Toeplitz Operators on the Weighted Pluriharmonic Bergman Space

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2011, Article ID 210596, 15 pages doi:10.1155/2011/210596

Research Article Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols Zhi Ling Sun1, 2 and Yu Feng Lu1 1 2

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028000, China

Correspondence should be addressed to Zhi Ling Sun, [email protected] Received 3 June 2011; Revised 20 July 2011; Accepted 21 July 2011 Academic Editor: Natig Atakishiyev Copyright q 2011 Z. L. Sun and Y. F. Lu. 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. We construct an operator R whose restriction onto weighted pluriharmonic Bergman Space bμ2 Bn is an isometric isomorphism between bμ2 Bn and l2# . Furthermore, using the operator R we prove that each Toeplitz operator Ta with radial symbols is unitary to the multication operator γa,μ I. Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about the operator, providing its spectral decomposition.

1. Introduction Let Bn be the open unit ball in the complex vector space Cn . For any z z1 , . . . , zn and √ ξ ξ1 , . . . , ξn in Cn , let z·ξ nj1 zj ξj , where ξj is the complex conjugate of ξj and |z| z · z. For a multi-index α α1 , . . . , αn and z z1 , . . . , zn ∈ Cn , we write zα zα1 1 · · · zαnn , where αk ∈ Z N ∪ {0}, and |α| α1 · · · αn is its length, α! α1 ! · · · αn !. The weighted pluriharmonic Bergman space bμ2 Bn is the subspace of the weighted space L2μ Bn consisting of all pluriharmonic functions on Bn . A pluriharmonic function in the unit ball is the sum of a holomorphic function and the conjugate of a holomorphic functions. μ It is known that bμ2 Bn is a closed subspace of L2μ Bn and hence is a Hilbert space. Let QBn be the Hilbert space orthogonal projection from L2μ Bn onto bμ2 Bn . For a function u ∈ L2μ Bn , the Toeplitz operator Tu : bμ2 Bn → bμ2 Bn with symbol u is the linear operator defined by μ Tu f QBn uf ,

f ∈ bμ2 Bn .

Tu is densely defined and not bounded in general.

1.1

2

Abstract and Applied Analysis

The boundedness and compactness of Toeplitz operators on Bergman type spaces have been studied intensively in recent years. The fact that the product of two harmonic functions is no longer harmonic adds some mystery in the study of Toeplitz operators on harmonic Bergman space. Many methods which work for the operator on analytic Bergman spaces lost their effectiveness on harmonic Bergman space. Therefore new ideas and methods are needed. We refer to 1–3 for references about the results of Toeplitz operator on harmonic Bergman space. The paper 3 characterizes compact Toeplitz operators in the case of the unit disk D. In 2, the authors consider Toeplitz operators acting on the pluriharmonic Bergman space and study the problem of when the commutator or semicommutator of certain Toeplitz operators is zero. Lee 1 proved that two Toplitz operators acting on the pluriharmonic Bergman space with radial symbols and pluriharmonic symbol, respectively, commute only in an obvious case. The authors in 4 analyze the influence of the radial component of a symbol to spectral, compactness and Fredholm properties of Toeplitz operators on Bergman space on unit disk D. In 5, they are devoted to study Toeplitz operators with radial symbols on the weighted Bergman spaces on the unit ball in Cn . In this paper, we will be concerned with the question of Toeplitz operators with radial symbols on the weighted pluriharmonic Bergman space. Based on the techniques in 4–6, we construct an operator R whose restriction onto weighted pluriharmonic Bergman space bμ2 Bn is an isometric isomorphism between bμ2 Bn and l2# , and RR∗ I : l2# −→ l2# , μ

R∗ R QBn : L2μ Bn −→ bμ2 Bn ;

1.2

where l2# is the subspace of l2 . Using the operator R we prove that each Toeplitz operator Ta with radial symbols is unitary to the multication operator γa,μ I acting on l2# . Next, we use the Berezin concept of Wick and anti-Wick symbols. It turns out that in our particular radial symbols case the Wick symbols of a Toeplitz operator give complete information about the operator, providing its spectral decomposition.

2. Pluriharmonic Bergman Space and Orthogonal Projection We start this section with a decomposition of the space L2μ Bn . Consider a nonnegative measurable function μr, r ∈ 0, 1, such that mes{r ∈ 0, 1 : μr > 0} 1, and Bn

1 2n−1 μ|z|dvz S μrr 2n−1 dr < ∞,

2.1

0

where |S2n−1 | 2π n−1/2 Γ−1 n − 1/2 is the surface area of unit sphere S2n−1 and Γz is the Gamma function. Introduce the weighted space L2μ Bn

2 f : f L2μ Bn

Bn

fz2 μ|z|dvz < ∞ ,

2.2


3

where dvz is the usual Lebesgue volume measure and L2 S2n−1 is the space with the usual Lebesgue surface measure. The space L2 S2n−1 is the direct sum of mutually orthogonal spaces Hk , that is, ∞

Hk , L2 S2n−1

2.3

k0

where Hk denotes the space of spherical harmonics of order k. Meanwhile, each space Hk is the direct sum under the identification Cn R2n of the mutually orthogonal spaces Hp,q see, e.g., 7: Hk

Hp,q ,

k ∈ Z ,

2.4

pqk p,q∈Z

where Hp,q , for each p, q 0, 1, . . ., is the space of harmonic polynomials their restrictions to the unit sphere of complete order p in the variable z and complete order q in the conjugate variable z z1 , . . . , zn . Thus, we can get

L2 S2n−1 Hp,q .

2.5

p,q∈Z

The Hardy space H 2 Bn in the unit ball Bn is a closed subspace of L2 S2n−1 . Denote by PS2n−1 the Szego¨ orthogonal projection of L2 S2n−1 onto the Hardy space H 2 Bn . It is well ∞

known that H 2 Bn Hp,0 . The standard orthonormal base in H 2 Bn has the form p0

eα ω dn,α ω , α

dn,α

n − 1 |α|! |S2n−1 |n − 1!α!

for |α| 0, 1, . . . .

2.6

Fix an orthonormal basis {eα,β ω}α,β , α, β ∈ Zn , in the space L2 S2n−1 so that eα,0 ω ≡ eα ω, e0,α ω ≡ eα,0 ω ≡ eα w, |α| 0, 1, . . .. Passing to the spherical coordinates in the unit ball we have

L2μ Bn L2 0, 1, μrr 2n−1 dr ⊗ L2 S2n−1 .

2.7

For any function fz ∈ L2μ Bn have the decomposition

fz

∞

cα,β reα,β ω,

|α||β|0

r |z|, ω

z , r

2.8

4


with the coefficients cα,β r satisfying the condition 2 f 2 n Lμ B

1 ∞ cα,β r2 μrr 2n−1 dr < ∞. |α||β|0 0

2.9

According to the decomposition 2.7, 2.8 together with Parseval’s equality, we can define the unitary operator

U1 : L2 0, 1, μrr 2n−1 dr ⊗ L2 S2n−1 −→ L2 0, 1, μrr 2n−1 dr ⊗ l2 2.10

≡ l2 L2 0, 1, μrr 2n−1 dr ,

by the rule U1 : fz → {cα,β r}, and ∞ 2 f 2 n cα,β r2 cα,β r2 . Lμ B l2 L2 0,1,μrr 2n−1 dr L2 0,1,μrr 2n−1 dr

2.11

|α||β|0

Let fz be a pluriharmonic in the unit ball Bn and write f g h, where the functions g, h are holomorphic in Bn . Suppose

gz

∞

hz

cα zα ,

|α|0

∞

2.12

cβ zβ

|β|0

are their power series representations of g and h, respectively. We have

fz

∞

cα zα

|α|0

∞

cβ zβ

|β|0

∞

∞

cα reα ω

|α|0

cβ reβ ω,

2.13

|β|0

−1 |β| −1 |α| where cα r cα dn,α r , cβ r cβ dn,β r , r |z|, ω z/r.

μ

Let bμ2 Bn be the pluriharmonic Bergman space in Bn from L2μ Bn . Denote by QBn the pluriharmonic Bergman orthogonal projection of L2μ Bn onto the Bergman space bμ2 Bn . From the above it follows that to characterize a function fz ∈ bμ2 Bn and considering its decomposition according to 2.13, one can restrict to the function having the representation

fz gz hz

∞

cα,0 reα,0 ω

|α|0

∞

c0,β re0,β ω.

|β|0

2.14


5

Now let us take an arbitrary fz from bμ2 Bn in the form 2.14. It will satisfy the CauchyRiemann equations, that is, ∂ 1 ∂ ∂ gz ≡ i gz 0, 2 ∂xk ∂yk ∂zk ∂ ∂ 1 ∂ hz ≡ −i hz 0, ∂zk 2 ∂xk ∂yk

k 1, . . . , n, z ∈ Bn , 2.15 k 1, . . . , n, z ∈ B . n

Applying ∂/∂zk , ∂/∂zk to g and h, respectively, we have ∞ ∞ ∂ d zk |α| cα,0 r − cα,0 r eα,0 ω, cα,0 reα,0 ω 2r |α|0 dr r ∂zk |α|0 ∞ ∞ β ∂ d zk c0,β r − c0,β r e0,β ω, c0,β re0,β ω ∂zk 2r dr r |β|0 |β|0

2.16

where k 0, . . . , n, and we come to the infinite system of ordinary linear differential equations d |α| cα,0 r − cα,0 r 0, dr r β d c0,β r − c0,β r 0, dr r

|α| 0, 1, . . . β 0, 1, . . . .

2.17

Their general solution has the form cα,0 bα r |α| λn, |α|cα,0 r |α| , c0,β bβ r |β| λn, |β|c0,β r |β| , 1 with λn, m 0 t2m2n−1 μtdt−1/2 . Hence, for any fz ∈ bμ2 Bn we have fz

∞

cα,0 λn, |α|r |α| eα,0

μ

c0,β λ n, β r |β| e0,β .

2.18

|β|0

|α|0

And, it is easy to verify f 2L2 Bn

∞

∞

|α|0

|cα,0 |2

∞

|β|0

2 |c0,β |2 . Thus the image b1,μ Bn

U1 bμ2 Bn is characterized as the closed subspace of

L2 0, 1, μrr 2n−1 dr ⊗ l2 l2 L2 0, 1, μrr 2n−1 dr

2.19

which consists of all sequences {cα,β r} of the form ⎧ ⎪ λn, |α|cα,0 r |α| , ⎪ ⎪ ⎨ cα,β r λ n, β c0,β r |β| , ⎪ ⎪ ⎪ ⎩ 0,

β 0 |α| 0

0. 0, β / |α| /

2.20

6

Abstract and Applied Analysis For each m ∈ Z introduce the function ϕm ρ λn, m1/n

ρ

1/2n r

2m2n−1

μrdr

,

ρ ∈ 0, 1.

2.21

0

Obviously, there exists the inverse function for the function ϕm ρ on 0, 1, which we will denote by φm r. Introduce the operator

um f r

√

2n −m φ rf φm r . λn, m m

2.22

By Proposition 2.1 in 5, the operator um maps unitary L2 0, 1, μrr 2n−1 onto L2 0, 1, r 2n−1 dr in such a way that um λn, mr m

√ 2n,

m ∈ Z .

2.23

Intoduce the unitary operator

U2 : l2 L2 0, 1, μrr 2n−1 dr −→ l2 L2 0, 1, r 2n−1 dr ,

2.24

U2 : cα,β r −→ u|α||β| cα,β r .

2.25

where

2 2 By 2.23, we can get the space b2,μ U2 b1,μ coincides with the space of all sequences {kα,β } for which

kα,β

⎧√ ⎪ 2ncα,0 , ⎪ ⎪ ⎨√ 2nc0,β , ⎪ ⎪ ⎪ ⎩ 0,

β 0 |α| 0

0. 0, β / |α| /

2.26

√ Let l0 r 2n. We have l0 r ∈ L2 0, 1, r 2n−1 dr and l0 L2 0,1,r 2n−1 1. Denote by L0 the one-dimensional subspace of L2 0, 1, r 2n−1 dr generated by l0 r. The orthogonal projection P0 of L2 0, 1, r 2n−1 dr onto L0 has the form √ P0 f r f, l0 l0 2n

1

√ f ρ 2nρ2n−1 dρ.

2.27

0

√ Let dα,β kα,β 2n−1 . Denote by l2# the subspace of l2 consisting of all sequences {dα,β }. And let p# be the orthogonal projections of l2 onto l2# , then p# χ α, βI, where χ α, β 0, if |α β| > 0 and χ α, β 1, if |α β| 0.


7

2 L0 ⊗ l2# and the orthogonal projection B2 of Observe that b2,μ

≡ L2 0, 1r 2n−1 dr ⊗ l2 l2 L2 0, 1, r 2n−1 dr

2.28

2 has the form B2 P0 ⊗ p# . This leads to the following theorem. onto b2,μ

Theorem 2.1. The unitary operator U U1 U2 gives an isometric isomorphism of the space L2μ Bn onto l2 L2 0, 1, r 2n−1 dr ≡ L2 0, 1, r 2n−1 dr ⊗ l2 such that (1) the pluriharmonic Bergman space bμ2 Bn is mapped onto L0 ⊗ l2# , U : bμ2 Bn −→ L0 ⊗ l2# ,

2.29

where L0 is the one-dimensional subspace of L2 0, 1, r 2n−1 dr, generated by the function l0 r √ 2n; μ (2) the pluriharmonic Bergman projection QBn is unitary equivalent to μ

UQBn U−1 P0 ⊗ p# ,

2.30

where P0 is the one-dimensional projection 2.27 of L2 0, 1, r 2n−1 dr onto L0 . Introduce the operator

R0 : l2# −→ L2 0, 1, r 2n−1 dr ⊗ l2

2.31

R0 : dα,β −→ l0 r dα,β .

2.32

by the rule

The mapping R0 is an isometric embedding, and the image of R0 coincides with the space 2 . The adjoint operator b2,μ

R∗0 : L2 0, 1, r 2n−1 dr ⊗ l2 −→ l2#

2.33

is given by R∗0

: cα,β r −→

χ α, β

1

√ 2n−1 cα,β ρ 2nρ dρ ,

0

R∗0 R0 I : l2# −→ l2# ,

2 R0 R∗0 B2 : L2 0, 1, r 2n−1 dr ⊗ l2 −→ b2,μ .

2.34

8


Meanwhile the operator R R∗0 U maps the space L2μ Bn onto l2# , and its restriction 2.35

R | bμ2 Bn : bμ2 Bn −→ l2# is an isometric isomorphism. The adjoint operator R∗ U∗ R0 : l2# −→ bμ2 Bn ⊂ L2μ Bn

2.36

is isometric isomorphism of l2# onto the subspace bμ2 Bn of L2μ Bn . Remark 2.2. We have RR∗ I : l2# −→ l2# ,

μ

R∗ R QBn : L2μ Bn −→ bμ2 Bn .

2.37

Theorem 2.3. The isometric isomorphism R∗ U∗ R0 : l2# → bμ2 Bn is given by ∞ ∞ R∗ : dα,β −→ λn, |α|cα,0 r |α| eα,0 ω λ n, β c0,β r |β| e0,β ω. |α|0 |β|1

2.38

Proof. Let {dα,β } ∈ l2# , we can get

√ R∗ U1∗ U2∗ R0 : dα,β −→ U1∗ U2∗ 2ndα,β U1∗

λn, |α|cα,0 r |α| λ n, β c0,β r |β|

2.39

∞

∞ λn, |α|cα,0 r |α| eα,0 ω λ n, β c0,β r |β| e0,β ω. |α|0 |β|1

Corollary 2.4. The inverse isomorphism R : bμ2 Bn → l2# is given by √ −1 R : ϕz −→ dα,β 2n kα,β , μ

where cα,0 ϕ, e!α,0 λn, |α|dn,α dvz, |α|, |β| ∈ Z , and space bμ2 Bn ; that is,

μ {! eα,0 }∞ |α|0

μ

∪

2.40 μ

ϕzzα dvz, c0,β ϕ, e!0,β λn, |β|dn,β

Bn μ {! e0,β }∞ |β|1

e!α,0 dn,α λn, |α|zα ,

Bn

ϕzzβ

is the standard basis for the pluriharmonic Bergman

μ e!0,β dn,β λ n, β zβ .

2.41


9

3. Toeplitz Operator with Radial Symbols on bμ2 Bn μ

μ

In this section we will study the Toeplitz operators Ta QBn a : ϕ ∈ bμ2 Bn → QBn aϕ ∈ bμ2 Bn with radial symbols a ar. Theorem 3.1. Let ar be a measurable function on the segment 0, 1. Then the Toeplitz operator Ta acting on bμ2 Bn is unitary equivalent to the multication operator γa,μ I acting on l2# . The sequence γa,μ {χ α, βγa,μ |α| |β|} is given by

γa,μ m λ2 n, m

1

arr 2m2n−1 μrdr,

m ∈ Z .

3.1

0

Proof. By means of Remark 2.2, the operator Ta is unitary equivalent to the operator μ

μ

RTa R∗ RQBn aQBn R∗ RR∗ RaR∗ RR∗ RR∗ RaR∗ RR∗ RaR∗ R∗0 U2 U1 arU1−1 U2−1 R0 R∗0 U2 arU2−1 R0 R∗0 χ α, β a φ|α||β| r R0 .

3.2

Further, let {dα,β } be a sequence from l2# . By 2.21, we have R∗0 χ α, β a φ|α||β| r R0 dα,β √ R∗0 2ndα,β χ α, β a φ|α||β| r

1

χ α, β a φ|α||β| r 2ndα,β r 2n−1 dr

0

1

χ α, β dα,β 0

χ α, β dα,β λ n, |α| β

2

3.3

a y dϕ2n y |α||β|

1

a y y2|α||β|2n−1 μ y dy

0

χ α, β dα,β γa,μ |α| β . Corollary 3.2. i The Toeplitz operator Ta with measurable radial symbol ar is bounded on bμ2 Bn if and only if supm∈Z |γa,μ m| < ∞. Moreover,

Ta sup γa,μ m. m∈Z

3.4

10


ii The Toeplitz operator Ta is compact if and only if limm → ∞ γa,μ m 0. The spectrum of the bounded operator Ta is given by spTa γa,μ m : m ∈ Z ,

3.5

and its essential spectrum ess-spTa coincides with the set of all limits points of the sequence {γa,μ m}m∈Z . Let H be a Hilbert space and {ϕg }g∈G a subset of elements of H parameterized by elements g of some set G with measure dμ. Then {ϕg }g∈G is called a system of coherent states, if for all ϕ ∈ H, 2 ϕ ϕ, ϕ

ϕ, ϕg 2 dμ,

3.6

G

or equivalently, if for all ϕ1 , ϕ2 ∈ H,

ϕ1 , ϕ2

ϕ1 , ϕg

ϕ2 , ϕg dμ.

3.7

G

We define the isomorphic inclusion V : H → L2 G by the rule V : ϕ ∈ H −→ f f g ϕ, ϕg ∈ L2 G.

3.8

By 3.7 we have ϕ1 , ϕ2 f1 , f2 , where ·, · and ·, · are the scalar products on H and L2 G, respectively, and fh g fg h. Let H2 G V H ⊂ L2 G. A function f ∈ L2 G is an element of H2 G if and only if for all h ∈ G, f, fh fh. The operator P fg ϕt , ϕg ftdμt is the orthogonal projection of L2 G onto H2 G. G The function ag, g ∈ G, is called the anti-Wick or contravariant symbol of an operator T : H → H if V T V −1 | H2 G P a g P P a g I | H2 G : H2 G −→ H2 G,

3.9

or, in other terminology, the operator V AV −1 | H2 G is the Toeplitz operator Tag P a g I | H2 G : H2 G −→ H2 G

3.10

with the symbols ag. Given an operator T : H → H, introduce the Wick function T ϕh , ϕg a! g, h , ϕh , ϕg

g, h ∈ G.

3.11


11

If the operator T has an anti-Wick symbols, that is, V T V −1 Tag for some function a ag, then Ta fh , fg , a! g, h

fh , fg

g, h ∈ G.

3.12

And the operator Ta admits the following representation in terms of its Wick function:

Ta f

g

atftft g dμt G

fhdμh

G

atft g dμt G

fhfh tdμh G

atft g fh tdμt

G

3.13

fh g fhdμh atfh tfg tdμt fh , fg G G a! g, h fhfh g dμh.

G

Interchanging the integrals above, we understand them in a weak sense. The restriction of the function a!g, h onto the diagonal T ϕg , ϕg a! g a! g, g , ϕg , ϕg

g ∈ G,

3.14

is called the Wick or covariant or Berezin symbols of the operator T : H → H. The Wick and anti-Wick symbols of an operator T : H → H are connected by the Berezin transform 2 atfg t dμt

Ta fg , fg T ϕg , ϕg G . a! g a! g, g

fg , fg fg t2 dμt ϕg , ϕg G

3.15

The pluriharmonic Bergman reproducing kernel in the space bμ2 Bn has the form 2 λ2 n, 0 Rz w Kz w Kz w − dn,0

∞

μ

μ

e!α w! eα z

|α|0

∞

μ

μ

2 e!α z! eα w − dn,0 λ2 n, 0,

|α|0

3.16 where α 0 0, . . . , 0. For f ∈ bμ2 Bn , the reproducing property fz

μ QBn f

z

Bn

fwRz wμ|w|dvw

3.17

12


shows that the system of functions Rz w, w ∈ Bn , forms a system of coherent states in the space bμ2 Bn . In our context, we have G Bn , dμ μ|z|dxdy, H H2 G bμ2 Bn , L2 G L2μ Bn , ϕg fg Rg , where g z ∈ Bn . Lemma 3.3. Let Ta be the Toeplitz operator with a radial symbol a ar. Then the corresponding Wick function 3.11 has the form ⎛ a!z, w

∞

μ μ ⎝ e!α w! R−1 eα zγa,μ |α| w z |α|0

⎞

∞

μ μ e!α z! eα wγa,μ |α| |α|0

2 λ2 n, 0γa,μ 0⎠. − dn,0

3.18

Proof. By 3.11 and 3.16, we have

Ta Rw , Rz R−1 w z aRw , Rz

Rw , Rz

' & 4 4 R−1 − d , K , K λ 0 a, 1 aK z n,

aK w z w z w n,0

a!z, w

⎛

μ μ ⎝ R−1 e!α w! eα z w z |α|0

⎛

∞

⎞ ∞ & μ μ' & μ μ' μ μ 4 a! eα , e!α eα , e!α − dn,0 e!α z! λ4 n, 0 a, 1⎠ eα w a! |α|0

∞

μ μ ⎝ R−1 e!α w! eα zγa,μ |α| w z |α|0

⎞

∞

μ μ e!α z! eα wγa,μ |α| |α|0

2 λ2 n, 0γa,μ 0⎠. − dn,0

3.19 μ

μ e!α z,

Denote by Lα the one-dimensional subspace of bμ2 Bn generated by the base element μ μ |α| ∈ Z . Then the one-dimensional projection Pα of bμ2 Bn onto Lα has obviously the

form

μ Pα f

&

μ f, e!α

'

μ e!α

μ e!α z

μ

Bn

fw! eα wμ|w|dvw.

3.20

μ

In the similar method, Lα denote the one-dimensional subspace of bμ2 Bn generated by the μ

μ

μ

base element e!α z. Let Pα be the projection from bμ2 Bn onto Lα , and the projection can be rewritten as

μ Pα fz

&

μ f, e!α

'

μ e!α z

μ e!α z

μ

Bn

fw! eα wμ|w|dvw.

3.21


13

Theorem 3.4. Let Ta be a bounded Toeplitz operator having radial symbol ar. Then one can get the spectral decomposition of the operator Ta : ∞

Ta

μ

γa,μ |α|Pα

|α|0

∞

μ

μ

γa,μ |α|Pα − γa,μ 0P0 .

3.22

|α|0

Proof. According to 3.13, 3.20, 3.21, and Lemma 3.3, we get

Ta f z

a!z, wfwRw zμ|w|dvw.

Bn

⎛

⎝

Bn

∞

μ

μ

e!α w! eα zγa,μ |α|

|α|0

∞

μ

μ

e!α z! eα wγa,μ |α|

|α|0

⎞

3.23

2 −dn,0 λ2 n, 0γa,μ 0 ⎠fwμ|w|dvw.

∞

μ

γa,μ |α|Pα fz

|α|0

∞

μ

μ

γa,μ |α|Pα fz − γa,μ 0P0 fz.

|α|0

The value γa,μ |α| depends only on |α|. Collecting the terms with the same |α| and using the formula z · wm

m! zα wα α! |α|m

3.24

we obtain ( a!z, w

R−1 w z

∞

m

lm, nγa,μ m z · w w · z

m

) −

2 λ2 n, 0γa,μ 0 dn,0

,

3.25

m0

where lm, n m n − 1!/|S2n−1 |m!n − 1!λ2 n, m. The orthogonal projection of bμ2 Bn μ onto the subspace generated by all element e!α with |α| m, m ∈ Z can be written as

μ Pm f

z lm, n

Bn

fwz · wm μ|w|dvw;

3.26

fww · zm μ|w|dvw

3.27

similarly,

μ Pm f z lm, n

Bn

14


denotes the orthogonal projection from bμ2 Bn onto the subspace generated by all elements μ

e!α with |α| m. Therefore, 3.22 has the form

Ta

∞

μ

γa,μ mPm

m0

∞

μ

μ

γa,μ mPm − γa,μ 0P0 .

3.28

m0

In view of 3.25, we can get the following useful corollary. Corollary 3.5. Let Ta be a bounded Toeplitz operator having radial symbol ar. Then the Wick symbol of the operator Ta is radial as well and is given by the formula

a!z a!r

R−1 z z

2

∞

lm, nγa,μ mr

2m

−

2 dn,0 λ2 n, 0γa,μ 0

3.29

,

m0

where Rz z 2

∞

m0

2 lm, nr 2m − dn,0 λ2 n, 0.

In terms of Wick function the composition formula for Toeplitz operators is quite transparent. Corollary 3.6. Let Ta , Tb be the Toeplitz operators with the Wick function

( a!z, w

∞

R−1 w z

m

m

m

m

lm, nγa,μ m z · w w · z

) −

2 λ2 n, 0γa,μ 0 dn,0

−

2 λ2 n, 0γb,μ 0 dn,0

m0

( ! w bz,

R−1 w z

∞

lm, nγb,μ m z · w w · z

,

)

3.30

,

m0

respectively. Then the Wick function c!z, w of the composition T Ta Tb is given by

( c!z, w

R−1 w z

∞

m

m

lm, nγb,μ mγa,μ m z · w w · z

m0

) −

2 λ2 n, 0γb,μ 0γa,μ 0 dn,0

.

3.31


15

Proof. According to Lemma 3.3 and 3.25, we have

Ta Tb Rw , Rz R−1 w z Tb Rw , aRz

Rw , Rz R−1 z Tb Rw uRz ua|u|μ|u|dvu w

c!z, w

Bn

R−1 w z

Bn

⎛ ⎝ R−1 w z

Tb Rw , Ru Rz ua|u|μ|u|dvu

∞

μ

μ

e!α weα zγb,μ |α|γa,μ |α|

|α|0

∞

μ

μ

e!α zeα wγb,μ |α|γa,μ |α|

|α|0

⎞ 2 −dn,0 λ2 n, 0γb,μ 0γa,μ 0⎠

(

R−1 w z

∞

m

m

lm, nγb,μ mγa,μ m z · w w · z

m0

) −

2 λ2 n, 0γb,μ 0γa,μ 0 dn,0

.

3.32

Acknowledgment This work was supported by the National Science Foundation of China Grant no. 10971020.

References 1 Y. J. Lee, “Commuting Toeplitz operators on the pluriharmonic Bergman space,” Czechoslovak Mathematical Journal, vol. 54, no. 2, pp. 535–544, 2004. 2 Y. J. Lee and K. Zhu, “Some differential and integral equations with applications to Toeplitz operators,” Integral Equations and Operator Theory, vol. 44, no. 4, pp. 466–479, 2002. 3 K. Guo and D. Zheng, “Toeplitz algebra and Hankel algebra on the harmonic Bergman space,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 213–230, 2002. 4 S. Grudsky and N. Vasilevski, “Bergman-Toeplitz operators: radial component influence,” Integral Equations and Operator Theory, vol. 40, no. 1, pp. 16–33, 2001. 5 S. Grudsky, A. Karapetyants, and N. Vasilevski, “Toeplitz operators on the unit ball in Cn with radial symbols,” Journal of Operator Theory, vol. 49, no. 2, pp. 325–346, 2003. 6 N. L. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space, vol. 185 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2008. 7 W. Rudin, Function Theory in the Unit Ball of Cn , Classics in Mathematics, Springer, Berlin, Germany, 2008.

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