Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 343578, 13 pages
doi:10.1155/2010/343578
Research Article
The Essential Norm of the Generalized
Hankel Operators on the Bergman Space of
the Unit Ball in Cn
Luo Luo and Yang Xuemei
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Correspondence should be addressed to Luo Luo, lluo@ustc.edu.cn
Received 26 August 2010; Revised 1 December 2010; Accepted 31 December 2010
Academic Editor: Simeon Reich
Copyright q 2010 L. Luo and Y. Xuemei. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
In 1993, Peloso introduced a kind of operators on the Bergman space A2 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.
1. Introduction
Let B be the open unit ball in Cn , m the Lebesgue measure on Cn normalized so that mB 1,
HB denotes the class of all holomorphic functions on B. The
space A2 B is the
Bergman
2
Banach space of all holomorphic functions f on B such that B |fz| dmz < ∞. It is easy to
show that A2 B is a closed subspace of L2 B, dm.
There is an orthogonal projection of L2 B, dm onto A2 B, denoted by P and
P fz Kz, wfwdmw,
1.1
B
where Kz, w 1/1 − z, wn1 is the Bergman kernel on B.
For a function f ∈ HB, define the Hankel operator Hf : A2 B → A2 B⊥ with
symbol f by
1.2
fz − fw Kz, wgwdmw,
Hf g I − P f g B
where I is the identity operator.
2
Abstract and Applied Analysis
Since the Hankel operator Hf 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 ∈ βB and Hf is compact A2 B if and only if f ∈ β0 B, where
βB f ∈ HB : sup 1 − |z| Rfz < ∞ ,
2
z∈B
β0 B f ∈ HB : 1 − |z|2 Rfz −→ 0, as |z| −→ 1 .
1.3
Rf is the radial derivative of f defined by
Rfz n
zj
j1
∂fz
.
∂zj
1.4
βB is called the Bloch space, and β0 B is called the little Bloch space.
For n 1, f ∈ HD 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
Bp D f ∈ HD : f z 1 − |z|2 ∈ Lp dλ ,
p > 1,
1.5
−2
and dλz 1 − |z|2 dmz 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.
For n > 1, f ∈ HB, 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
Bp B f ∈ HB : 1 − |z|2 Rfz ∈ Lp dλ ,
p > n,
1.6
−n1
dmz is the invariant volume measure on B. Bp B is called the
and dλz 1 − |z|2 Besov space on B. Then, the cutoff phenomenon of Hf appears at p 2n. If cn denotes the
value of “cutoff,” then
cn ⎧
⎨1,
n 1,
1.7
⎩2n, n > 1.
Obviously, cn depends on the dimension n of the unit ball.
In 1993, Peloso 3 replaced fz − fw with
Δj fw, z fw −
Dα fz
|α|<j
α!
w − zα
1.8
Abstract and Applied Analysis
3
to define a kind of generalized Hankel operator:
Hf,j gz −Δj fw, zKz, wgwdmw,
B
Hf,j
gz 1.9
Δj fz, wKz, wgwdmw.
B
Here, Dα fz ∂|α| fz/∂zα1 1 · · · ∂zαnn . 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 ∈ HB, 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 cn depends not only on the dimension but also on the
degree of the polynomial
Dα fz
|α|<j
α!
w − zα ,
1.10
and we are able to lower the cutoff constant by increasing j.
The cutoff phenomenon expressed that the generalized Hankel operator Hf,j defined
by Peloso and the classical Hankel operator Hf are different.
In the present paper, we will consider the generalized Hankel operators Hf,j defined
p
by Peloso on the Bergman
spacep A B which is the Banach space of all holomorphic
functions f on B such that B |fz| dmz < ∞, for p > 1.
For fz ∈ HB, j is a positive integer, and we define the generalized Hankel
operators Hf,j and Hf,j
of order j with symbol f by
Hf,j gz −Δj fw, zKz, wgwdmw,
B
gz
Hf,j
1.11
Δj fz, wKz, wgwdmw,
B
where g ∈ Ap B,
Δj fw, z fw −
Dα fz
|α|<j
Δj fz, w fw −
α!
w − zα ,
Dα fw
|α|<j
α!
1.12
α
z − w .
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.
4
Abstract and Applied Analysis
We will study the essential norm of this kind of generalized Hankel operators Hf,j and
. We recall that the essential norm of a bounded linear operator T is the distance from T
Hf,j
to the compact operators; that is,
Tess inf T − K : K is a compact operator .
1.13
The essential norm of a bounded linear operator T is connected with the compactness
of the operator T and the spectrum of the operator T.
We know that Tess 0 if and only if T is compact, so that estimates on Tess 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 point a ∈ B − {0}, z ∈ B, define the Möbius transformation ϕa by
ϕa z a − Pa z − sa Qa z
,
1 − z, a
2.1
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
Pa z z, a
|a|2
Qa z z −
z ∈ Cn ,
a,
z, a
|a|2
2.2
a,
z ∈ B.
Lemma 2.1. For every a ∈ 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. For s > −1, t real, define
It z B
1 − |w|2
s
|1 − z, w|n1st
dmw,
z ∈ B.
2.3
Abstract and Applied Analysis
5
Then,
1 t < 0, It z is bounded in B,
2 t 0, It z ∼ log1/1 − |z|2 as |z| → 1− ,
−t
3 t > 0, It z ∼ 1 − |z|2 as |z| → 1− .
Here, the notation az ∼ bz means that the ratio az/bz has a positive finite limit
as |z| → 1− .
Proof. This is in 7, Theorem 1.12.
Lemma 2.3. Let kξ z Kz, ξ/Kξ Lp dm , where Kξ z Kz, ξ 1/1 − z, ξn1 , 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. Let Kξ z Kz, ξ. Then, for any positive integer j,
1 Hf,j Kξ −Δj fξ, ·Kξ ,
2 Hf,j
Kξ Δj f·, ξKξ .
Proof. The proof is obtained by the definition of Hf,j and Hf,j
and the reproducing property
of Kz, ξ, through the direct computation to get them.
Lemma 2.5. Let j be any positive integer, f ∈ HB, and 0 < q < ∞, then there is a constant C
independent of f, such that
j
1/q
1 1 − |z|2 |Rj fz| ≤ C{ B |Δj fϕz w, z|q dmw} ,
j
1/q
2 1 − |z|2 |Rj fz| ≤ C{ B |Δj fz, ϕz w|q dmw} ,
where Rj f is the jth order radial derivative of f,
Rj fz ∞
2.4
kj fk z,
k1
and fz ∞
k0
fk z is the homogeneous expansion.
Proof. This is in 3, Proposition 3.2.
Lemma 2.6. Let j be any positive integer, f ∈ HB, and 0 < ρ < 1, p > 1, then
−ρ
−ρ
≤
C1 − |z|2 supz∈B 1 −
1 B |Δj fw, z|p 1 − |w|2 /|1 − z, w|n1 dmw
j
p
|z|2 |Rj fz| ,
−ρ
2 B |Δj fz, w|p 1 − |w|2 /|1 − z, w|n1 dmw
j
p
|z|2 |Rj fz| .
≤
−ρ
C1 − |z|2 supz∈B 1 –
6
Abstract and Applied Analysis
Proof. 1 Write Fw, z for Δj fw, z. Using the change of variables w ϕz ξ, we obtain
|Fw, z|p
B
1 − |w|2
−ρ
dmw
|1 − z, w|n1
n1
2 −ρ 1 − |z|2
p 1 − ϕz ξ
F ϕz ξ, z dmξ
1 − z, ϕz ξ n1 |1 − ξ, z|2n1
B
−ρ
2
1
−
|ξ|
−ρ F ϕz ξ, z p
1 − |z|2
dmξ.
|1 − ξ, z|n1−2ρ
B
∗
Let
1 < q < min
1 n1
,
ρ n1−ρ
2.5
and set q q /q − 1. Then, applying Hölder’s inequality to ∗, we obtain
|Fw, z|p
B
1 − |w|2
−ρ
|1 − z, w|n1
dmw
⎛
1/q −ρ ⎜
F ϕz ξ, z pq dmξ
≤ 1 − |z|2
⎝
B
B
1 − |ξ|2
⎞1/q
−ρq
|1 − ξ, z|n1−2ρq
⎟
dmξ⎠
.
2.6
Because of our choice of q , it follows that −ρq > −1 and n 1 − 2ρq < n 1 − ρq . Now,
Lemma 2.2 implies that
B
1 − |ξ|2
−ρq
|1 − ξ, z|n1−2ρq
2.7
dmξ
is bounded by a constant. Therefore, applying 3, Theorem 3.4, we get
Δj fw, zp
B
1 − |w|2
−ρ
|1 − z, w|n1
dmw ≤ C 1 − |z|
2 The proof of 2 is similar to that of 1.
2
−ρ
j sup 1 − |z| Rj fz
z∈B
2
p
.
2.8
Abstract and Applied Analysis
7
3. The Main Result and Its Proof
Theorem 3.1. Let f ∈ HB, j any positive integer, p > 1, and the generalized Hankel operators
defined on Ap B by
Hf,j , Hf,j
Hf,j gz −Δj fw, zKz, wgwdmw,
B
3.1
Hf,j
gz Δj fz, wKz, wgwdmw.
B
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 j
2 lim|z| → 1− 1 − |z| |Rj fz|,
3 lim|z| → 1− 1 − |z|2 |Rfz|.
Particularly, Hf,j and Hf,j
are compact on Ap B if and only if lim|z| → 1− 1 −
|z|2 |Rfz| 0.
j
Proof. First, we will prove that Hf,j ess ≥ C lim|z| → 1− 1 − |z|2 |Rj fz|. By the definition of
kξ z of Lemmas 2.3 and 2.4, we have
!
!
!Hf,j kξ !p p
L dm
Hf,j kξ zp dmz
B
Kz, ξ
!
Hf,j !
!
Kξ ! p
B
L dm
1
! !p
!Kξ ! p
L dm
1
! !p
!Kξ ! p
Hf,j Kz, ξp dmz
B
L dm
p
dmz
3.2
Δj fξ, zp |Kz, ξ|p dmz
B
1
! !p
· I,
!Kξ ! p
L dm
here I B |Δj fξ, z|p |Kz, ξ|p dmz.
Use the change of variables z ϕξ τ in the integral I, and recall that
dmz 1 − |ξ|2
|1 − τ, ξ|2
n1
dmτ.
3.3
8
Abstract and Applied Analysis
Thus
I
B
Δj f ξ, ϕξ τ p
1 − |ξ|2
·
1 − ϕξ τ, ξ pn1
|1 − τ, ξ|2
1 − |ξ|2
1
n1p−1
1
≥ n1p−1
1 − |ξ|2
B
dmτ
Δj f ξ, ϕξ τ p
|1 − τ, ξ|2−pn1
B
dmτ
p
Δj f ξ, ϕξ τ dmτ
B
3.4
×
n1
1−p
1
2−pn1/1−p
|1 − τ, ξ|
C
≥ n1p−1
1 − |ξ|2
dmτ
p
Δj f ξ, ϕξ τ dmτ
B
"
# p
j C
2 j
≥ n1p−1 1 − |ξ| R fξ .
2
1 − |ξ|
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,
!
!
!Hf,j kξ !p p
L dm
1
≥ ! !p
!K ξ ! p
L dm
"
# p
j C
2 j
1
−
fξ
|ξ|
R
n1p−1
1 − |ξ|2
n1p−1
≥ C 1 − |ξ|2
C
"
1 − |ξ|2
#p
j 1 − |ξ|2 Rj fξ .
So Hf,j kξ Lp dm ≥ C1 − |ξ|2 j |Rj fξ|.
"
#p
j 1
2 j
1
−
fξ
|ξ|
R
n1p−1
3.5
Abstract and Applied Analysis
9
For any compact operator T, by 2 of Lemma 2.3, we have Tkξ Lp dm → 0 as |ξ| →
1− . Then,
!
!
!
!
!Hf,j − T ! ≥ lim ! Hf,j − T kξ ! p
L dm
−
|ξ| → 1
!
!
!
!
≥ lim − !Hf,j kξ !Lp dm − !Tkξ !Lp dm
|ξ| → 1
!
!
lim − !Hf,j kξ !Lp dm
3.6
|ξ| → 1
j ≥ C lim − 1 − |ξ|2 Rj fξ.
|ξ| → 1
j
Thus, Hf,j ess ≥ C lim|z| → 1− 1 − |z|2 |Rj fz|.
j
Now, we will prove that Hf,j ess ≤ C lim|z| → 1− 1 − |z|2 |Rj fz|.
Write Fz, w for −Δj fz, w. For 0 < ρ < 1 and g ∈ Ap B, let B0, ρ and B0; ρ, 1
denote the ball |z| ≤ ρ and the ring ρ < |z| < 1, respectively, then we have
Hf,j gz χB0,ρ z
Fw, zKz, wgwdmw
B
χB0;ρ,1 z
Fw, zKz, wgwdmw
3.7
B
T1 gz T2 gz.
Here,
T1 gz χB0,ρ z
Fw, zKz, wgwdmw,
B
3.8
T2 gz χB0;ρ,1 z
Fw, zKz, wgwdmw.
B
We first show that T1 is compact. Let {gl } be a sequence weakly converging to 0 and
p p/p − 1, by Hölder’s inequality, then we have
p
T1 gl zp χB0,ρ z Fw, zKz, wgl wdmw
B
p
gl w
≤ χB0,ρ z
dmw
|Fw, z|
|1 − z, w|n1
B
⎞p/p
⎛
|Fw, z|p 1 − |w|2 −1/p
⎟
⎜
≤ χB0,ρ z⎝
dmw⎠
|1 − z, w|n1
B
×
1/p
gl wp 1 − |w|2
B
|1 − z, w|n1
dmw.
3.9
10
Abstract and Applied Analysis
By Lemma 2.6, we get
j T1 gl zp ≤ CχB0,ρ z sup 1 − |z|2 Rj fz
p
z∈B
× 1 − |z|2
1/p
p −1/p gl w 1 − |w|2
|1 − z, w|n1
B
3.10
dmw.
Thus,
!
!
!T1 gl !p p
L dm
T1 gl zp dmz
B
≤C
×
|z|<ρ
j sup 1 − |z|2 Rj fz
p
1 − |z|2
−1/p
z∈B
1/p
gl wp 1 − |w|2
dmwdmz
|1 − z, w|n1
j 2 −1/p
≤C 1− ρ
sup 1 − |z|2 Rj fz
B
p
z∈B
×
gl wp 1 − |w|2 1/p
dmwdmz
|1 − z, w|n1
p
j 2 −1/p
2 j
C 1− ρ
sup 1 − |z| R fz
B
z∈B
1/p gl wp 1 − |w|2
·
B
2 −1/p
≤ C 1 − ρ
1
B
|1 − z, w|n1
j sup 1 − |z| Rj fz
dmzdmw
p
2
z∈B
1/p
gl wp 1 − |w|2
log 1 − |w|2 dmw
×
B
−→ 0,
as l −→ ∞.
3.11
So, T1 is compact.
For g ∈ Ap and p p/p − 1, by Hölder’s inequality,
p
T2 gzp χB0;ρ,1 z Fw, zKz, wgwdmw
B
≤
χB0;ρ,1 z|Fw, z|
B
gw
|1 − z, w|
p
dmw
n1
Abstract and Applied Analysis
11
⎛
−1/p
|Fw, z|p 1 − |w|2
⎜
≤⎝
χB0;ρ,1 z
|1 − z, w|n1
B
×
1/p
gwp 1 − |w|2
|1 − z, w|n1
B
⎞p/p
⎟
dmw⎠
dmw.
3.12
So
!
!
!T2 g !p p
L dm
T2 gzp dmz ≤
⎛
B
⎜
⎝
B
·
χB0;ρ,1 z
−1/p
|Fw, z|p 1 − |w|2
|1 − z, w|
B
gwp 1 − |w|2 1/p
B
|1 − z, w|n1
n1
⎞p/p
⎟
dmw⎠
dmwdmz.
3.13
Change the variables w ϕz ξ, let
1 < q < min p,
n1
,
n 1 − 1/p
3.14
and set q q /q − 1, by Lemmas 2.1 and 2.2, then we obtain
χB0;ρ,1 z
|Fw, z|p 1 − |w|2 −1/p
dmw
|1 − z, w|n1
B
n1
−1/p F ϕz ξ, z p 1 − ϕz ξ2
1 − |z|2
χB0;ρ,1 z
dmξ
1 − z, ϕz ξ n1
B
|1 − ξ, z|2n1
−1/p F ϕz ξ, z p
χB0;ρ,1 z 1 − |z|2
1 − |ξ|2
−1/p
|1 − ξ, z|n1−2/p
1/q
−1/p p q
2
≤ 1 − |z|
χB0;ρ,1 z F ϕz ξ, z
dmξ
B
B
⎛
⎜
×⎝
B
1 − |ξ|2
|1 − ξ, z|n1−2/pq
≤ C 1 − |z|
2
⎞1/q
−q /p
−1/p ⎟
dmξ⎠
p q
χB0;ρ,1 zF ϕz ξ, z dmξ
B
1/q
.
dmξ
3.15
12
Abstract and Applied Analysis
By the same argument of 3, Theorem 3.4, we know that
p q
χB0;ρ,1 zF ϕz ξ, z dmξ
1/q
j sup 1 − |z| Rj fz
≤C
B
p
2
.
3.16
ρ<|z|<1
Applying Fubini’s theorem and Lemma 2.2, we have
!
!
!T2 g !p p
L dm
j sup 1 − |z|2 Rj fz
≤C
p
ρ<|z|<1
×
1 − |z|2
B
≤C
1/p
p −1/p gw 1 − |w|2
B
|1 − z, w|n1
j sup 1 − |z|2 Rj fz
p
! !p
!g ! p
L dm
ρ<|z|<1
3.17
dmwdmz
.
So
j T2 ≤ C sup 1 − |z|2 Rj fz.
3.18
ρ<|z|<1
Thus, by the definition of the essential norm, we have
!
!
!Hf,j !
ess
j ≤ T1 T2 ess ≤ T2 ≤ C sup 1 − |z|2 Rj fz.
3.19
ρ<|z|<1
As ρ → 1, we have
j !
!
!Hf,j ! ≤ C lim 1 − |z|2 Rj fz.
ess
−
3.20
|z| → 1
Similarly, we get the equality of Hf,j
j
ess
and lim|z| → 1− 1 − |z|2 |Rj fz|.
j
By 7, Theorems 3.4 and 3.5, we obtain the equality of lim|z| → 1− 1 − |z|2 |Rj fz| and
lim|z| → 1− 1 − |z|2 |Rfz|.
We complete the proof of Theorem 3.1.
Acknowledgments
The authors would like to thank the referee for the careful reading of the first version of this
paper and for the several suggestions made for improvement. The paper was supported by
the NNSF of China no. 10771201 and the Natural Science Foundation of Anhui Province
no. 090416233.
Abstract and Applied Analysis
13
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