈ CN , we denote by 〈z, w〉 the Euclidean inner product ofz and w, i.e., 〈z, w〉 = w∗z = z1w1 + · · · + zNwN . The associated norm is

|z| =√|z1|2 + · · ·+ |zN |2.

We write N for the set of all non-negative integers. An element α =(α1, . . . , αN ) ∈ NN is called a multi-index. We use the following multi-indexnotations: zα = zα1

1 · · · zαN

N , |α| = α1 + · · ·+ αN and α! = α1! . . . αN !.

Recall that a function f : CN → C is said to be entire if f is entire ineach variable zj , i.e. ∂f/∂zj = 0 for all j ∈ {1, . . . , N}. We may then writef in the homogeneous expansion centered at the origin f(z) =

∑∞n=0pn(z),

where each pn is a homogeneous polynomial of order n, that is, pn(z) =∑|α|=n cαz

α for some (cα) ⊂ C.

We denote the unit ball of CN by BN and SN as its boundary, the unitsphere. Let σN be the normalized rotation-invariant positive Borel measureon SN . The space of all square integrable measurable functions on the unitsphere L2 = L2(SN , dσN ) is a Hilbert space with inner product

〈f, g〉L2 =

∫SNf(z)g(z)dσN (z), f, g ∈ L2.

For two different multi-indices µ and γ, the monomials zµ and zγ areorthogonal in L2, and the norm of the monomial zµ is given by (see, e.g. [12,Proposition 1.4.9])

νµ := ‖zµ‖2L2 =

∫SN|zµ|2dσN (z) =

(N − 1)!µ!

(N − 1 + |µ|)!. (2.1)

This implies that for a homogeneous polynomial pn(z) =∑|α|=n cαz

α, we

have

‖pn‖2L2 =∑|α|=n

|cα|2να. (2.2)


Denote νn =(N−1+n

n

)−1= (N−1)!n!

(N−1+n)! for each n ∈ N. Direct calculations

verify that

ν(n,0,...,0) = νn, ν(n,1,0,...) =νn+1

n+ 1, (2.3)

and more generally,

νµµ!

=ν|µ|

|µ|!. (2.4)

We have the following multinomial expansion∑|α|=n

|zα|2

να=

1

νn

∑|α|=n

n!

α!|zα|2 =

1

νn

∑|α|=n

n!

α!|z1|2α1 · · · |zN |2αN (2.5)

=1

νn

(|z1|2 + · · ·+ |zN |2

)n=|z|2n

νn.

The following inequality, which is probably well known, will be usefulfor us.

Lemma 2.1. Let m and n be two non-negative integers. Let α be a multi-indexin NN with |α| = m+ n. We then have∑

µ+γ=α|µ|=m

α!

µ! γ!=

(m+ n)!

m!n!. (2.6)

Consequently, for any multi-indices µ and γ in NN ,

(µ+ γ)!

µ! γ!≤ (|µ|+ |γ|)!|µ|! |γ|!

.

Indeed, from the identity

(1 + x)α1 · · · (1 + x)αN = (1 + x)m+n

computing the coefficients of xm on both sides yields (2.6).We recall the Singular Value Decomposition Theorem, whose proof can

be found in many sources on Matrix Theory, see, e.g. [11, Theorem 2.6.3].

Theorem 2.2. (Singular Value Decomposition) Let A be an N × N matrixof rank m ≤ N . Then there are N × N unitary matrices U and V and asquare diagonal matrix A = diag(σ1, . . . , σN ) such that A = UAV . Here,σ1 ≥ · · · ≥ σm ≥ σm+1 = · · · = σN = 0 are the singular values of A, whichare non-negative square roots of the eigenvalues of A∗A.

The following corollary [2, Lemma 1] will also be useful for us later.

Corollary 2.3. Let A be an N × N matrix and b be a vector in CN suchthat ‖A‖ ≤ 1 and 〈A∗b, ζ〉 = 0 for all ζ ∈ CN satisfying |Aζ| = |ζ|. WriteA = UAV as described in Theorem 2.2 and b = U∗b. Suppose σ1 = σ2 =· · · = σp = 1 for some p ≤ N . Then the first p coordinates of b are all zeros.


A mapping F = (f1, . . . , fN ) : CN → CN is entire if each componentfj : CN → C is entire for j = 1, . . . , N . Let ϕ : CN → CN be an entiremapping andH be a Hilbert space of entire functions on CN . The compositionoperator Cϕ (induced by ϕ) acting on H is defined as

Cϕf = f ◦ ϕ for f ∈ H.

Let ω > 0. The well-known Fock (also known as Segal–Bargmann) spaceF2ω is the Hilbert space of entire functions on CN that are square integrable

with respect to the Gaussian measure dVω(z) = e−ω|z|2

dV (z), where dV isthe Lebesgue volume measure on CN . More specifically, an entire function fon CN belongs to F2

ω if and only if

‖f‖F2ω

:=

((ωπ

)N ∫CN

|f(z)|2e−ω|z|2

dV (z)

)1/2

<∞.

Note that the set {ω|α|(α!)−1/2zα} forms an orthonormal basis for F2ω. The

paper [2] provides us with the following important theorem for the bounded-ness and compactness of composition operator on F2

ω.

Theorem 2.4. Let ϕ be an entire mapping on CN . The composition operatorCϕ acting on a Fock space F2

ω is

(i) bounded if and only if ϕ(z) = Az + b, where A is a linear operator onCN with ‖A‖ ≤ 1, b ∈ CN and 〈A∗b, ζ〉 = 0 whenever |Aζ| = |ζ|,

(ii) compact if and only if ϕ(z) = Az + b, where A is a linear operator onCN with ‖A‖ < 1 and b ∈ CN .

We close this section by recalling the notion of Hilbert–Schmidt andSchatten class operators. A bounded linear operator L on a Hilbert space His called a Hilbert–Schmidt operator if it has finite Hilbert–Schmidt norm(also known as Frobenius norm) ‖L‖HS. That is, for some orthonormal basis{ej}j∈J of H, we have

‖L‖HS :=(∑j∈J‖Lej‖2

)1/2

< +∞.

It is well known that ‖L‖HS does not depend on the choice of the orthonormalbasis and that if L is Hilbert–Schmidt, then it is compact. For 0 < p < ∞,the Schatten p-class consists of all bounded linear operators L on H for which(L∗L)p/4 is a Hilbert–Schmidt operator. The set of Schatten p-class operatorsforms an ideal in the algebra of all bounded linear operators on H. If L isdiagonal with respect to an orthonormal basis {ej}j∈J , that is, Lej = aj ejfor all j ∈ J , then it is well known that L belongs to the Schatten p-class ifand only if

∑j∈J |aj |p < ∞. For more details on these classes of operators,

see, for example, [16, Chapter 1].


3. Hilbert spaces of entire functions in several variables

3.1. Hilbert spaces H(β) of formal power series

Recall that a polynomial p in CN is homogeneous of degree n if

p(tz) = tnp(z) for all t ∈ C, z ∈ CN .

Note that the zero polynomial is considered, by definition, homogeneous ofdegree n for any integer n ≥ 0.

Let β = (βn) be a sequence of positive real numbers. For each integern ≥ 0, let Pn(β) be the Hilbert space of all homogeneous polynomials ofdegree n equipped with the following inner product

〈p, q〉β := 〈p, q〉L2 β2n,

for all p, q in Pn(β). Recall that 〈p, q〉L2 denotes the inner product of p andq in L2(SN , dσN ). If p(z) =

∑|α|=n cαz

α, then (2.2) gives

‖p‖β =( ∑|α|=n

|cα|2‖zα‖2L2

)1/2

βn =( ∑|α|=n

|cα|2 να β2|α|

)1/2

.

We then define H(β) to be the Hilbert space direct sum

H(β) =

∞⊕n=0

Pn(β). (3.1)

The space H(β) can be considered as the space of formal (not necessarilyconvergent) homogeneous expansions f =

∑∞n=0pn with

‖f‖H(β) =( ∞∑n=0

‖pn‖2β)1/2

=( ∞∑n=0

‖pn‖2L2 β2n

)1/2

<∞,

where pn ∈ Pn(β). Writing each pn as a linear combination of monomials∑|α|=n cαz

α, we see that f can also be considered as a formal power series

f =∑α cαz

α with

‖f‖H(β) =(∑

α

|cα|2 ναβ2|α|

)1/2

. (3.2)

The inner product in H(β) is then given by

〈f, g〉H(β) =∑α

cαdα να β2|α|,

if f =∑α cαz

α and g =∑α dαz

α. It follows that the set of multiples of

monomials{

(β|α|√να)−1zα : α ∈ NN

}forms an orthonormal basis for H(β).

We shall refer to it as the standard orthonormal basis.The space H(β) is called a weighted Hardy space in [7, Chapter 2].

With appropriate choices of the weight sequence, H(β) can be identified withthe classical Hardy, Bergman and weighted Bergman spaces of holomorphicfunctions over the unit ball. On the other hand, with certain choices of β,the formal power series of an element in H(β) may diverge. In the following


theorem, we obtain a characterization of the weight sequence β for which allelements of H(β) are entire functions over CN .

Theorem 3.1. Let β = (βn) be a sequence of positive numbers and H(β) bethe associated Hilbert space of formal power series. Then all elements of H(β)are entire functions over CN if and only if

limn→∞

β1/nn =∞. (3.3)

Proof. • Necessity: Assume that all elements of H(β) are entire functionsover CN . Consider the formal power series

f =

∞∑n=1

zn1nβn

.

We have

‖f‖2H(β) =

∞∑n=1

1

n2 β2n

‖zn1 ‖2L2 β2n ≤

∞∑n=1

1

n2<∞.

Therefore, f belongs to H(β) and hence it is an entire function over CN . Itthen follows that

limn→∞

( 1

nβn

)1/n

= 0.

Since limn→∞ n1/n = 1, we conclude that (3.3) holds.• Sufficiency: Suppose (3.3) holds. Let f =

∑α cαz

α be a formal powerseries belonging to H(β). We have∑

α

|cα|2 να β2|α| = ‖f‖2H(β) <∞.

For any z ∈ CN , the Cauchy–Schwarz Inequality gives∑α

|cαzα| ≤(∑

α

|cα|2 να β2|α|

)1/2(∑α

|zα|2

να β2|α|

)1/2

= ‖f‖H(β)

( ∞∑n=0

β−2n

∑|α|=n

|zα|2

να

)1/2

= ‖f‖H(β)

( ∞∑n=0

ν−1n β−2

n |z|2n)1/2

(by (2.5)). (3.4)

Since limn→∞ β−1/nn = 0 by assumption and limn→∞ ν

−1/nn = 1, the series

in (3.4) converges. This shows that the formal power series of f actuallyconverges absolutely for any z in CN and it defines an entire function. �

Example 3.2. Let ω > 0. For βn = ω−n( (N−1+n)!

(N−1)!

)1/2, the space H(β) can be

identified with the Fock space F2ω discussed in the previous section. One can

verify that the condition (3.3) holds.


Example 3.3. Consider N = 1. Let β be any weight sequence for which the

sequence of ratios nβn−1

βnis bounded and that lim infn→∞

log βn

n logn < +∞. It is

shown in [9] that condition (3.3) holds.Every element f in the induced space H(β) has finite growth order ρf ,

and at least one element g ∈ H(β) has positive growth order. Moreover, everytranslation operator Tu : f(z) 7→ f(z+ u) (u ∈ C) is bounded on H(β). Suchspace is denoted by H(β+

ρ , T ). This class of spaces and actions of compositionoperators on them are well studied in [9].

3.2. Reproducing kernels

Recall that a Hilbert space H of functions from a non-empty set X to Cis called a reproducing kernel Hilbert space (RKHS) if for every y ∈ X, theevaluation functional δy : f 7→ f(y) is bounded. By the Riesz RepresentationTheorem, there exists a unique element Ky ∈ H such that f(y) = 〈f,Ky〉 forevery f ∈ H. We call Ky the reproducing kernel at the point y. The functionK : X ×X → C defined by

K(x, y) = 〈Ky,Kx〉 = Ky(x), x, y ∈ X,is called the reproducing kernel for H. It is well known that if a collection ofelements {ej : j ∈ J} is an orthonormal basis for H, then

K(x, y) =∑j∈J

ej(x)ej(y),

where the convergence is pointwise for x, y ∈ X. The theory of reproducingkernel Hilbert spaces was investigated in great details by Aronszajn in hisseminal paper [1] in the fifties.

We show in the following proposition that if all elements of H(β) areentire functions, then H(β) is a reproducing kernel Hilbert space.

Proposition 3.4. Let β = (βn) satisfy the condition in Theorem 3.1. Then thespace H(β) induced by β is a reproducing kernel Hilbert space over CN withthe reproducing kernel K : CN × CN → C given by

K(z, w) =

∞∑n=0

〈z, w〉n

νnβ2n

.

The convergence is uniform on compact subsets of CN × CN .

Proof. From (3.4) in the proof of Theorem 3.1, we see that for any z ∈ CN ,there exists a constant Mz > 0 such that |f(z)| ≤ Mz‖f‖H(β) for all f ∈H(β). Hence, each evaluation functional δz is bounded, which shows thatH(β) is an RKHS.

Recall that the set of monomials{

(β|α|√να)−1zα : α ∈ NN

}forms an

orthonormal basis of H(β). Using the multinomial expansion and (2.4), wethen have

K(z, w) =∑α

zαwα

ναβ2|α|

=

∞∑n=0

1

νnβ2n

∑|α|=n

n!

α!zαwα =

∞∑n=0

〈z, w〉n

νnβ2n

.


Note that since limn→∞(νnβ2n)−1/n = 0, the series converges uniformly on

compact subsets of CN×CN . This completes the proof of the proposition. �

One well-known property of RKHSs that is useful to us is that if acomposition operator Cϕ maps an RKHS H into itself (i.e. Cϕ(H) ⊆ H),then it is bounded (cf. the proof of [15, Proposition 6.2]). This follows fromthe closed graph theorem and the fact that norm convergence in H impliespointwise convergence.

Remark 3.5. We sketch here an alternative approach to the construction ofH(β). Define for ζ ∈ C,

Φ(ζ) =

∞∑n=0

ζn

νn β2n

. (3.5)

Proposition 3.4 shows that H(β) is a reproducing kernel Hilbert space withkernel K(z, w) = Φ(〈z, w〉). Consequently, H(β) coincides with the spaceΦ(CN ) in [15, Section 3]. Now, [15, Theorem 7.2] provides a unitary operatorbetween Φ(CN ) and the direct sum

⊕n≥0(CN )�n, where (CN )�n denotes

the symmetric tensor product of n copies of CN . From the standard orthonor-mal basis of CN , via the well-known procedure (see e.g., [14]), one constructsthe corresponding orthonormal basis for (CN )�n, hence recovers the standardbasis

{(β|α|√να)−1zα : α ∈ NN

}for H(β).

Convention. From here to the end of the paper, unless otherwise stated,whenever we write β or H(β), we implicitly assume that β satisfies condition(3.3) in Theorem 3.1 so that all elements of H(β) are entire functions.

4. Bounded composition operators

In this section we investigate the boundedness of composition operators onH(β). It turns out that the characterization of the boundedness dependsstrongly on the behavior of the weight sequence β.

4.1. Composition operators with linear symbols

If ϕ(z) = Az, where A : CN → CN is a linear map, then we denote the com-position operator Cϕ by CA and call this operator composition operator withlinear symbols. Note that A can be considered as an N ×N complex matrix.We establish some elementary properties of CA in the following proposition.

Proposition 4.1. The following statements hold.

(a) For any unitary map V : CN → CN , the composition operator CV is aunitary operator on H(β).

(b) For any linear map A : CN → CN with ‖A‖ ≤ 1, the compositionoperator CA is bounded on H(β) and ‖CA‖ = 1. If ‖A‖ < 1, then CAbelongs to any Schatten p-class for 0 < p <∞.


Remark 4.2. By Remark 3.5, (a) follows from [15, Corollary 8.4] and the firstpart of (b) is a particular case of [15, Theorem 8.2(ii)]. We present here adifferent approach which is more closely related with our construction of thespace H(β) in Section 3.

Proof. (a) Suppose V is a unitary map on CN . Let f =∑∞n=0pn be an

element in H(β). Then

CV f =

∞∑n=0

pn ◦ V

is a homogeneous expansion of CV f . It follows that

‖CV f‖2H(β) =

∞∑n=0

‖pn ◦ V ‖L2 β2n =

∞∑n=0

‖pn‖L2 β2n = ‖f‖2H(β).

In the second equality, we have used the fact that the norm on L2(SN , dσN )is unitarily invariant. We have shown that CV is an isometry on H(β). Inaddition, it can be checked directly that CV is an invertible operator withinverse CV −1 . We conclude that CV is a unitary operator.

(b) By the Singular Value Decomposition Theorem (Theorem 2.2), anylinear operator A on CN may be written as A = UAV , where U and V areunitary operators and A is diagonal of the form

A(z1, . . . , zN ) = (σ1z1, . . . , σNzN ) for z = (z1, . . . , zN ) ∈ CN .

Note that ‖A‖ = ‖A‖ = σ1. Since

CA = CUAV = CV CACU

and CV , CU are unitary, any statement about the boundedness and Schattenmembership of CA holds if and only if the same statement for CA holds.Consequently, it suffices to prove (b) for A.

For any multi-index α, we compute CAzα = σαzα, where we have used

the multi-index notation σα = σα11 · · ·σ

αN

N . This shows that CA is a diagonaloperator with respect to the standard orthonormal basis of H(β). Since 0 ≤σj ≤ 1 for all 1 ≤ j ≤ N , we conclude that

‖CA‖ = sup{σα : α ∈ NN

}= 1.

If ‖A‖ < 1 then 0 ≤ σj < 1 for all 1 ≤ j ≤ N . It follows that for any0 < p <∞, the pth power of the Schatten p-norm of CA is given by∑

α

σpα =

N∏j=1

∞∑αj=0

σpαj

j =

N∏j=1

1

1− σpj<∞.

This shows that CA belongs to any Schatten p-class for 0 < p <∞. �

Using the results in Proposition 4.1, we obtain estimates for the normof several polynomials in H(β). These estimates will be useful when we in-vestigate other properties of composition operators.


Lemma 4.3. Let v be a vector in CN and δ be a complex number. For eachinteger n ≥ 1, put hn(z) = (〈z, v〉)n and gn(z) = (〈z, v〉+ δ)n. For all n ≥ 0,we have

‖hn‖H(β) = |v|n√νn βn, (4.1)

and

‖gn‖2H(β) =

n∑k=0

(n

k

)2

|δ|2(n−k)|v|2k νk β2k. (4.2)

Consequently, for each 0 ≤ k ≤ n,

‖gn‖H(β) ≥(n

k

)|δ|n−k|v|k

√νk βk. (4.3)

Proof. Choose a unitary operator W on CN such that W ∗v = |v|e1, wheree1 = (1, 0, . . . , 0). Then CWhn = |v|nzn1 . By (3.2),

‖hn‖H(β) = ‖CWhn‖H(β) = |v|n‖zn1 ‖H(β) = |v|n√νn βn.

Equation (4.2) for the norm of gn follows from the binomial expansion, theorthogonality of homogeneous polynomials with distinct degrees inH(β), and(4.1). �

4.2. Composition operators with general symbols

Now we study the boundedness of Cϕ on H(β), where ϕ : CN → CN is anentire mapping. We see from Proposition 4.1 that if ϕ is a linear map thenCϕ is bounded. On the other hand, if ϕ is a constant map, ϕ(z) = b for allz ∈ CN , then Cϕ can be identified as the bounded rank-one operator 1⊗Kb,where Kb denotes the reproducing kernel at b. That is, for f ∈ H(β),

Cϕf = f(b) = 〈f,Kb〉 · 1 = (1⊗Kb)(f).

See [15, Proposition 4.4] for a detailed treatment of such Cϕ in a more generalsetting.

In this section, we obtain several necessary conditions and sufficientconditions for the boundedness of more general composition operator Cϕ onH(β).

We begin with a well-known formula on the action of the adjoint of abounded composition operator on reproducing kernel functions. The formulain fact holds on any reproducing kernel Hilbert space.

Lemma 4.4 (cf. [15, Theorem 4.2]). Suppose a composition operator Cϕ isbounded on H(β). Then

C∗ϕKw = Kϕ(w), ∀w ∈ CN .

To obtain a necessary condition on the symbol ϕ for which Cϕ is abounded operator on H(β), we first present a preparatory result. The follow-ing lemma is a refined version of [15, Lemma 6.1] (cf. [13, Lemma 2.1]).


Lemma 4.5. Let Φ be a non-polynomial function given in the form of a seriesΦ(t) =

∑∞n=0 ant

n, where an ≥ 0 for all n ≥ 0 and the series converges forall t ≥ 0. Then for any real number γ > 1, we have

limt→∞

Φ(γt)

Φ(t)=∞. (4.4)

Consequently, if {sm} and {tm} are two sequences of non-negative numberssuch that limm→∞ sm =∞ and supm≥0{Φ(tm)/Φ(sm)} <∞, then

lim supm→∞

tmsm≤ 1.

Proof. Fix an integer M ≥ 1. Let hM (t) =∑M−1n=0 ant

n. We have

Φ(γt) ≥∞∑

n=M

anγntn ≥ γM

( ∞∑n=M

antn)

= γM (Φ(t)− hM (t)).

It follows that

Φ(γt)

Φ(t)≥ γM

(1− hM (t)

Φ(t)

).

Since Φ is not a polynomial and an ≥ 0 for all n ≥ 0, we have hM (t)/Φ(t)→ 0as t→∞. It follows that

lim inft→∞

Φ(γt)

Φ(t)≥ γM .

Since M was arbitrary and γ > 1, we obtain (4.4).Now suppose {sm} and {tm} are two sequences of non-negative numbers

such that limm→∞ sm =∞ and M = supm≥0{Φ(tm)/Φ(sm)} <∞. To arriveat a contradiction, assume that lim supm→∞ tm/sm > γ > 1. Then thereexists a strictly increasing sequence {mk} of integers such that tmk

≥ γ smk

for all k ≥ 1. It then follows that

Φ(γ smk)

Φ(smk)≤ Φ(tmk

)

Φ(smk)≤M <∞.

Letting k →∞ and using (4.4), we obtain a contradiction. �

In our characterization of the boundedness of Cϕ on H(β), we needan auxiliary sequence of positive real numbers associated with each weight

sequence β. We define χ(β) = (χ(β)n ) by the ratio

χ(β)n =

√n(n+N − 1)

βn−1

βn, n ≥ 1. (4.5)

To simplify the notation when there is no danger of confusion, we shall drop

the superscript and write χn instead of χ(β)n . We have the following useful

identities:

χk+1 · · ·χn(n− k)!

=

√(n

k

)(n+N − 1

k +N − 1

)βkβn

=

(n

k

)√νk βk√νn βn

, (4.6)


for all k ≥ 0 and n ≥ k + 1; and√|µ|!

(|µ|+ |γ|)!χ|µ|+1 · · ·χ|µ|+|γ| =

√(µ+ γ)!

µ!

√νµ β|µ|

√νµ+γ β|µ|+|γ|

, (4.7)

for any multi-indices µ, γ ∈ NN .We now describe several necessary conditions for a composition operator

Cϕ to be bounded onH(β). In the case of the classical Fock space over CN , werecover the result in [2]. The existence of the linear operator A and the vectorb in the result below was established in [15, Proposition 6.2] in a more generalcontext. However, in the case of H(β), our theorem provides a refinement ofthat result.

Theorem 4.6. Let ϕ : CN → CN be an entire mapping. If the compositionoperator Cϕ is bounded on the space H(β), then ϕ(z) = Az+b for some linearoperator A on CN with operator norm ‖A‖ ≤ 1, and b ∈ CN . Furthermore,for any unit vector v ∈ CN , we have{

supm≥1, k≥0

(χk+1 · · ·χk+m

m!|A∗v|k

)1/m}|〈b, v〉| <∞, (4.8)

and

lim supm→∞

{supk≥0

(χk+1 · · ·χk+m

m!|A∗v|k

)1/m}|〈b, v〉| ≤ 1. (4.9)

Proof. Since Cϕ is bounded, Lemma 4.4 shows that

‖Kϕ(z)‖2

‖Kz‖2=‖C∗ϕKz‖2

‖Kz‖2≤ ‖C∗ϕ‖2 = ‖Cϕ‖2, ∀z ∈ CN ,

which is equivalent to,

Φ(|ϕ(z)|2)

Φ(|z|2)≤ ‖Cϕ‖2, ∀z ∈ CN .

Here, we define Φ(t) =∑∞n=0 ν

−1n β−2

n tn, which is convergent for all t ≥ 0,

since limn→∞ β−1/nn = 0 and limn→∞ ν

−1/nn = 1. Lemma 4.5 shows

lim sup|z|→∞

|ϕ(z)||z|

≤ 1.

By the argument as in the proof of [2, Theorem 1], we have ϕ(z) = Az + bfor some linear operator A on CN with ‖A‖ ≤ 1 and b ∈ CN .

Now let v be a unit vector in CN . Define hn(z) = 〈z, v〉n for each integern ≥ 1. By Lemma 4.3,

‖hn‖H(β) =√νn βn

and for any 0 ≤ k ≤ n,

‖Cϕhn‖H(β) = ‖〈Az + b, v〉n‖H(β)

= ‖(〈z,A∗v〉+ 〈b, v〉)n‖H(β)

≥(n

k

)|〈b, v〉|n−k|A∗v|k

√νk βk.


The boundedness of Cϕ gives ‖Cϕhn‖H(β) ≤ ‖Cϕ‖ ‖hn‖H(β). It follows that(n

k

)|〈b, v〉|n−k|A∗v|k

√νk βk ≤ ‖Cϕ‖

√νn βn.

Using the identity (4.6) and writing m = n− k yield

χk+1 · · ·χk+m

m!|A∗v|k · |〈b, v〉|m ≤ ‖Cϕ‖.

Taking mth root then gives{χk+1 · · ·χk+m

m!|A∗v|k

}1/m

|〈b, v〉| ≤ ‖Cϕ‖1/m.

Taking supremum in both k and m yields (4.8). Taking supremum in k, thenlimsup in m, and using the fact that ‖Cϕ‖1/m → 1 as m → ∞ we obtain(4.9). �

By Theorem 4.6, the study of bounded composition operators Cϕ on thespace H(β) reduces to investigating the case where ϕ is an affine mapping ofthe form ϕ(z) = Az+ b. By making use of the Singular Value Decomposition(Theorem 2.2), we may further restrict our attention to a special class ofsymbols which are described in the following lemma (cf. [15, Proposition 6.4]).

Lemma 4.7. Suppose ϕ(z) = Az + b, where ‖A‖ ≤ 1. Write A = UAV asdescribed in Theorem 2.2. Let ψ(z) = Az+b with b = U∗b. Then on H(β), theoperator Cψ is bounded (respectively, compact) if and only if Cϕ is bounded(respectively, compact) and ‖Cϕ‖ = ‖Cψ‖. Furthermore, for any 0 < p <∞,the Schatten p-norms of Cϕ and of Cψ are equal.

Proof. Since ϕ = U ◦ ψ ◦ V , it follows that

Cϕ = CV CψCU .

Because CV and CU are unitary operators on H(β) by Proposition 4.1, theconclusions of the lemma follow. �

We have seen that linear and constant maps ϕ : CN → CN always giverise to bounded composition operators Cϕ on any Hilbert spaceH(β) of entirefunctions. As an application of Theorem 4.6, we derive a characterizationof the weight sequence β for which there are other bounded compositionoperators on H(β).

Theorem 4.8. Let β be a weight sequence and χ be the associated sequencedefined in (4.5). The following statements are equivalent.

(a) There exists an entire map ϕ on CN that is neither linear nor constantsuch that Cϕ is bounded on H(β).

(b) There exists a number r ∈ (0, 1] such that

supm≥1, k≥0

(χk+1 · · ·χk+m

m!rk)1/m

<∞. (4.10)


In this case, the limit

τ = lim supm→∞

{supk≥0

(χk+1 · · ·χk+m

m!rk)1/m}

is finite, and for any map ϕ(z) = Az+ b with ‖A‖ < r and τ |b| < 1, theoperator Cϕ is bounded on H(β) and it belongs to any Schatten p-classfor 0 < p <∞.

Proof. We first show that (a) implies (b). Assume that there exists an entiremapping ϕ which is neither linear nor constant such that Cϕ is bounded onH(β). Theorem 4.6 implies that ϕ(z) = Tz + d, where T is a non-zero linearoperator with ‖T‖ ≤ 1 and d ∈ CN \ {0}. There then exists a unit vectorv such that T ∗v 6= 0 and 〈d, v〉 6= 0 (to see this, one can use, for instance,Theorem 2.2). Put r = |T ∗v|. We have 0 < r ≤ ‖T‖ ≤ 1 and by inequality(4.8) in Theorem 4.6,

supm≥1, k≥0

(χk+1 · · ·χk+m

m!rk)1/m

|〈d, v〉| <∞.

Since 〈d, v〉 6= 0, we obtain (4.10).

It is clear that 0 ≤ τ <∞. We now show that whenever ϕ(z) = Az + bwith ‖A‖ < r and τ |b| < 1, the operator Cϕ is bounded on H(β) and itbelongs to any Schatten p-class for 0 < p <∞. By Lemma 4.7, we only needto consider the case A(z1, . . . , zN ) = (σ1z1, . . . , σNzN ) where 0 ≤ σj < r forall 1 ≤ j ≤ N . We first show that Cϕ is a Hilbert–Schmidt operator.

Define ω0 = 1 and ωm = supk≥0

(χk+1 · · ·χk+mr

k)

for each integer

m ≥ 1. Then χk+1 · · ·χk+m ≤ ωm r−k for all m ≥ 1 and k ≥ 0. Identity (4.7)yields

(µ+ γ)!

µ!

νµ β2|µ|

νµ+γ β2|µ|+|γ|

=|µ|!

(|µ|+ |γ|)!χ2|µ|+1 · · ·χ

2|µ|+|γ|

≤ |µ|!(|µ|+ |γ|)!

ω2|γ| r

−2|µ| (4.11)

for all µ and γ in NN .


Recall that {eα(z) = (√να β|α|)

−1 zα : α ∈ NN} is an orthonormal basisfor H(β). For any multi-index α, the binomial expansion gives

(Cϕeα)(z) = eα(ϕ(z)) =1

√να β|α|

N∏j=1

(σjzj + bj)αj

=1

√να β|α|

α1∑γ1=0

· · ·αN∑γN=0

α!

γ! (α− γ)!σα−γ bγ zα−γ

=∑

µ+γ=α

(µ+ γ)!

γ!µ!σµ bγ

zµ√νµ+γ β|µ|+|γ|

(by setting µ = α− γ)

=∑

µ+γ=α

(µ+ γ)!

γ!µ!σµ bγ

√νµ β|µ|

√νµ+γ β|µ|+|γ|

eµ(z).

We then have

‖Cϕeα‖2H(β) =∑

µ+γ=α

|σµ|2 |bγ |2 (µ+ γ)!2

γ!2 µ!2

νµ β2|µ|

νµ+γ β2|µ+γ|

≤∑

µ+γ=α

|σµ|2 |bγ |2 1

γ!

(µ+ γ)!

γ!µ!

|µ|!(|µ|+ |γ|)!

ω2|γ| r

−2|µ|

(by (4.11))

≤∑

µ+γ=α

|(r−1σ)µ|2 |bγ |2ω2|γ|

γ! |γ|!.

The last inequality follows from Lemma 2.1. We now compute∑α

‖Cϕeα‖2H(β) ≤∑µ

∑γ

|(r−1σ)µ|2 |bγ |2ω2|γ|

γ! |γ|!

=(∑

µ

|(r−1σ)µ|2)(∑

γ

|bγ |2ω2|γ|

γ! |γ|!

)

=( N∏j=1

1

1− r−2σ2j

){ ∞∑m=0

ω2m

m!2

( ∑|γ|=m

m!

γ!|bγ |2

)}

=( N∏j=1

1

1− r−2σ2j

){ ∞∑m=0

ω2m

m!2|b|2m

}. (4.12)

Since

τ = lim supm→∞

(ωmm!

)1/m

and τ |b| < 1, the series in (4.12) converges. This shows that Cϕ is a Hilbert–Schmidt operator on H(β). In particular, Cϕ is bounded on H(β).

Now choose a constant 0 < λ < 1 such that λ−1‖A‖ < r. Define ϕ1(z) =λz and ϕ2(z) = λ−1Az + b. Then ϕ = ϕ2 ◦ ϕ1 and hence,

Cϕ = Cϕ1Cϕ2

.


The above argument shows that Cϕ2is bounded on H(β). In addition, Propo-

sition 4.1 tells us that Cϕ1 belongs to any Schatten p-class for 0 < p < ∞.The same conclusion for Cϕ then follows.

Finally, it is clear that (b) implies (a). This completes the proof of thetheorem. �

Theorem 4.8 immediately provides a condition on the weight sequencefor which the only bounded composition operators Cϕ on H(β) are inducedby ϕ(z) = Az with ‖A‖ ≤ 1, or ϕ(z) = b.

Corollary 4.9. Suppose that for all 0 < r ≤ 1,

supm≥1, k≥0

(χk+1 · · ·χk+m

m!rk)1/m

=∞. (4.13)

If Cϕ is bounded on H(β), then ϕ is either linear or constant on CN .

We now exhibit several examples of weight sequences β satisfying con-dition (4.13).

Example 4.10. Define β0 = 1 and for n ≥ 1,

βn =

{nn if n is odd,

n2n if n is even.

For any r > 0 and any even integer k ≥ 2, we have

χk+1rk =

√(k + 1)(k +N)

βkβk+1

rk ≥ (k + 1)k2k

(k + 1)k+1rk =

( r k2

k + 1

)k.

This shows that

supk≥0

(χk+1r

k)

=∞

and hence condition (4.13) is satisfied (in fact, even with m = 1, the supre-mum taken in k is already infinite).

In order to obtain more examples, it is useful to work directly with thesequence χ instead of the weight sequence β. We provide here the constructionof β from χ and offer a condition on χ to ensure that β satisfies (3.3) inTheorem 3.1.

Lemma 4.11. Let {χn}n≥1 be a sequence of positive numbers. Then thereexists a weight sequence β = (βn) such that

χn =√n(n+N − 1)

βn−1

βnfor all n ≥ 1. (4.14)

Furthermore, limn→∞ β1/nn =∞ if and only if

limn→∞

( n!

χ1 · · ·χn

)1/n

=∞. (4.15)


Proof. Define β0 =√

(N − 1)! and for each n ≥ 1,

βn =

√n! (n+N − 1)!

χ1 · · ·χn.

A direct calculation shows that (4.14) holds. In addition,

n!

χ1 · · ·χn≤ βn ≤

(n+N)N−1 · n!

χ1 · · ·χn, n ≥ 1.

Since limn→∞(n + N)(N−1)/n = 1, we conclude that limn→∞ β1/nn = ∞ if

and only if (4.15) holds. �

Example 4.12. Condition (4.13) does seem complicated to verify so one maywonder if there exists an equivalent but simpler condition, which only involvesa finite value of m. In fact, in Example 4.10, we only need m = 1. Unfortu-nately, there exists an example which shows this may not be possible. Morespecifically, we shall construct a weight sequence for which condition (4.13)is satisfied but for each 0 < r < 1 and each integer m ≥ 1, we have

supk≥0

(χk+1 · · ·χk+m r

k)<∞.

For any integer n ≥ 1, there exists a unique pair of non-negative integers sand ` for which n = 2s + ` and 0 ≤ ` < 2s. We then define

χn =

{(2s!)−2 if ` = 0,

`2 if 1 ≤ ` < 2s.

A first few terms of the sequence χ are( 1

1!

)2

,( 1

2!

)2

, 12,( 1

4!

)2

, 12, 22, 32,( 1

8!

)2

, 12, 22, 32, 42, 52, 62, 72,( 1

16!

)2

, . . . .

Note that χ1 · · ·χn ≤ 1 for all n ≥ 1 so this sequence satisfies condition (4.15)in Lemma 4.11. Furthermore, we have χn ≤ n2 for all n ≥ 1. This impliesthat for each 0 < r < 1 and each fixed m ≥ 1,

supk≥0

(χk+1 · · ·χk+m r

k)≤ sup

k≥0

{(k +m)2m rk

}<∞.

However, for 0 < r < 1, k = 2s and m = k − 1 = 2s − 1, we have(χk+1 · · ·χk+m

m!rk)1/m

=( (m!)2

m!rm+1

)1/m

= (m!)1/m r1+1/m.

Since (m!)1/m →∞ as m = 2s − 1→∞, we conclude that

supm≥1, k≥0

(χk+1 · · ·χk+m

m!rk)1/m

=∞.

We discuss here another consequence of Theorem 4.8 and its examplesthat will be useful for Section 6 and Section 7.

Theorem 4.13. The following statements are equivalent.

(a) The operator Cϕ is bounded on H(β) whenever ϕ(z) = Az + b with‖A‖ < 1 and b ∈ CN .


(b) For each 0 < r < 1,

supm≥1

supk≥0

(χk+1 · · ·χk+m

m!rk)1/m

<∞, (4.16)

and

limm→∞

{supk≥0

(χk+1 · · ·χk+m

m!rk)1/m}

= 0. (4.17)

Moreover, if one, hence both, of the statements hold, then any Cϕ in (a)belongs to every Schatten p-class for 0 < p <∞.

Proof. Assume that (a) holds. Let 0 < r < 1 be given. Take b to be anynonzero vector in CN and define ϕ(z) = rz + b (so A = rI, where I is theidentity operator on CN ). Then Cϕ is bounded on H(β). Applying Theo-rem 4.6 with v = b/|b| yields

supm≥1, k≥0

{(χk+1 · · ·χk+m

m!|A∗v|k

)1/m}|b| <∞

and

lim supm→∞

{supk≥0

(χk+1 · · ·χk+m

m!|A∗v|k

)1/m}≤ 1

|b|.

Since |A∗v| = r and b 6= 0 was arbitrary, (4.16) and (4.17) follow. Thus, (a)implies (b).

The implication (b) → (a) and the “moreover part” follow from Theo-rem 4.8. �

Example 4.14. The following weight sequence satisfies conditions (4.16) and(4.17). Consider for n ≥ 0,

βn =

{(n+N − 1)! if n is even,

(n+N − 2)! if n is odd.

For m ≥ 1 and k ≥ 0, formula (4.6) gives

χk+1 · · ·χk+m

m!=

√(k +m

k

)(k +m+N − 1

k +N − 1

)βk

βk+m

≤(k +m+N − 1

k +N − 1

)βk

βk+m

≤(k +m+N − 1

k +N − 1

)(k +N − 1)!

(k +m+N − 2)!

=k +m+N − 1

m!

≤ (k +N)m

m!=

k +N

(m− 1)!.


In the third inequality, we have used the facts that βk ≤ (k + N − 1)! andβk+m ≥ (k +m+N − 2)!. For 0 < r < 1, we then have(χk+1 · · ·χk+m

m!rk)1/m

≤( (k +N)rk

(m− 1)!

)1/m

≤ (k +N)rk ·( 1

(m− 1)!

)1/m

.

It follows that both conditions (4.16) and (4.17) are satisfied.

Example 4.15. We also provide an example of a weight sequence that satisfiescondition (4.16) but does not satisfy (4.17) in Theorem 4.13. For any integern ≥ 1, write n = 2s + ` with s ≥ 0 and 0 ≤ ` < 2s. We then define

χn =

{(2s!)−1 if ` = 0,

` if 1 ≤ ` < 2s.

A first few terms of the sequence are

1

1!,

1

2!, 1,

1

4!, 1, 2, 3,

1

8!, 1, 2, 3, 4, 5, 6, 7,

1

16!, 1, 2, 3, 4, . . . , 15,

1

32!, . . . .

Note that χ1 · · ·χn ≤ 1 for all n ≥ 1 so this sequence satisfies condition (4.15)in Lemma 4.11. Furthermore, we have χn ≤ n for all n ≥ 1. This implies thatfor each 0 < r < 1 and m ≥ 1, k ≥ 0,

χk+1 · · ·χk+m

m!rk ≤ (k + 1) · · · (k +m)

m!rk =

(m+ 1) · · · (m+ k)

k!rk

≤∞∑`=0

(m+ 1) · · · (m+ `)

`!r` = (1− r)−m−1.

The last identity follows from the binomial expansion of (1− r)−m−1. It thenfollows that

supm≥1, k≥0

(χk+1 · · ·χk+m

m!rk)1/m

≤ supm≥1

(1− r)−1−1/m ≤ (1− r)−2 <∞,

which is (4.16).On the other hand, for any integer s ≥ 0, put m = 2s − 1. Then

supk≥0

(χk+1 · · ·χk+m

m!rk)≥ χ2s+1 · · ·χ2s+m

m!rm+1 (choosing k = 2s = m+ 1)

=m!

m!rm+1 = rm+1.

It then follows that

lim supm→∞

{supk≥0

(χk+1 · · ·χk+m

m!rk)1/m}

≥ limm=2s−1→∞

r1+1/m = r,

which shows that (4.17) does not hold.

By Theorem 4.13, there exists ϕ(z) = Az + b with ‖A‖ < 1 such thatCϕ is not bounded on H(β). This is possible due to (4.8) of Theorem 4.6. Infact, for each 0 < r < 1 and |b| > r−1, the operator Crz+b is not bounded onH(β).


5. Boundedness of composition, differential, and translationoperators

Theorem 4.6 asserts that for Cϕ to be bounded on H(β), the mapping ϕ mustbe of the form ϕ(z) = Az + b with ‖A‖ ≤ 1 and b ∈ CN . In this section, wecharacterize weight sequences β for which all such mappings induce boundedcomposition operators. It turns out differential and translation operators playimportant roles.

Let Dj denote the operator of differentiating in the variable zj in CN ,for 1 ≤ j ≤ N . We denote by ∇ = (D1, . . . , DN ) the gradient operator. Foreach vector b ∈ CN , we use Tb to denote the translation operator by b, thatis, (Tbf)(z) = f(z + b) for any function f on CN . It is clear that Tb = Cz+b.We first observe a close relation between Tb and ∇.

Lemma 5.1. For any holomorphic polynomial p in CN , we have

e〈∇,b〉p = Tbp (5.1)

where 〈∇, b〉 = b1D1 + · · · + bNDN and e〈∇,b〉 is considered as the formalpower series

e〈∇,b〉 =

∞∑n=0

〈∇, b〉n

n!.

Remark 5.2. Since p is a polynomial, the left hand side of (5.1) is a finitesum so no convergence issue arises. Lemma 5.1 in the one dimensional caseappeared in [4, Corollary 1.2]. We provide here a short argument that worksfor several variables as well.

Proof. Fix z ∈ CN and consider the holomorphic polynomial of one complexvariable g(ζ) = p(z + ζ b) for ζ ∈ C. Taylor formula gives

g(1) =

∞∑n=0

g(n)(0)

n!,

where the series on the right hand side is in fact a finite series. Since g(1) =p(z+b) and g(n)(0) = 〈∇, b〉np(z) for all n ≥ 0 by the chain rule, we concludethat

p(z + b) =

∞∑n=0

〈∇, b〉np(z)n!

= e〈∇,b〉p(z)

as required. �

Definition 5.3. We denote by C1 the set of all weight sequences β for whichCϕ is bounded on H(β) whenever ϕ(z) = Az + b with ‖A‖ ≤ 1 and b ∈ CN .

The following theorem offers several equivalent characterizations of theweight sequences in C1. These results are generalization to higher dimensionsof [9, Proposition 2.9] and [10, Proposition 2.4].

Theorem 5.4. Let β be a weight sequence and χ be the associated ratio se-quence as in (4.5). The following statements are equivalent.


(a) χ is a bounded sequence.(b) The differential operators Dj (1 ≤ j ≤ N) are all bounded on H(β).(c) Tb is bounded on H(β) for any b ∈ CN .(d) β belongs to C1.(e) Tb is bounded on H(β) for some b ∈ CN\{0}.

Proof. We shall prove (a)→ (b)→ (c)→ (d)→ (e)→ (a).Suppose (a) holds, that is, there exists a positive number κ such that

χn ≤ κ for all n ≥ 1. We first show that for any 1 ≤ j ≤ N and anymulti-index α with |α| > 0,

‖Dj(zα)‖2H(β) =

αj|α|

χ2|α| ‖z

α‖2H(β). (5.2)

When αj = 0 both sides are zero so they equal. Now assume αj > 0. Wecompute

‖Dj(zα)‖2H(β) = |αj |2 ‖z(α1,...,αj−1,...,αN )‖2H(β)

= |αj |2(N − 1)!α1! · · · (αj − 1)! · · ·αN !

(N − 2 + |α|)!β2|α|−1

= αj(N − 1)!α!

(N − 2 + |α|)!β2|α|−1

and

‖zα‖2H(β) =(N − 1)!α!

(N − 1 + |α|)!β2|α|.

Since χ2|α| = |α|(|α| + N − 1)β2

|α|−1/β2|α|, we obtain the identity (5.2). Now

let f =∑α cαz

α be a function in H(β). Since f is an entire function, Djf isalso entire and we have the power expansion Djf =

∑|α|>0 cαDj(z

α). The

orthogonality of monomials of different degrees in H(β) and formula (5.2)show that

‖Djf‖2H(β) =∑|α|>0

|cα|2 ‖Dj(zα)‖2H(β) =

∑|α|>0

αj|α|

χ2|α| |cα|

2 ‖zα‖2H(β).

Summing j from 1 to N , changing the order of summation and using the factthat |α| = α1 + · · ·+ αN , we obtain

N∑j=1

‖Djf‖2H(β) =∑|α|>0

χ2|α| |cα|

2 ‖zα‖2H(β)

≤ κ2∑|α|>0

|cα|2 ‖zα‖2H(β) ≤ κ2‖f‖2H(β).

The second inequality follows from the assumption χn ≤ κ for all n ≥ 1. Weconclude that each Dj is bounded on H(β) with ‖Dj‖ ≤ κ, for 1 ≤ j ≤ N .

Now assume that (b) holds. Let b be any vector in CN . Then the operator

〈∇, b〉 is bounded on H(β), which implies that e〈∇,b〉 is bounded as well.Lemma 5.1 together with the density of polynomials in H(β) then show thatTb is bounded on H(β).


Next suppose (c) holds. Let ϕ(z) = Az + b for some A with ‖A‖ ≤ 1and b ∈ CN . By Proposition 4.1, the operator CA is bounded on H(β) andby assumption, Tb is also bounded. Since Cϕ = TbCA, we conclude that Cϕis bounded on H(β) as well.

The implication (d) → (e) is trivial since Tb = Cϕ if ϕ(z) = z + b forz ∈ CN .

Finally, assume that (e) holds. Then, as above, Cϕ is a bounded operatoron H(β), where ϕ(z) = z + b. Since b 6= 0, Theorem 4.6 with A = I andv = b/|b| shows that

τ = supm≥1, k≥0

(χk+1 · · ·χk+m

m!

)1/m

<∞.

By considering m = 1, we see that χk+1 ≤ τ for all k ≥ 0, which shows that(a) holds. The proof of the theorem is now complete. �

Example 5.5. An example of a weight sequence belonging to C1 is βn = n!for n = 0, 1, . . .. We see that

χn =√n(n+N − 1)

βn−1

βn=

√n+N − 1

n≤√N,

for all n ≥ 1. Therefore, this weight sequence β is in C1.

Example 5.6. The admissible comparison weights β introduced in [4], inwhich {nβn−1/βn} is monotonically decreasing, satisfy statement (a) in The-orem 5.4, hence belong to C1.

We summarize here the criterion for the boundedness of compositionoperators on H(β) when β belongs to the class C1. The necessity follows fromTheorem 4.6 while the sufficiency is the implication (a)→ (d) in Theorem 5.4.

Corollary 5.7. Let β be a weight sequence such that the associated ratio se-quence χ is bounded. A composition operator Cϕ is bounded on H(β) if andonly if ϕ(z) = Az+ b, where A is a linear operator on CN with ‖A‖ ≤ 1 andb ∈ CN .

Example 5.8. For the case of Fock spaces F2ω over CN ,

χn = ω√n→∞ as n→∞.

The generating weight sequence of F2ω thus does not belong to the class C1.

As a result, a different characterization of bounded composition operators isneeded. This was investigated in details in [2]. In the next section, we developa criterion of the weight sequences β for which the same phenomenon happenson those H(β).


6. Fock-like spaces and their composition operators

Corollary 5.7 characterizes the boundedness of composition operators onH(β) when the weight sequence β belongs to the class C1. When β doesnot belong to C1, for Cϕ to be bounded on H(β), an additional necessarycondition must be satisfied, as shown in the following proposition.

Proposition 6.1. Suppose β /∈ C1, or equivalently, the associated sequence χis not bounded. If Cϕ is a bounded operator on H(β), then ϕ(z) = Az + b,where A is linear and b ∈ CN such that the following condition is satisfied:

‖A‖ ≤ 1, and 〈A∗b, ζ〉 = 0 whenever |Aζ| = |ζ|. (6.1)

Proof. By Theorem 4.6, we have ϕ(z) = Az + b, where A is linear with‖A‖ ≤ 1 and b ∈ CN . Now let ζ be a unit vector in CN such that |Aζ| = 1.Put v = Aζ. Then |v| = 1 and hence, |A∗v| ≤ ‖A∗‖ |v| = 1. On the otherhand,

|A∗v| = |A∗Aζ| ≥ |〈A∗Aζ, ζ〉| = |Aζ|2 = 1.

Consequently, |A∗v| = 1. Applying inequality (4.8) with m = 1 gives

supk≥0

(χk+1|A∗v|k|〈b, v〉|

)<∞.

Since (χn) is unbounded and |A∗v| = 1, we conclude that 〈b, v〉 = 0, whichimplies that 〈A∗b, ζ〉 = 0. �

Remark 6.2. For ‖A‖ ≤ 1, it can be verified that the set of all ζ such that|Aζ| = |ζ| is exactly ker(I − A∗A). Condition (6.1) can be then rewritten inthe form

‖A‖ ≤ 1 and b ∈ ran(I −AA∗), (6.2)

since {A(

ker(I −A∗A))}⊥

={

ker(I −AA∗)}⊥

= ran(I −AA∗).

One of the main results in [2] as stated in Theorem 2.4 asserts that inany dimension N ≥ 1, a composition operator Cϕ is bounded on the Fockspace F2

ω if and only if ϕ(z) = Az + b and condition (6.1) is satisfied. In thissection, we study the weight sequences β for which this remains true. Webegin with a definition.

Definition 6.3. Define C2 to be the set all of weight sequences β for which thefollowing holds: for any ϕ(z) = Az+b satisfying condition (6.1), the operatorCϕ is bounded on H(β).

Let β be a weight sequence belonging to C2\C1. Proposition 6.1 and thedefinition of C2 show that a composition operator Cϕ is bounded on H(β)if and only if ϕ(z) = Az + b, where ‖A‖ ≤ 1 and 〈A∗b, ζ〉 = 0 whenever|Aζ| = |ζ|. By Theorem 2.4, this is equivalent to the statement that Cϕ isbounded on any Fock space F2

ω. Due to this fact, we call any such H(β) aFock-like space.

We obtain the following important property of the class C2.


Proposition 6.4. Let ω = (ωn) be an element in C2. Suppose β = (βn) is asequence of positive real numbers so that there exists a positive number κ > 0for which

βn−1

βn≤ κωn−1

ωn(6.3)

for all n ≥ 1. Then β belongs to C2 as well.

Proof. We wish to show that any mapping ϕ(z) = Az + b satisfying (6.1)induces a bounded composition operator on H(β). By Lemma 4.7 and Corol-lary 2.3, it suffices to consider the case ϕ(z) = (σ1z1 + b1, . . . , σNzN + bN ),where 1 ≥ σ1 ≥ · · · ≥ σN ≥ 0 and for each 1 ≤ j ≤ N , either 0 < σj < 1,or σj = 1 and bj = 0. Let f be a function in H(β) with power expansion

f(z) =∑α cαβ

−1|α| z

α so that ‖f‖2H(β) =∑α |cα|2 να. Using the binomial

expansion, we compute

(Cϕf)(z) =∑α

cαβ|α|

N∏j=1

(σjzj + bj)αj =

∑µ,γ

cµ+γ

β|µ+γ|

(µ+ γ)!

µ! γ!σµzµbγ

=∑µ

(∑γ

cµ+γ

β|µ+γ|

(µ+ γ)!

µ! γ!σµbγ

)zµ.

This implies that

‖Cϕf‖2H(β) =∑µ

∣∣∣∑γ

cµ+γ

β|µ+γ|

(µ+ γ)!

µ! γ!σµbγ

∣∣∣2νµ β2|µ|

≤∑µ

(∑γ

|cµ+γ |β|µ|

β|µ+γ|

(µ+ γ)!

µ! γ!σµ |bγ |

)2

νµ.

From (6.3) it follows that β|µ|/β|µ|+|γ| ≤ κ|γ|ω|µ|/ω|µ|+|γ|. Consequently,

‖Cϕf‖2H(β) ≤∑µ

(∑γ

|cµ+γ |κ|γ|ω|µ|

ω|µ+γ|

(µ+ γ)!

µ! γ!σµ |bγ |

)2

νµ

=∑µ

(∑γ

|cµ+γ |ω|µ|

ω|µ+γ|

(µ+ γ)!

µ! γ!σµ |(κb)γ |

)2

νµ

=∑µ

(∑γ

|cµ+γ |ω|µ+γ|

(µ+ γ)!

µ! γ!σµ |(κb)γ |

)2

νµ ω2|µ|.

The last quantity can be recognized as ‖CψWf‖2H(ω), where ψ(z) = (σ1 z1 +

κ|b1|, . . . , σN zN + κ|bN |) and

(Wf)(z) =∑α

|cα|zα

ω|α|.

Note that

‖Wf‖2H(ω) =∑α

|cα|2 να = ‖f‖2H(β).


Since ψ satisfies condition (6.1) and ω belongs to the class C2, the operatorCψ is bounded on H(ω). Therefore, ‖Cψ‖H(ω)→H(ω) <∞ and

‖CψWf‖H(ω) ≤ ‖Cψ‖H(ω)→H(ω)‖Wf‖H(ω).

We have used ‖Cψ‖H(ω)→H(ω) to denote the norm of Cψ as an operator onH(ω). It then follows that

‖Cϕf‖H(β) ≤ ‖CψWf‖H(ω) ≤ ‖Cψ‖H(ω)→H(ω)‖Wf‖H(ω)

= ‖Cψ‖H(ω)→H(ω)‖f‖H(β).

Since f was arbitrary in H(β), we conclude that Cϕ is bounded on H(β). �

The weight sequence ω with ωn =( (n+N−1)!

(N−1)!

)1/2gives rise to the Fock

space F21 over CN . By Theorem 2.4, ω belongs to C2. Proposition 6.4 shows

that β belongs to C2 if it satisfies condition (6.3): there exists a positivenumber κ such that

βn−1

βn≤ κωn−1

ωn=

κ√n+N − 1

for all n ≥ 1.

Using formula (4.5), we see that this condition is equivalent toχn√n≤ κ for all n ≥ 1.

This motivates the following definition.

Definition 6.5. Define C to be the set of all weight sequences β satisfying

limn→∞ β1/nn =∞ such that the sequence (χn/

√n)n≥1 is bounded.

Example 6.6. We exhibit here an example of a weight sequence which doesnot belong to C. Recall the weight sequence in Example 4.14 defined forn ≥ 0,

βn =

{(n+N − 1)! if n is even,

(n+N − 2)! if n is odd.

When n is an odd positive integer, a direct calculation shows that χn =√n(n+N − 1), which implies that β does not belong to C.

The equivalence of (a) and (d) in Theorem 5.4 together with the dis-cussion preceding Definition 6.5 shows that C1 C ⊆ C2. It is extremelysurprising, as the following result shows, that C and C2 coincide in severalvariables but are different in the one variable case.

Theorem 6.7. Consider three statements:

(a) There exists ϕ(z) = Az + b with ‖A‖ = 1 and b 6= 0 such that Cϕ isbounded on H(β).

(b) β belongs to C.(c) β belongs to C2.

One always has (a) → (b) → (c). In addition, depending on the number ofvariables N , one has the following:

(i) If N = 1, (b) does not imply (a); and (c) implies neither (a) nor (b).


(ii) If N ≥ 2, (c)→ (a), so the three statements are equivalent.

Proof. The implication (b) → (c) holds regardless of N since C ⊆ C2 as wehave seen above. We now consider separately two cases.

The case N = 1: Suppose (a) holds. A map ϕ satisfying the conditionin (a) must be of the form ϕ(z) = λz + b for some |λ| = 1 and b 6= 0. SinceCϕ is bounded, the inequality (4.8) in Theorem 4.6 with m = 1 and v = 1gives supk

(χk |λ| |b|

)< ∞. It follows that β belongs to C1 and hence C. We

have thus shown that (a) implies (b).To see that (b) does not imply (a), we simply observe that C \ C1 6= ∅.Finally we prove that there exists β ∈ C2 \ C, for which (c) holds true

while both (a) and (b) are false.Consider β in Example 6.6, which is not in C, so (a) and (b) fail to hold.

Consider ϕ(z) = λz + b with |λ| ≤ 1 and b ∈ C. The statement “〈A∗b, ζ〉 = 0whenever |Aζ| = |ζ|” implies b = 0 when |λ| = 1. If ϕ(z) = λz with |λ| = 1,Cϕ is bounded by Theorem 4.1(b). If ϕ(z) = λz+b with |λ| < 1, Theorem 4.13shows that Cϕ is bounded. Therefore, (c) holds as desired.

The case N ≥ 2: The implication (c)→ (a) follows by choosing ϕ(z) =(z1, 1, 0, . . . , 0) for z ∈ CN . Note that this choice of ϕ is only possible whenN ≥ 2.

Suppose now (a) holds. By Lemma 4.7, we may assume that Az =(σ1z1, . . . , σNzN ) for z ∈ CN , where 1 = σ1 ≥ · · · ≥ σN ≥ 0. We now havetwo subcases to consider.

• Subcase 1: b1 6= 0. Applying inequality (4.8) in Theorem 4.6 withm = 1 and v = (1, 0, . . . , 0) ∈ CN yields

supk≥0

(χk+1|A∗v|k

)|b1| <∞.

Since |A∗v| = 1 and b1 6= 0, we conclude that (χk+1)k≥0 is a bounded se-quence, which certainly implies that β belongs to C.

• Subcase 2: b1 = 0 and bj 6= 0 for some 2 ≤ j ≤ N . Without loss ofgenerality, we may assume that b2 6= 0. For any integer k ≥ 0, put fk(z) =zk1z2. (We have also made use of the fact that N ≥ 2 here.) Then

Cϕfk(z) = zk1 · (σ2z2 + b2) = σ2zk1z2 + b2z

k1 .

We now compute the norms

‖fk‖H(β) = ‖zk1z2‖H(β) =

√(N − 1)! k! 1!

(N + k)!βk+1

and

‖Cϕfk‖H(β) ≥ |b2| ‖zk1‖H(β) = |b2|

√(N − 1)! k!

(N − 1 + k)!βk.

The boundedness of Cϕ implies ‖Cϕfk‖H(β) ≤ ‖Cϕ‖‖fk‖H(β) for all k ≥ 0.A direct calculation then gives

|b2|√N + k

βkβk+1

≤ ‖Cϕ‖.


Consequently,

χk+1/√k + 1 ≤ ‖Cϕ‖/|b2| for all k ≥ 0,

which shows that β belongs to C. We have thus shown that (a) implies (b).This completes the proof of the theorem. �

The relation between the classes C1, C and C2 is summarized in thefollowing corollary to Theorem 6.7.

Corollary 6.8. The inclusion relation C1 ( C ⊆ C2 holds. In addition, C = C2

if and only if the number of variables N is greater than 1.

7. Compact composition operators

We first obtain a necessary condition for Cϕ to be compact on H(β).

Theorem 7.1. Let ϕ : CN → CN be an entire mapping. If the compositionoperator Cϕ is compact on H(β), then ϕ(z) = Az+b for some linear operatorA on CN with operator norm ‖A‖ < 1 and b ∈ CN .

Proof. Since Cϕ is compact on H(β), it is bounded. Theorem 4.6 shows thatϕ(z) = Az + b for some linear operator A : CN → CN with ‖A‖ ≤ 1 andb ∈ CN . Fix a unit vector v ∈ CN . Consider the sequence of functions (fn) ⊂H(β), where fn(z) = (〈z, v〉)n/(βn

√νn). By Lemma 4.3, ‖fn‖ = 1. Also, for

any homogeneous polynomial pm of degree m ≥ 0, we have 〈fn, pm〉H(β) = 0for all sufficiently large n. Since the space of homogeneous polynomials isdense in H(β) by definition (3.1), we conclude that fn → 0 weakly as n→∞.The compactness of Cϕ implies ‖Cϕfn‖ → 0 as n→∞. Now for z ∈ CN ,

Cϕfn(z) = (〈Az + b, v〉)n/(βn√νn) =

(〈z,A∗v〉+ 〈b, v〉

)n/(βn√νn).

Inequality (4.3) in Lemma 4.3 with k = n gives

‖Cϕfn‖H(β) ≥ |A∗v|n.Since ‖Cϕfn‖ → 0 as n→∞, we conclude that |A∗v| < 1. Because v was anarbitrary unit vector on CN , we then have

‖A‖ = ‖A∗‖ = sup|v|=1

|A∗v| = max|v|=1

|A∗v| < 1. �

It turns out that Theorem 4.13 provides us with a characterization ofweight sequences β for which the necessary condition obtained in Theorem 7.1is also sufficient for the compactness of Cϕ on H(β). We state this result moreexplicitly below.

Proposition 7.2. Suppose for each 0 < r < 1,

supm≥1, k≥0

(χk+1 · · ·χk+m

m!rk)1/m

<∞,

and

limm→∞

{supk≥0

(χk+1 · · ·χk+m

m!rk)1/m}

= 0.


Then Cϕ is compact on H(β) if and only if ϕ(z) = Az+ b for some ‖A‖ < 1and b ∈ CN .

Proof. The necessity is given by Theorem 7.1. The sufficiency follows fromthe implication (b)→ (a) in Theorem 4.13. �

Example 7.3. By the implication (a) → (b) in Theorem 4.13, any weightsequence β ∈ C2 satisfies the hypotheses of Proposition 7.2. Consequently, acomposition operator Cϕ is compact on suchH(β) if and only if ϕ(z) = Az+bfor some ‖A‖ < 1 and b ∈ CN . This is consistent with the criterion forcompactness in the case of Fock spaces obtained in [2].

8. Concluding remarks and related open questions

We would like to summarize our findings in the following remarks, and pro-pose related questions for further investigations.

Remark 8.1. Any composition operator Cϕ that is bounded on a space H(β)must necessarily be induced by an affine mapping ϕ(z) = Az+b, with ‖A‖ ≤ 1(Theorem 4.6). This condition is sufficient if the weight β belongs to classC1 (Definition 5.3 and Theorem 5.4). On the other hand, by Proposition 6.1,if β /∈ C1, then the boundedness of Cϕ implies an additional condition thatb ∈ ran(I − AA∗), as in (6.2). This condition is sufficient for compositionoperators to be bounded on Fock-like spaces, i.e., spaces H(β) generated byweights β ∈ C2 (Definition 6.3).

By the above results and Theorem 6.7, we observe the differences be-tween the case of one variable and the case of several variables as follows.

• The case N ≥ 2: We obtain C2 = C, which provides a simple descriptionof weights in C2 (Definition 6.5). In addition, Theorem 6.7 shows thatthe statement “for any ϕ(z) = Az + b, the boundedness of Cϕ on H(β)implies ‖A‖ < 1 or b = 0” is equivalent to “β /∈ C2”.• The case N = 1: On one hand, C 6= C2. On the other hand, Corollary 5.7

together with Proposition 6.1 shows that the statement “for any ϕ(z) =az + b, the boundedness of Cϕ on H(β) implies |a| < 1 or b = 0” isequivalent to “β /∈ C1”.

Theorem 4.13 provides conditions on β for which Cϕ is bounded when-ever ϕ(z) = Az + b with ‖A‖ < 1. However, the conditions are quite difficultto verify. We propose the following question.

Question 1. Find a simpler description of β that satisfies conditions (4.16)and (4.17) in Theorem 4.13.

Remark 8.2. Let β be the weight sequence in Example 4.15. We have alreadyseen that for any 0 < r < 1 and b ∈ CN with |b| > r−1, the operator Crz+b isnot bounded on H(β). More generally, Theorem 4.8 provides some examplesof non-constant affine symbols which give rise to bounded operators but thesemay not be all. The following question is natural and seems difficult.


Question 2. Characterize all entire mappings ϕ for which Cϕ is bounded orcompact on an arbitrary H(β), where

lim supm→∞

{supk≥0

(χk+1 · · ·χk+m

m!rk)1/m}

< +∞

for each 0 < r < 1.

Remark 8.3. For the Fock spaces, [2, Theorem 4] provides an exact formulafor the norm of any bounded composition operator. The Gaussian measureand the simple formula of the reproducing kernels on the Fock spaces playcrucial roles in [2]. The situation becomes highly non-trivial for compositionoperators acting on a more general space H(β). The difficulty comes from thefacts that the norm (or any equivalent norm) on such H(β) may not be givenby a measure and the kernel does not admit a simple closed-form formula.

The proof of Theorem 6.4 and the comment before Definition 6.5 showthat ‖Cϕ‖ is dominated by the norm of a related composition operator actingon some Fock space. It is also obvious that ‖Cϕ‖ ≥ 1. However, these upperand lower bounds seem to be very loose and they do not provide any usefulstrategy to study other related problems, for instance, computing the essentialnorm ‖Cϕ‖e.

We believe the interested readers will find it intriguing to investigatethe following question.

Question 3. Compute or provide a good approximation for the norm andessential norm of any given bounded composition operator Cϕ on a spaceH(β).

Acknowledgment

The authors would like to thank the referee for a careful reading and helpfulsuggestions that improved the presentation of the paper.

References

[1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950),337–404.

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Minh Luan DoanDivision of Mathematical SciencesSchool of Physical and Mathematical SciencesNanyang Technological University (NTU)637371 Singaporee-mail: [email protected]

Le Hai KhoiDivision of Mathematical SciencesSchool of Physical and Mathematical SciencesNanyang Technological University (NTU)637371 Singaporee-mail: [email protected]

Trieu LeDepartment of Mathematics and StatisticsUniversity of ToledoToledo, OH 43606USAe-mail: [email protected]

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