# The Essential Norm of the Generalized Hankel Operators on the Bergman Space of the Unit Ball inCn

Authors: Luo Luo; Yang Xuemei
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

In 1993, Peloso introduced a kind of operators on the Bergman spaceA2(B) of the
unit ball that generalizes the classical Hankel operator. In this paper, we estimate the
essential norm of the generalized Hankel operators on the Bergman space Ap(B)(p>1)
of the unit ball and give an equivalent form of the essential norm.


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## Body

## 1. Introduction

LetB be the open unit ball in Cn, m the Lebesgue measure on Cn normalized so that m(B)=1, H(B) denotes the class of all holomorphic functions on B. The Bergman space A2(B) is the Banach space of all holomorphic functions f on B such that ∫B|f(z)|2dm(z)<∞. It is easy to show that A2(B) is a closed subspace of L2(B,dm).There is an orthogonal projection ofL2(B,dm) onto A2(B), denoted by P and (1.1)Pf(z)=∫BK(z,w)f(w)dm(w),
where K(z,w)=1/(1-〈z,w〉)n+1 is the Bergman kernel on B.For a functionf∈H(B), define the Hankel operator Hf:A2(B)→A2(B)⊥ with symbol f by (1.2)Hfg=(I-P)(f¯g)=∫B(f(z)-f(w))¯K(z,w)g(w)dm(w),
where I is the identity operator.Since the Hankel operatorHf is connected with the Toeplitz operator, the commutator, the Bloch space, and the Besov space, it has been extensively studied. Important papers in this context are [1, 2] for the case n=1 and [3–5] for the case n>1. It is known that Hf is bounded on A2(B) if and only if f∈β(B) and Hf is compact A2(B) if and only if f∈β0(B), where (1.3)β(B)={f∈H(B):sup⁡z∈B(1-|z|2)|Rf(z)|<∞},β0(B)={f∈H(B):(1-|z|2)|Rf(z)|⟶0,as|z|⟶1}.Rf is the radial derivative of f defined by (1.4)Rf(z)=∑j=1nzj∂f(z)∂zj.β(B) is called the Bloch space, and β0(B) is called the little Bloch space.Forn=1, f∈H(D)  (D is the open unit disc), Hf is in the Schatten class Sp(1<p<∞) if and only if f∈Bp(D); Hf∈Sp(0<p≤1) if and only if f is a constant, where (1.5)Bp(D)={f∈H(D):f′(z)(1-|z|2)∈Lp(dλ)},p>1,
and dλ(z)=(1-|z|2)-2dm(z) is the invariant volume measure on D, Bp(D) is called the Besov space on D. This theorem expresses that there is a cutoff of Hf at p=1.Forn>1, f∈H(B), Hf∈Sp(2n<p<∞) if and only if f∈Bp(B), Hf∈Sp(0<p≤2n) if and only if f is a constant, where (1.6)Bp(B)={f∈H(B):(1-|z|2)Rf(z)∈Lp(dλ)},p>n,
and dλ(z)=(1-|z|2)-(n+1)dm(z) is the invariant volume measure on B. Bp(B) is called the Besov space on B. Then, the cutoff phenomenon of Hf appears at p=2n. If c(n) denotes the value of “cutoff,” then (1.7)c(n)={1,n=1,2n,n>1.
Obviously, c(n) depends on the dimension n of the unit ball.In 1993, Peloso [3] replaced f(z)-f(w) with (1.8)Δjf(w,z)=f(w)-∑|α|<jDαf(z)α!(w-z)α
to define a kind of generalized Hankel operator: (1.9)Hf,jg(z)=∫B-Δjf(w,z)¯K(z,w)g(w)dm(w),Hf,j′g(z)=∫BΔjf(z,w)¯K(z,w)g(w)dm(w).
Here, (Dαf)(z)=(∂|α|f(z))/(∂z1α1⋯∂znαn). Clearly, if j=1, Hf,1 and Hf,1′ are just the classical Hankel operator Hf. He proved that Hf,j has the same boundedness and compactness properties as Hf, but the Schatten class property of Hf,j is different from that of Hf. If n≥2, f∈H(B), Hf,j∈Sp((2n/j)<p<∞) if and only if f∈Bp(B); if 0<p≤(2n/j), Hf,j∈Sp if and only if f is a polynomial of degree at most j-1. So the value of “cutoff” of Hf,j is 2n/j; this means that the cutoff constant c(n) depends not only on the dimension but also on the degree of the polynomial (1.10)∑|α|<jDαf(z)α!(w-z)α,
and we are able to lower the cutoff constant by increasing j.The cutoff phenomenon expressed that the generalized Hankel operatorHf,j defined by Peloso and the classical Hankel operator Hf are different.In the present paper, we will consider the generalized Hankel operatorsHf,j defined by Peloso on the Bergman space Ap(B) which is the Banach space of all holomorphic functions f on B such that ∫B|f(z)|pdm(z)<∞, for p>1.Forf(z)∈H(B), j is a positive integer, and we define the generalized Hankel operators Hf,j and Hf,j′ of order j with symbol f by (1.11)Hf,jg(z)=∫B-Δjf(w,z)¯K(z,w)g(w)dm(w),Hf,j′g(z)=∫BΔjf(z,w)¯K(z,w)g(w)dm(w),
where g∈Ap(B), (1.12)Δjf(w,z)=f(w)-∑|α|<jDαf(z)α!(w-z)α,Δjf(z,w)=f(w)-∑|α|<jDαf(w)α!(z-w)α.Luo and Ji-Huai [6] studied the boundedness, compactness, and the Schatten class property of the generalized Hankel operator Hf,j on the Bergman space Ap(B)(p>1), which extended the known results.We will study the essential norm of this kind of generalized Hankel operatorsHf,j and Hf,j′. We recall that the essential norm of a bounded linear operator T is the distance from T to the compact operators; that is, (1.13)‖T‖ess=inf⁡{‖T-K‖:Kisacompactoperator}.The essential norm of a bounded linear operatorT is connected with the compactness of the operator T and the spectrum of the operator T.We know that∥T∥ess=0 if and only if T is compact, so that estimates on ∥T∥ess lead to conditions for T to be compact. Thus, we will obtain a different proof of the compactness of the generalized Hankel operators Hf,j and Hf,j′.Throughout the paper,C denotes a positive constant, whose value may change from one occurrence to the next one.


## 2. Preliminaries

For any fixed pointa∈B-{0}, z∈B, define the Möbius transformation φa by (2.1)φa(z)=a-Pa(z)-saQa(z)1-〈z,a〉,
where sa=1-|a|2 and Pa is the orthogonal projection from Cn onto the one-dimensional subspace [a] generated by a, Qa is the orthogonal projection from Cn onto Cn![a]. It is clear that
(2.2)Pa(z)=〈z,a〉|a|2a,z∈Cn,Qa(z)=z-〈z,a〉|a|2a,z∈B.Lemma 2.1.
For everya∈B, φa has the following properties:(1)
φa(0)=a and φa(a)=0,(2)
φa∘φa(z)=z,z∈B,(3)
1/(1-〈φa(z),a〉)=(1-〈z,a〉)/(1-|a|2),z∈B.Proof.
The proofs can be found in [7].Lemma 2.2.
Fors>-1, t real, define
(2.3)It(z)=∫B(1-|w|2)s|1-〈z,w〉|n+1+s+tdm(w),z∈B.
Then,(1)
t<0, It(z) is bounded in B,(2)
t=0, It(z)~log(1/(1-|z|2)) as |z|→1-,(3)
t>0, It(z)~(1-|z|2)-t as |z|→1-.Here, the notationa(z)~b(z) means that the ratio a(z)/b(z) has a positive finite limit as |z|→1-.Proof.
This is in [7, Theorem  1.12].Lemma 2.3.
Letkξ(z)=K(z,ξ)/∥Kξ∥Lp(dm), where Kξ(z)=K(z,ξ)=1/(1-〈z,ξ〉)n+1, then kξ(z) has the following properties:(1)
∥kξ∥Lp(dm)=1,(2)
kξ(z)→0 at every point z∈B as |ξ|→1-.Proof.
It is obvious.Lemma 2.4.
LetKξ(z)=K(z,ξ). Then, for any positive integer j,(1)
Hf,jKξ=-Δjf(ξ,·)¯Kξ,(2)
Hf,j′Kξ=Δjf(·,ξ)¯Kξ.Proof.
The proof is obtained by the definition ofHf,j and Hf,j′ and the reproducing property of K(z,ξ), through the direct computation to get them.Lemma 2.5.
Letj be any positive integer, f∈H(B), and 0<q<∞, then there is a constant C independent of f, such that(1)
(1-|z|2)j|Rjf(z)|≤C{∫B|Δjf(φz(w),z)|qdm(w)}1/q,(2)
(1-|z|2)j|Rjf(z)|≤C{∫B|Δjf(z,φz(w))|qdm(w)}1/q,whereRjf is the jth order radial derivative of f, (2.4)Rjf(z)=∑k=1∞kjfk(z),
and f(z)=∑k=0∞fk(z) is the homogeneous expansion.Proof.
This is in [3, Proposition  3.2].Lemma 2.6.
Letj be any positive integer, f∈H(B), and 0<ρ<1,  p>1, then (1)
∫B|Δjf(w,z)|p((1-|w|2)-ρ/|1-〈z,w〉|n+1)dm(w)≤C(1-|z|2)-ρ(supz∈B(1-|z|2)j|Rjf(z)|)p,(2)
∫B|Δjf(z,w)|p((1-|w|2)-ρ/|1-〈z,w〉|n+1)dm(w)≤C(1-|z|2)-ρ(supz∈B(1-|z|2)j|Rjf(z)|)p.Proof.
(1) WriteF(w,z) for Δjf(w,z). Using the change of variables w=φz(ξ), we obtain
(*)∫B|F(w,z)|p(1-|w|2)-ρ|1-〈z,w〉|n+1dm(w)=∫B|F(φz(ξ),z)|p(1-|φz(ξ)|2)-ρ|1-〈z,φz(ξ)〉|n+1(1-|z|2)n+1|1-〈ξ,z〉|2(n+1)dm(ξ)=(1-|z|2)-ρ∫B|F(φz(ξ),z)|p(1-|ξ|2)-ρ|1-〈ξ,z〉|n+1-2ρdm(ξ).
Let
(2.5)1<q′<min(1ρ,n+1n+1-ρ)
and set q=q′/(q′-1). Then, applying Hölder's inequality to (*), we obtain
(2.6)∫B|F(w,z)|p(1-|w|2)-ρ|1-〈z,w〉|n+1dm(w)≤(1-|z|2)-ρ(∫B|F(φz(ξ),z)|pqdm(ξ))1/q(∫B(1-|ξ|2)-ρq′|1-〈ξ,z〉|(n+1-2ρ)q′dm(ξ))1/q′.
Because of our choice of q′, it follows that -ρq′>-1 and (n+1-2ρ)q′<n+1-ρq′. Now, Lemma 2.2 implies that
(2.7)∫B(1-|ξ|2)-ρq′|1-〈ξ,z〉|(n+1-2ρ)q′dm(ξ)
is bounded by a constant. Therefore, applying [3, Theorem  3.4], we get
(2.8)∫B|Δjf(w,z)|p(1-|w|2)-ρ|1-〈z,w〉|n+1dm(w)≤C(1-|z|2)-ρ(sup⁡z∈B(1-|z|2)j|Rjf(z)|)p.(2) The proof of (2) is similar to that of (1).


## 3. The Main Result and Its Proof

Theorem 3.1.
Letf∈H(B), j any positive integer, p>1, and the generalized Hankel operators Hf,j, Hf,j′ defined on Ap(B) by
(3.1)Hf,jg(z)=∫B-Δjf(w,z)¯K(z,w)g(w)dm(w),Hf,j′g(z)=∫BΔjf(z,w)¯K(z,w)g(w)dm(w).
Suppose that Hf,j and Hf,j′ are bounded on Ap(B), then the following quantities are equivalent:(1)
∥Hf,j∥ess and ∥Hf,j′∥ess,(2)
lim|z|→1-̅(1-|z|2)j|Rjf(z)|,(3)
lim|z|→1-̅(1-|z|2)|Rf(z)|.Particularly,Hf,j and Hf,j′ are compact on Ap(B) if and only if lim|z|→1-̅(1-|z|2)|Rf(z)|=0.Proof.
First, we will prove that∥Hf,j∥ess≥Clim|z|→1-̅(1-|z|2)j|Rjf(z)|. By the definition of kξ(z) of Lemmas 2.3 and 2.4, we have
(3.2)‖Hf,jkξ‖Lp(dm)p=∫B|Hf,jkξ(z)|pdm(z)=∫B|Hf,jK(z,ξ)‖Kξ‖Lp(dm)|pdm(z)=1‖Kξ‖Lp(dm)p∫B|Hf,jK(z,ξ)|pdm(z)=1‖Kξ‖Lp(dm)p∫B|Δjf(ξ,z)|p|K(z,ξ)|pdm(z)=1‖Kξ‖Lp(dm)p⋅I,
here I=∫B|Δjf(ξ,z)|p|K(z,ξ)|pdm(z).
Use the change of variablesz=φξ(τ) in the integral I, and recall that
(3.3)dm(z)=(1-|ξ|2|1-〈τ,ξ〉|2)n+1dm(τ).
Thus
(3.4)I=∫B|Δjf(ξ,φξ(τ))|p|1-〈φξ(τ),ξ〉|p(n+1)⋅(1-|ξ|2|1-〈τ,ξ〉|2)n+1dm(τ)=1(1-|ξ|2)(n+1)(p-1)∫B|Δjf(ξ,φξ(τ))|p|1-〈τ,ξ〉|(2-p)(n+1)dm(τ)≥1(1-|ξ|2)(n+1)(p-1)(∫B|Δjf(ξ,φξ(τ))|dm(τ))p×(∫B1|1-〈τ,ξ〉|(2-p)(n+1)/(1-p)dm(τ))1-p≥C(1-|ξ|2)(n+1)(p-1)(∫B|Δjf(ξ,φξ(τ))|dm(τ))p≥C(1-|ξ|2)(n+1)(p-1)[(1-|ξ|2)j|Rjf(ξ)|]p.
Here, we have used (3) of Lemma 2.1, Hölder’s inequality for the indexes p and p/(p-1), (1) of Lemma 2.2, and (2) of Lemma 2.5.
Therefore,(3.5)‖Hf,jkξ‖Lp(dm)p≥1‖Kξ‖Lp(dm)pC(1-|ξ|2)(n+1)(p-1)[(1-|ξ|2)j|Rjf(ξ)|]p≥C(1-|ξ|2)(n+1)(p-1)1(1-|ξ|2)(n+1)(p-1)[(1-|ξ|2)j|Rjf(ξ)|]p=C[(1-|ξ|2)j|Rjf(ξ)|]p.
So ∥Hf,jkξ∥Lp(dm)≥C(1-|ξ|2)j|Rjf(ξ)|.
For any compact operatorT, by (2) of Lemma 2.3, we have ∥Tkξ∥Lp(dm)→0 as |ξ|→1-. Then,
(3.6)‖Hf,j-T‖≥lim⁡|ξ|→1-¯‖(Hf,j-T)kξ‖Lp(dm)≥lim⁡|ξ|→1-¯(‖Hf,jkξ‖Lp(dm)-‖Tkξ‖Lp(dm))=lim⁡|ξ|→1-¯‖Hf,jkξ‖Lp(dm)≥Clim⁡|ξ|→1-¯(1-|ξ|2)j|Rjf(ξ)|.
Thus, ∥Hf,j∥ess≥Clim|z|→1-̅(1-|z|2)j|Rjf(z)|.
Now, we will prove that∥Hf,j∥ess≤Clim|z|→1-̅(1-|z|2)j|Rjf(z)|.
WriteF(z,w) for -Δjf(z,w). For 0<ρ<1 and g∈Ap(B), let B(0,ρ) and B(0;ρ,1) denote the ball |z|≤ρ and the ring ρ<|z|<1, respectively, then we have
(3.7)Hf,jg(z)=χB(0,ρ)(z)∫BF(w,z)¯K(z,w)g(w)dm(w)+χB(0;ρ,1)(z)∫BF(w,z)¯K(z,w)g(w)dm(w)=T1g(z)+T2g(z).
Here,
(3.8)T1g(z)=χB(0,ρ)(z)∫BF(w,z)¯K(z,w)g(w)dm(w),T2g(z)=χB(0;ρ,1)(z)∫BF(w,z)¯K(z,w)g(w)dm(w).
We first show thatT1 is compact. Let {gl} be a sequence weakly converging to 0 and p′=p/(p-1), by Hölder’s inequality, then we have
(3.9)|T1gl(z)|p=|χB(0,ρ)(z)∫BF(w,z)¯K(z,w)gl(w)dm(w)|p≤χB(0,ρ)(z)(∫B|F(w,z)||gl(w)||1-〈z,w〉|n+1dm(w))p≤χB(0,ρ)(z)(∫B|F(w,z)|p′(1-|w|2)-1/p|1-〈z,w〉|n+1dm(w))p/p′×∫B|gl(w)|p(1-|w|2)1/p′|1-〈z,w〉|n+1dm(w).
By Lemma 2.6, we get
(3.10)|T1gl(z)|p≤CχB(0,ρ)(z)(sup⁡z∈B(1-|z|2)j|Rjf(z)|)p×(1-|z|2)-1/p′∫B|gl(w)|p(1-|w|2)1/p′|1-〈z,w〉|n+1dm(w).
Thus,
(3.11)‖T1gl‖Lp(dm)p=∫B|T1gl(z)|pdm(z)≤C∫|z|<ρ(sup⁡z∈B(1-|z|2)j|Rjf(z)|)p(1-|z|2)-1/p′×∫B|gl(w)|p(1-|w|2)1/p′|1-〈z,w〉|n+1dm(w)dm(z)≤C(1-|ρ|2)-1/p′(sup⁡z∈B(1-|z|2)j|Rjf(z)|)p×∫∫B|gl(w)|p(1-|w|2)1/p′|1-〈z,w〉|n+1dm(w)dm(z)=C(1-|ρ|2)-1/p′(sup⁡z∈B(1-|z|2)j|Rjf(z)|)p⋅∫B|gl(w)|p(1-|w|2)1/p′∫B1|1-〈z,w〉|n+1dm(z)dm(w)≤C(1-|ρ|2)-1/p′(sup⁡z∈B(1-|z|2)j|Rjf(z)|)p×∫B|gl(w)|p(1-|w|2)1/p′log⁡⁡(1-|w|2)dm(w)⟶0,asl⟶∞.
So,T1 is compact.
Forg∈Ap and p′=p/(p-1), by Hölder’s inequality,
(3.12)|T2g(z)|p=|χB(0;ρ,1)(z)∫BF(w,z)¯K(z,w)g(w)dm(w)|p≤(∫BχB(0;ρ,1)(z)|F(w,z)||g(w)||1-〈z,w〉|n+1dm(w))p≤(∫BχB(0;ρ,1)(z)|F(w,z)|p′(1-|w|2)-1/p|1-〈z,w〉|n+1dm(w))p/p′×∫B|g(w)|p(1-|w|2)1/p′|1-〈z,w〉|n+1dm(w).
So
(3.13)‖T2g‖Lp(dm)p=∫B|T2g(z)|pdm(z)≤∫B(∫BχB(0;ρ,1)(z)|F(w,z)|p′(1-|w|2)-1/p|1-〈z,w〉|n+1dm(w))p/p′⋅∫B|g(w)|p(1-|w|2)1/p′|1-〈z,w〉|n+1dm(w)dm(z).
Change the variables w=φz(ξ), let
(3.14)1<q′<min(p,n+1n+1-1/p),
and set q=q′/(q′-1), by Lemmas 2.1 and 2.2, then we obtain
(3.15)∫BχB(0;ρ,1)(z)|F(w,z)|p′(1-|w|2)-1/p|1-〈z,w〉|n+1dm(w)=∫BχB(0;ρ,1)(z)|F(φz(ξ),z)|p′(1-|φz(ξ)|2)-1/p|1-〈z,φz(ξ)〉|n+1(1-|z|2)n+1|1-〈ξ,z〉|2(n+1)dm(ξ)=∫BχB(0;ρ,1)(z)(1-|z|2)-1/p|F(φz(ξ),z)|p′(1-|ξ|2)-1/p|1-〈ξ,z〉|n+1-2/pdm(ξ)≤(1-|z|2)-1/p(∫BχB(0;ρ,1)(z)|F(φz(ξ),z)|p′qdm(ξ))1/q×(∫B(1-|ξ|2)-q′/p|1-〈ξ,z〉|(n+1-2/p)q′dm(ξ))1/q′≤C(1-|z|2)-1/p(∫BχB(0;ρ,1)(z)|F(φz(ξ),z)|p′qdm(ξ))1/q.
By the same argument of [3, Theorem  3.4], we know that
(3.16)(∫BχB(0;ρ,1)(z)|F(φz(ξ),z)|p′qdm(ξ))1/q≤C(sup⁡ρ<|z|<1(1-|z|2)j|Rjf(z)|)p′.
Applying Fubini’s theorem and Lemma 2.2, we have
(3.17)‖T2g‖Lp(dm)p≤C(sup⁡ρ<|z|<1(1-|z|2)j|Rjf(z)|)p×∫B(1-|z|2)-1/p′∫B|g(w)|p(1-|w|2)1/p′|1-〈z,w〉|n+1dm(w)dm(z)≤C(sup⁡ρ<|z|<1(1-|z|2)j|Rjf(z)|)p‖g‖Lp(dm)p.
So
(3.18)‖T2‖≤Csup⁡ρ<|z|<1(1-|z|2)j|Rjf(z)|.
Thus, by the definition of the essential norm, we have
(3.19)‖Hf,j‖ess≤‖T1+T2‖ess≤‖T2‖≤Csup⁡ρ<|z|<1(1-|z|2)j|Rjf(z)|.
As ρ→1, we have
(3.20)‖Hf,j‖ess≤Clim⁡|z|→1-¯(1-|z|2)j|Rjf(z)|.
Similarly, we get the equality of∥Hf,j′∥ess and lim|z|→1-̅(1-|z|2)j|Rjf(z)|.
By [7, Theorems  3.4 and 3.5], we obtain the equality of lim|z|→1-̅(1-|z|2)j|Rjf(z)| and lim|z|→1-̅(1-|z|2)|Rf(z)|.
We complete the proof of Theorem3.1.



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