1Introduction Compositionoperatorsbetween N -spacesintheball

North-Western European Journal of Mathematics

W N

M

E J

Composition operators between N

p -spaces in the ball

Bingyang Hu¹ Le Hai Khoi² Trieu Le³

Received: January 28, 2016/Accepted: June 6, 2016/Online: July 22, 2016

Abstract

We study composition operators acting betweenN_p-spaces in the unit ball inCⁿ. We obtain characterizations of the boundedness and compactness of C_ϕ:N_p−→ N_qforp, q >0.

Keywords:N_p-space, composition operator, boundedness, compactness.

msc: 32A36, 47B33.

1 Introduction

LetBbe the open unit ball in Cⁿ. The space O(B) consists of all holomorphic functions inB. For any holomorphic self-mappingϕof the unit ballB, the linear operatorCϕ:O(B)−→ O(B) defined by

C_ϕ(f) =f ◦ϕ, f ∈ O(B),

is called thecomposition operatorwith symbolϕ. We are often interested in the study ofC_ϕacting between Banach spaces contained inO(B).

Composition operators on spaces of holomorphic functions in the unit disk Dand the unit ballBsuch as the Hardy, Bergman, Bloch spaces, just to name a few, have been studied intensively in many settings. We refer the reader to the monographs of Cowen and MacCluer⁴and Shapiro⁵for detailed information. In this paper we would like to investigate composition operators acting on a different class of Banach spaces, theN_p-spaces.

1Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA

2Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Techno- logical University (NTU), 637371 Singapore

3Department of Mathematics and Statistics, Mail Stop 942, University of Toledo, Toledo, OH 43606, USA

4Cowen and MacCluer, 1995,Composition operators on spaces of analytic functions.

5Shapiro, 1993,Composition operators and classical function theory.

TheN_p-spaces on the unit disk were introduced and studied by Palmberg⁶. For p >0, the spaceN_p(D) consists of functions inO(D) such that

sup

a∈D

Z

D

|f(z)|²(1− |σ_a(z)|²)^pdA(z)<∞.

The study of such spaces was motivated byQ_p-spaces, which have been of interests by many researchers. The book of Xiao⁷provides an excellent source of information aboutQ_p-spaces.

Some important properties ofN_p-spaces are: forp >1,N_p-spaces all coincide withA⁻¹(D) and forp∈(0,1], theN_p-spaces are all different. HereA^−p(D) denotes the Bergman-type space that consists of functions inO(D) such that sup_z∈D|f(z)|(1−

|z|²)^p<∞.

In Palmberg (2007), Palmberg studied the boundedness and compactness of composition operators acting betweenN_p-spaces and from anN_p-space intoA^−q(D).

Certain characterizations of boundedness and compactness were obtained. In Ueki (2012), Ueki investigated weighted composition operators betweenN_p-spaces and A^−q-spaces.

The first two authors⁸introduced and studied properties ofN_p-spaces in higher dimensions. Forp >0, theN_p-space ofBis defined as follows:

N_p=N_p(B) :=

(

f ∈ O(B) :kfk_p= sup

a∈B

Z

B

|f(z)|²(1− |Φ_a(z)|²)^pdV(z) 1/2

<∞ )

,

where dV is the normalized Lebesgue volume measure overBandΦ_ais the involu- tive automorphism ofBthat interchanges 0 anda. We also defined the little space ofN_pas

N_p⁰=N_p⁰(B) :=

(

f ∈ N_p: lim

|a|→1⁻

Z

B

|f(z)|²(1− |Φ_a(z)|²)^pdV(z) = 0 )

, which is a closed subspace ofN_p.

Several basic properties ofN_p-spaces are proved, in connection with the Bergman- type spacesA^−q, which, similar to the one dimensional cases, consists of functions f ∈ O(B) for which

|f|_p= sup

z∈B

|f(z)|(1− |z|²)^p<∞.

Recall thatH^∞denotes the Banach space of all bounded functions inO(B) with the normkfk_∞= sup_z∈B|f(z)|.

6Palmberg, 2007, “Composition operators acting onN_p-spaces”.

7Xiao, 2001,HolomorphicQclasses.

8Hu and Khoi, 2013, “Weighted composition operators onN_p-spaces in the ball”.

1. Introduction

Theorem 1 (Hu and Khoi⁹) – The following statements hold:

1. Forp > q >0, we haveH^∞,→ N_q,→ N_p,→A⁻ⁿ⁺¹² .

2. Forp >0, ifp >2k−1, k∈(0,ⁿ⁺¹₂ ], thenA^−k ,→ N_p. In particular, whenp > n, N_p=A⁻ⁿ⁺¹² .

3. N_p is a functional Banach space with the norm k · k_p, and moreover, its norm topology is stronger than the compact-open topology.

4. For0< p <∞,B,→ N_p, whereBis the Bloch space inB.

Considering weighted composition operators betweenN_pand Bergman-type spacesA^−q

W_u,ϕ(f) =u·(f ◦ϕ), f ∈ O(B),

whereu:B→Cis a holomorphic function, the properties above allow us to prove criteria for boundedness and compactness of these operators¹⁰. In Hu and Khoi (2015) and Hu, Khoi, and Le (2016a), compact differences and the estimate for essential norm of weighted composition operators acting from anN_p-space to an A^−q-space were investigated. In these works, various properties stated in Theorem 1 were used.

All of the aforementioned results concern weighted composition operatorsW_u,ϕ acting between the spacesN_pandA^−q. A natural question one may ask is: what is the situation like when we consider composition operators acting betweenN_p- spaces themselves? The aim of the present paper is to investigate this question. More precisely, we study the boundedness and compactness of composition operators Cϕacting betweenN_pandN_qforp, q >0. To our knowledge, this problem has not been treated before, neither in one variable nor several variables.

The structure of this paper is as follows. Section 2 on the next page is devoted to the boundedness ofC_ϕ. We obtain sufficient conditions and necessary conditions forC_ϕ to be bounded fromN_p intoN_q. In Section 3 on p. 118, we investigate the compactness ofC_ϕ. Different characterizations for compactness are provided.

We also show in this section that the range of any compact composition operator C_ϕ:N_p −→ N_qmust be contained inN_q⁰. Consequently, the compactness ofC_ϕ fromN_pintoN_qand fromN_pintoN_q⁰are equivalent.

Throughout the paper,dσdenotes the normalized surface measure on the sphere S, the boundary ofB. For two quantitiesaandbof a certain variable, we writea.b (respectively,a&b) if there exists a positive numberCindependent of the variable under consideration such thata≤Cb(respectively,a≥Cb). Moreover, if botha.b anda&bhold, then we writea'b.

9Hu and Khoi, 2013, “Weighted composition operators onN_p-spaces in the ball”.

10Ibid., Theorems 3.2 and 3.4.

2 Boundedness of composition operators from N

p

to N

In this section we investigate the boundedness of composition operators between N_p-spaces. The one dimensional case is considered by Palmberg¹¹. A sufficient condition and a necessary condition forCϕto be bounded onN_pof the unit disk were given there. These conditions involve the generalized Nevanlinna counting function introduced by Shapiro.

We begin with the caseϕis a univalent holomorphic self-mapping of the unit ballB.

Theorem 2 – Letϕbe a univalent holomorphic self-mapping ofBandpbe any positive real number. Suppose that

δ= inf{|Jϕ(z)|:z∈B}>0,

whereJϕthe complex Jacobian ofϕ. ThenC_ϕ:N_p−→ N_pis a bounded operator with kCϕk ≤δ⁻¹for allp >0.

Proof. Letf be any function inN_p. Fora∈B, andz∈B, the Schwarz-Pick lemma¹² gives|Φ_ϕ(a)(ϕ(z))| ≤ |Φ_a(z)|. Sinceϕ is univalent, it is biholomorphic fromBonto ϕ(B)¹³. This makes the change of variables possible in the following estimates:

δ² Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^pdV(z)

≤ Z

B

|f(ϕ(z))|²(1− |Φ_ϕ(a)(ϕ(z))|²)^p|Jϕ(z)|²dV(z)

= Z

ϕ(B)

|f(w)|²(1− |Φ_ϕ(a)(z)|²)^pdV(w) (by the change-of-variablesw=ϕ(z))

≤ Z

B

|f(w)|²(1− |Φ_ϕ(a)(z)|²)^pdV(w)

≤ kfk²

p.

Taking supremum over a ∈ B gives δ²kC_ϕfk²

p ≤ kfk²

p, which implies kC_ϕfk_p ≤ δ⁻¹kfk_p. Sincef was an arbitrary element inN_p, we conclude thatC_ϕis a bounded

operator onN_pwithkCϕk ≤δ⁻¹as desired.

11Palmberg, 2007, “Composition operators acting onN_p-spaces”, Theorem 4.2.

12Rudin, 1980,Function theory in the unit ball ofCⁿ, Theorem 8.1.4.

13See e.g. ibid., Theorem 15.1.8.

2. Boundedness of composition operators fromN_ptoN_q

Corollary 1 – SupposeA:Cⁿ−→Cⁿis an invertible linear operator andbis a vector inCⁿsuch thatϕ(z) =Az+bis a self-mapping ofB. ThenC_ϕ:N_p−→ N_pis bounded for allp >0.

Proof. Since|Jϕ(z)|=|det(A)|>0, Theorem 2 on the preceding page provides the

desired conclusion.

For general self-mappingsϕof the unit ball, we offer a necessary condition and a sufficient condition forCϕto be bounded fromN_pintoN_q. We shall make use of a sequence of homogeneous polynomials{Pm}which satisfy deg(P_m) =m,

kP_mk_∞= sup

|ζ|=1

|P_m(ζ)|= 1, and Z

S

|P_m(ζ)|²dσ(ζ)≥ π

4ⁿ. (1)

The existence of such a sequence was proved in Ryll and Wojtaszczyk (1983). We shall also need the inequality

1 +

∞

X

k=0

2^kγx^2k &(1−x)^−γ for 0≤x <1, (2)

whereγis a positive number. Indeed, it was showed¹⁴that

∞

X

k=0

2^kγx^2k&(1−x)^−γ forx >1 2.

On the other hand, it is clear that 2^γ≥(1−x)^−γfor 0≤x≤¹

2. Therefore, (2) holds.

The following preparatory result will be needed in obtaining the necessary condition for the boundedness ofC_ϕ.

Proposition 1 – Letpandqbe positive numbers andϕa holomorphic self-mapping ofB. LetB_N

p denotes the unit ball ofN_p. Supposeµis a positive number andEis a measurable subset ofBsuch that

sup

a∈B,f∈B_N

p

Z

E

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)≤µ.

Then for any0< ε≤p+ 1, we have sup

a∈B

Z

E

(1− |Φ_a(z)|²)^q

(1− |ϕ(z)|²)^p+1−εdV(z).µ.

14Ueki, 2012, “Weighted composition operators acting between theN_p-space and the weighted-type spaceH_α^∞”, p. 247.

Proof. The hypothesis implies that for anya∈Band anyf ∈ N_p, we have Z

E

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)≤µkfk²_p. (3) Put

F(z) = 1 +

∞

X

k=0

2^{k(p+1−ε)/2}P₂k(z), z∈B. By Hu, Khoi, and Le (2016b, Theorem 5.1),

kFk²_p≈

∞

X

k=0

2^k(p+1−ε) 2^k(p+1) =

∞

X

k=0

2^−kε<∞,

which shows thatFbelongs toN_p. LetU_ndenote the group of all unitary operators on the Hilbert spaceCⁿ. For anyU ∈U_n, the unitary invariant property of the volume measure shows thatF◦U also belongs toN_pandkF◦Uk_p=kFk_p. Setting f =F◦U in (3) gives

Z

E

|F(U ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)≤µkF◦Uk²_p=µkFk²_p, for eacha∈BandU∈U_n.

Now fix a∈B. Integration with respect to the Haar measure dU onU_n and Fubini’s Theorem yield

Z

E

Z

U_n

|F(U ϕ(z))|²dU

1− |Φ_a(z)|²q

dV(z)≤µkFk²_p. For anyz∈B, we compute

Z

U_n

|F(U ϕ(z))|²dU= Z

S

F

|ϕ(z)|ζ

2dσ(ζ)

(by Rudin (1980, Proposition 1.4.7))

= Z

S

1 +

∞

X

k=0

2^{k(p+1−ε)/2}P₂k(|ϕ(z)|ζ)

2dσ(ζ)

= 1 +

∞

X

k=0

2^k(p+1−ε)(|ϕ(z)|²)^2k Z

S

|P₂k(ζ)|²dσ(ζ)

(by the orthogonality and the homogeneity of{P₂k})

≥ π 4ⁿ

1 +

∞

X

k=0

2^k(p+1−ε)(|ϕ(z)|²)^2k

(by (1))

(Cont. next page)

2. Boundedness of composition operators fromN_ptoN_q

&

1− |ϕ(z)|²⁻(p+1−ε)

(by (2)).

Consequently, Z

E

(1− |Φ_a(z)|²)^q (1− |ϕ(z)|²)^p+1−εdV(z)

. Z

E

Z

U

|F(U ϕ(z))|²dU

(1− |Φ_a(z)|²)^qdV(z)≤µkFk²_p

as desired.

We are now ready to prove a necessary condition and a sufficient condition for the boundedness ofC_ϕ:N_p−→ N_q.

Theorem 3 – Letpandqbe two positive numbers andϕa holomorphic self-mapping of B. If

sup

a∈B

Z

B

(1− |Φ_a(z)|²)^q

(1− |ϕ(z)|2)ⁿ⁺¹dV(z)<∞, (4)

thenCϕ:N_p−→ N_qis bounded.

Conversely, ifCϕ:N_p−→ N_qis bounded, then for any0< ε≤p+ 1, sup

a∈B

Z

B

(1− |Φ_a(z)|²)^q

(1− |ϕ(z)|²)^p+1−εdV(z)<∞.

Proof. Suppose (4) holds. By Item 1 of Theorem 1 on p. 113, there is a constant C >0 such that for eachf ∈ N_p, we have

|f(z)|(1− |z|²)ⁿ⁺¹² ≤Ckfk_p, ∀z∈B. Hence,

sup

a∈B

Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)≤(Ckfk_p)²sup

a∈B

Z

B

(1− |Φ_a(z)|²)^q (1− |ϕ(z)|²)ⁿ⁺¹dV(z), which shows thatC_ϕis bounded fromN_pintoN_q.

Conversely, supposeC_ϕ:N_p−→ N_qis bounded. Then sup

a∈B,f∈B_N

p

Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z) = sup

f∈B_N

p

kCϕfk²_q≤ kCϕk².

The desired inequality now follows from Proposition 1 on p. 115.

An application of Theorem 3 immediately gives the following result. In fact, the operatorCϕin the corollary is compact, as we shall see in the next section.

Corollary 2 – Letϕ be a holomorphic self-mapping ofBsuch thatkϕk_∞<1. Then Cϕ:N_p−→ N_qis bounded for allp, q >0.

3 Compactness of composition operators from N

p

to N

3.1 General characterizations

In this section we study the compactness of composition operators betweenN_p- spaces. By standard argument¹⁵, we have the following criterion for compactness.

Lemma 1 – A bounded composition operatorCϕ:N_p→ N_qis compact if and only if for any bounded sequence{fm} ⊂ N_pconverging to zero uniformly on compact subsets of B, the sequencen

kf_m◦ϕk_qo

converges to zero asm→ ∞.

It turns out, as we shall see below, that the range of any compact composition operatorCϕ:N_p−→ N_qmust be contained in the little spaceN_q⁰. We first prove an important property of elements inN_q⁰.

Lemma 2 – Let h be an element in the spaceN_q⁰. Suppose {Ak}_k≥1 is a decreasing sequence of measurable subsets ofBwhose intersection is empty. Then

klim→∞

"

sup

a∈B

Z

A_k

|h(z)|²(1− |Φ_a(z)|²)^qdV(z)

#

= 0. (5)

Proof. Letε >0 be given. Sinceh∈ N_q⁰, there exists a positive number 0< δ <1 such that

sup

δ<|a|<1

Z

B

|h(z)|²(1− |Φ_a(z)|²)^qdV(z)< ε. (6) On the other hand, ifa∈Bwith|a| ≤δ, then forz∈B,

1− |Φ_a(z)|²=(1− |a|²)(1− |z|²)

|1− hz, ai|² ≤1 +|a|

1− |a|(1− |z|²)≤1 +δ

1−δ(1− |z|²).

Consequently, for any integerk≥1, we have sup

|a|≤δ

Z

A_k

|h(z)|²(1− |Φ_a(z)|²)^qdV(z)≤ 1 +δ

1−δ qZ

A_k

|h(z)|²(1− |z|²)^qdV(z).

Sinceh ∈ N⁰

q ⊂ A²_q, the function|h(z)|²(1− |z|²)^q belongs toL¹(B, dV). Because {A_k≥1}is a decreasing sequence of subsets whose intersection is empty, there exists a positive integerk_ε such that for allk≥k_ε,

1 +δ 1−δ

qZ

A_k

|h(z)|²(1− |z|²)^qdV(z)< ε.

15As in Cowen and MacCluer, 1995,Composition operators on spaces of analytic functions, Proposi- tion 3.11.

3. Compactness of composition operators fromN_ptoN_q

This implies that for suchk, sup

|a|≤δ

Z

A_k

|h(z)|²(1− |Φ_a(z)|²)^qdV(z)< ε. (7) Combining (6) and (7) yields

sup

a∈B

Z

A_k

|h(z)|²(1− |Φ_a(z)|²)^qdV(z)< ε

for allk≥kε. Sinceεwas arbitrary, (5) follows.

The following result provides a necessary condition forCϕ to be a compact operator.

Proposition 2 – Suppose C_ϕ : N_p −→ N_q is a compact composition operator. Let B_N

p={f ∈ N_p:kfk_p≤1}. Then the following statements are true.

1. Cϕ(N_p)⊂ N_q⁰and

|alim|→1⁻ sup

f∈B_N

p

Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z) = 0. (8)

2. For any decreasing sequence{A_k}_k≥1of measurable subsets of the unit ball whose intersection is empty, we have

klim→∞ sup

f∈B_N

p

"

sup

a∈B

Z

A_k

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

#

= 0. (9)

Proof. We first prove thatC_ϕ(N_p) is a subset ofN⁰

q. Letf be inN_p. For any integer m≥1, putf_m(z) =f(_m+1^m z). Then eachf_mbelongs toH^∞and the sequence{f_m}is bounded onN_pand converges tof uniformly on compact subsets ofB. By Lemma 1 on the preceding page, the sequence{C_ϕf_m}converges toC_ϕf inN_q asm→ ∞. Since each functionCϕfmbelongs toH^∞⊂ N_q⁰andN_q⁰is a closed subspace ofN_p, we conclude thatCϕf is an element inN_q⁰. Sincef was arbitrary, it follows that the image ofN_punderCϕis contained inN_q⁰.

Letε >0 be given. SinceCϕis compact and its range is contained inN_q⁰, the imageCϕ(B_N

p) is pre-compact inN_q⁰. Therefore,Cϕ(B_N

p) can be covered by finitely many

√ ε

2 -balls. That is, there exists a finite set{f₁, . . . , f_M} ⊂B_N

p such that for any f ∈B_N

p, there is a numberj∈ {1,2, . . . , M}for which kCϕ(f)−Cϕ(fj)k²_q< ε

4. (10)

On other hand, since{f₁◦ϕ, . . . , f_M◦ϕ}is contained inN⁰

q, there exists 0< r <1 such that for all 1≤j≤Mand|a|> r,

Z

B

|f_j(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)<ε

4. (11)

For eacha∈Bwith|a|> randf ∈ N_pwithkfk_p<1, choose 1≤j≤Msuch that (10) holds. Combining with (11), we have

Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

≤2 Z

B

|f(ϕ(z))−f_j(ϕ(z))|²+|f_j(ϕ(z))|²

(1− |Φ_a(z)|²)^qdV(z)

= 2kC_ϕ(f)−C_ϕ(f_j)k²_q+ 2 Z

B

|f_j(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

<2 ε

4+ε 4

=ε.

This shows that for allr <|a|<1, sup

f∈B_N

p

Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)≤ε, which implies (8).

Now let{Ak}be a decreasing sequence of measurable subsets ofBwhose inter- section is empty. Since{f1◦ϕ, . . . , fM◦ϕ}is contained inN_q⁰, Lemma 2 on p. 118 shows that there exists an integerk_εsuch that for anyk≥k_εand any 1≤j≤M,

sup

a∈B

Z

A_k

|fj(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)< ε

4. (12)

Inequalities (10) and (12) together give sup

a∈B

Z

A_k

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

≤2 sup

a∈B

Z

A_k

|f(ϕ(z))−fj(ϕ(z))|²+|fj(ϕ(z))|²

(1− |Φ_a(z)|²)^qdV(z)

≤2kC_ϕ(f)−C_ϕ(f_j)k²_q+ 2 sup

a∈B

Z

A_k

|f_j(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

<2 ε

4+ε 4

=ε, for anyk≥k_εandf ∈B_N

p. The limit (9) then follows.

3. Compactness of composition operators fromN_ptoN_q

Now we give the following characterization of the compactness ofC_ϕacting fromN_pintoN_q.

Theorem 4 – Letp, qbe positive numbers andϕa holomorphic self-mapping ofBsuch that the composition operatorC_ϕ:N_p−→ N_qis bounded. LetE₁⊂E₂⊂ · · · ⊂Bbe an increasing sequence of measurable sets such that∪_k≥1E_k=Band for eachk, the closure ϕ(Ek)is compact inB. ThenCϕ:N_p−→ N_qis compact if and only if

klim→∞ sup

f∈B_N

p

"

sup

a∈B

Z

B\E_k

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

#

= 0. (13)

We recall here thatB_N

p={f ∈ N_p:kfk_p≤1}is the unit ball ofN_p.

Proof. Necessity. SetA_k=B\E_k for all integersk≥1. Then{A_k}_k≥1is a decreasing sequence of measurable subsets ofBand

\

k≥1

A_k=B\[

k≥1

E_k

=∅.

IfC_ϕis a compact operator fromN_pintoN_q, then (13) follows from Item 2 of Proposition 2 on p. 119.

Sufficiency. Suppose (13) holds. Take any bounded sequence{f_m} ⊂ N_pconverging to zero uniformly on every compact subset ofB. By Lemma 1 on p. 118, it suffices to show that the sequence{kfm◦ϕk_q}converges to zero asm→ ∞. Letε >0 be given. By (13), there exists a positive integerksuch that for any m∈N,

sup

a∈B

Z

B\E_k

|f_m(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)<ε

2. (14)

On the other hand, since{fm}converges to zero uniformly on the compact set ϕ(Ek), for sufficiently largem, we have

sup

a∈B

Z

E_k

|f_m(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)≤ Z

E_k

|f_m(ϕ(z))|²dV(z)

≤sup

z∈E_k

|f_m(ϕ(z))|²

= sup

w∈ϕ(E_k)

|f_m(w)|²< ε 2. This estimate together with (14) immediately yields

kCϕfmk²_p= sup

a∈B

Z

B

|fm(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)< ε,

for sufficiently large integersm. Consequently,kCϕfmk_q→0 as desired.

By weakening condition (13) and adding an extra condition, we obtain an equiv- alent characterization for the compactness ofC_ϕas follows.

Theorem 5 – Letp, qbe positive numbers andϕa holomorphic self-mapping ofBsuch that the composition operatorC_ϕ:N_p−→ N_qis bounded. LetE₁⊂E₂⊂ · · · ⊂Bbe an increasing sequence of measurable sets such that∪_k≥1E_k=Band for eachk, the closure ϕ(E_k)is compact inB. Then the following statements are equivalent.

1. C_ϕis compact fromN_pintoN_q. 2. C_ϕis compact fromN_pintoN_q⁰.

3. The following two conditions are satisfied

klim→∞

sup

f∈B_N

p

Z

B\E_k

|f(ϕ(z))|²(1− |z|²)^qdV(z)

= 0; (15)

and

|alim|→1⁻ sup

f∈B_N

p

Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z) = 0. (16)

Proof. The equivalence of 1 and 2 comes from the fact thatN_q⁰is a subspace ofN_q and Item 1 of Proposition 2 on p. 119, which shows that wheneverC_ϕ:N_p−→ N_q is compact, the range ofC_ϕis actually contained inN⁰

q. We only need to prove the equivalence of 1 and 3.

Suppose 1 holds, that is,C_ϕ:N_p → N_q is compact. Then (15) follows from Item 2 of Proposition 2 on p. 119 withA_k=B\E_kanda= 0. In addition, (16) follows from Item 1 of Proposition 2 on p. 119.

Now suppose 3 hold, that is, both (15) and (16) are satisfied. Take any bounded sequence{f_m} ⊂ N_pconverging to zero uniformly on every compact subset ofB. We may assume thatkf_mk_p≤1 for eachm∈N. To showC_ϕ is compact, it suffices to show that

mlim→∞

kC_ϕ(f_m)k_q= 0.

Letε >0 be given. By (16), there exists 0< δ <1 such that sup

δ<|a|<1

sup

f∈B_N

p

Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

<ε

3. (17)

By (15), there exists an integerk≥1 such that sup

f∈B_N

p

"Z

B\E_k

|f(ϕ(z))|²(1− |z|²)^qdV(z)

#

<(1−δ)^2q·ε

3 . (18)

3. Compactness of composition operators fromN_ptoN_q

Ifa∈Bwith|a| ≤δ, then

1− |Φ_a(z)|²=(1− |a|²)(1− |z|²)

|1− hz, ai|² ≤ 1− |z|² (1−δ)². This together with (18) implies

sup

f∈B_N

p

sup

|a|≤δ

Z

B\E_k

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

≤ 1

(1−δ)^2q sup

f∈B_N

p

Z

B\E_k

|f(ϕ(z))|²(1− |z|²)^qdV(z)<ε 3.

Since{f_m}converges to zero uniformly on the compact setϕ(E_k), there existsm₀∈N such that wheneverm > m₀,

sup

a∈B

Z

E_k

|fm(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

≤ Z

E_k

|fm(ϕ(z))|²dV(z)

≤ sup

w∈ϕ(E_k)

|f_m(w)|<ε

3. (19)

Form > m₀, by (17), (18) and (19), we have kC_ϕ(f_m)k²

q= sup

a∈B

Z

B

|f_m(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

≤ sup

δ<|a|<1

Z

B

|fm(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z) + sup

|a|≤δ

Z

B

|f_m(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

≤ sup

δ<|a|<1

Z

B

|f_m(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z) + sup

|a|≤δ

Z

B\E_k

|f_m(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z) + sup

|a|≤δ

Z

E_k

|fm(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

< ε.

It follows thatkC_ϕf_mk_q→0 as required.

By using Theorem 4 on p. 121 with certain choices of the sets{Ek}_k≥1, we obtain somewhat more concrete criteria for the compactness ofCϕ.

Corollary 3 – Letp, qbe positive numbers andϕa holomorphic self-mapping ofBsuch that the composition operatorC_ϕ:N_p−→ N_qis bounded. Then the following statements are equivalent.

1. C_ϕis a compact operator fromN_pintoN_q. 2. limt→1⁻

sup_a∈B, f∈B_N

p

R

|z|>t|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

= 0.

3. lim_t→1⁻

sup_a∈B, f∈B_N

p

R

|ϕ(z)|>t|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

= 0.

Proof. Observe that statement 2 and respectively, statement 3, is equivalent to the statement that for any sequence{tk}_k≥1of positive numbers increasing to 1, we have

klim→∞

sup

a∈B,f∈B_N

p

Z

|z|>t_k

|f(z)|²(1− |Φ_a(z)|²)^qdV(z)

= 0,

and respectively,

klim→∞

sup

a∈B,f∈B_N

p

Z

|ϕ(z)|>t_k

|f(z)|²(1− |Φ_a(z)|²)^qdV(z)

= 0.

For each integerk ≥1, define E_k ={z :|z| ≤ t_k}in the case of statement 2 and Ek ={z:|ϕ(z)| ≤ tk}in the case of statement 3. Since {tk}_k≥1 is increasing to 1, we see that{Ek}_k≥1 is an increasing sequence of measurable sets and∪^∞

k=1Ek =B. Furthermore, it is clear that the setϕ(Ek) is compact for eachk. The equivalence of 1 and 2 and the equivalence of 1 and 3 now follow from Theorem 4 on p. 121.

Corollary 4 – Letϕ be a holomorphic self-mapping ofBsuch thatkϕk_∞ <1. Then C_ϕ:N_p−→ N_qis compact for allp, q >0.

Proof. By Corollary 2 on p. 117, the operatorC_ϕis bounded fromN_pintoN_q. In addition, condition (d) in Corollary 3 is clearly satisfied since the set{|ϕ(z)|> t}is empty for allkϕk_∞< t <1. Consequently,C_ϕis compact.

By changing the role ofN_ptoN_p⁰in the proofs of Theorems 4 and 5 on p. 121 and on p. 122, we immediately obtain the following result describing the compact- ness of composition operators acting betweenN_p⁰andN_q.

Theorem 6 – Letp, qbe positive numbers andϕa holomorphic self-mapping ofBsuch that the composition operatorCϕ:N_p⁰−→ N_qis bounded. LetE1⊂E2⊂ · · · ⊂Bbe an increasing sequence of measurable sets such that∪_k≥1Ek=Band for eachk, the closure ϕ(E_k)is compact inB. Then the following statements are equivalent.

1. Cϕis compact fromN_p⁰intoN_q.

3. Compactness of composition operators fromN_ptoN_q 2. C_ϕis compact fromN⁰

p intoN⁰

q. 3. lim

k→∞ sup

f∈B_N0

p

"

sup

a∈B

Z

B\E_k

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)

#

= 0.

4. The following two conditions are satisfied

klim→∞

sup

f∈B_N0

p

Z

B\E_k

|f(ϕ(z))|²(1− |z|²)^qdV(z)

= 0;

and

|alim|→1⁻ sup

f∈B

N0 p

Z

B

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z) = 0.

HereB_N0

p ={f ∈ N_p⁰:kfk_p≤1}is the unit ball ofN_p⁰.

3.2 Some simplifications

Although Theorems 4 to 6 on p. 121, on p. 122 and on the preceding page offer several characterizations of the compactness of composition operatorsCϕ:N_p−→

N_q, the conditions are rather abstract and difficult to check. We shall provide in this section a necessary condition and a sufficient condition for the compactness of Cϕdirectly in terms ofϕ. These conditions seem to be more useful in applications.

Theorem 7 – Letp, q∈(0, n]be two positive numbers andϕa holomorphic self-mapping ofBsuch thatC_ϕ:N_p−→ N_qis bounded. If

tlim→1⁻sup

a∈B

Z

|ϕ(z)|>t

(1− |Φ_a(z)|²)^q

(1− |ϕ(z)|²)ⁿ⁺¹dV(z) = 0, (20)

thenCϕis compact.

Conversely, ifCϕ:N_p→ N_qis compact, then for0< ε≤p+ 1,

tlim→1⁻sup

a∈B

Z

|ϕ(z)|>t

(1− |Φ_a(z)|²)^q

(1− |ϕ(z)|2)^p+1−εdV(z) = 0. (21) Proof. Suppose (20) holds. As in the proof of Theorem 3 on p. 117, there is a constantC >0 such that for anyf ∈ N_pwithkfk_p≤1,

|f(z)| ≤Ckfk_p(1− |z|²)^−(n+1)/2≤C(1− |z|²)^−(n+1)/2. It implies that for any 0< t <1 and anya∈B,

Z

|ϕ(z)|>t

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)≤C² Z

|ϕ(z)|>t

(1− |Φ_a(z)|²)^q (1− |z|²)ⁿ⁺¹ dV(z).

Now (20) shows that statement 3 in Corollary 3 on p. 124 is satisfied. As a result, C_ϕis a compact operator fromN_pintoN_q.

Now supposeC_ϕ:N_p−→ N_qis compact. Letµ >0 be given. By Corollary 3 on p. 124, there is a numbert_µ∈(0,1) such that for allt_µ< t <1,

sup

a∈B,f∈B_N

p

Z

|ϕ(z)|>t

|f(ϕ(z))|²(1− |Φ_a(z)|²)^qdV(z)≤µ.

An application of Proposition 1 on p. 115 withE={z∈B:|ϕ(z)|> t}yields sup

a∈B

Z

|ϕ(z)|>t

(1− |Φ_a(z)|²)^q

(1− |z|²)^p+1−εdV(z).µ,

for alltµ< t <1. Sinceµis arbitrary, (21) follows.

As applications of Theorems 3 and 7 on p. 117 and on the previous page, we have the following result.

Corollary 5 – Supposek >0, p, q, r∈(0, n), r≥q, ε∈(0, q+ 1)andϕis a holomorphic self-mapping ofB. The following statements hold.

1. IfC_ϕ :A^{−k(q+1−ε)}−→A^−k(n+1)is a bounded operator, thenC_ϕ :N_p−→ N_r is a bounded operator;

2. IfC_ϕ :A^{−k(q+1−ε)}−→A^−k(n+1) is a compact operator, thenC_ϕ:N_p −→ N_r is a compact operator.

Proof. The boundedness ofCϕ:A^{−k(q+1−ε)}−→A^−k(n+1)is equivalent to¹⁶ M:= sup

z∈B

(1− |z|²)^q+1−ε (1− |ϕ(z)|²)ⁿ⁺¹ <∞.

By Theorem 3 on p. 117, it suffices to show that sup

a∈B

Z

B

(1− |Φ_a(z)|²)^r

(1− |ϕ(z)|²)ⁿ⁺¹dV(z)<∞. Indeed, we have

sup

a∈B

Z

B

(1− |Φ_a(z)|²)^r (1− |ϕ(z)|²)ⁿ⁺¹dV(z)

≤Msup

a∈B

Z

B

(1− |Φ_a(z)|²)^r (1− |z|²)^q+1−εdV(z)

=Msup

a∈B

(1− |a|²)^r Z

B

(1− |z|²)^r−q−1+ε

|1− hz, ai|2r dV(z)

(Cont. next page)