Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols

and Applied Analysis 3 where dv z 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, L2 ( S2n−1 ) ∞ ⊕ k 0 Hk, 2.3 where Hk denotes the space of spherical harmonics of order k. Meanwhile, each space Hk is the direct sum under the identification C R2n of the mutually orthogonal spaces Hp,q see, e.g., 7 : Hk ⊕ p q k p,q∈Z Hp,q, k ∈ Z , 2.4 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 ) ⊕ p,q∈Z Hp,q. 2.5 The Hardy space H2 B in the unit ball B is a closed subspace of L2 S2n−1 . Denote by PS2n−1 the Szegö orthogonal projection of L2 S2n−1 onto the Hardy spaceH2 B . It is well known that H2 B ∞ ⊕ p 0 Hp,0. The standard orthonormal base in H2 B has the form eα ω dn,αω, 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 ∈ Z , 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 Lμ B n L2 ( 0, 1 , μ r r2n−1dr ) ⊗ L2 ( S2n−1 ) . 2.7 For any function f z ∈ Lμ B have the decomposition f z ∞ ∑ |α| |β| 0 cα,β r eα,β ω , r |z|, ω z r , 2.8 4 Abstract and Applied Analysis with the coefficients cα,β r satisfying the condition ∥ ∥f ∥ ∥2 Lμ Bn ∞ ∑ |α| |β| 0 ∫1 0 ∣ ∣cα,β r ∣ ∣μ r r2n−1dr < ∞. 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 , μ r r2n−1dr ) ⊗ L2 ( S2n−1 ) −→ L2 ( 0, 1 , μ r r2n−1dr ) ⊗ l2 ≡ l2 ( L2 ( 0, 1 , μ r r2n−1dr )) , 2.10 by the rule U1 : f z → {cα,β r }, and ∥ ∥f ∥ ∥2 Lμ Bn ∥ ∥cα,β r ∥ ∥2 l2 L2 0,1 ,μ r r2n−1dr ∞ ∑ |α| |β| 0 ∥ ∥cα,β r ∥ ∥2 L2 0,1 ,μ r r2n−1dr . 2.11 Let f z be a pluriharmonic in the unit ball B and write f g h, where the functions g, h are holomorphic in B. Suppose g z ∞ ∑ |α| 0 cαz, h z ∞ ∑ |β| 0 cβz β 2.12 are their power series representations of g and h, respectively. We have f z ∞ ∑ |α| 0 cαz α ∞ ∑ |β| 0 cβz ∞ ∑ |α| 0 cα r eα ω ∞ ∑ |β| 0 cβ r eβ ω , 2.13 where cα r cαd−1 n,αr |α|, cβ r cβd−1 n,βr |β|, r |z|, ω z/r . Let b2 μ B n be the pluriharmonic Bergman space in B from Lμ B n . Denote by Q Bn the pluriharmonic Bergman orthogonal projection of Lμ B n onto the Bergman space b2 μ B n . From the above it follows that to characterize a function f z ∈ b2 μ B and considering its decomposition according to 2.13 , one can restrict to the function having the representation f z g z h z ∞ ∑ |α| 0 cα,0 r eα,0 ω ∞ ∑ |β| 0 c0,β r e0,β ω . 2.14 Abstract and Applied Analysis 5 Now let us take an arbitrary f z from b2 μ B n in the form 2.14 . It will satisfy the CauchyRiemann equations, that is,and Applied Analysis 5 Now let us take an arbitrary f z from b2 μ B n in the form 2.14 . It will satisfy the CauchyRiemann equations, that is, ∂ ∂zk g z ≡ 1 2 ( ∂ ∂xk i ∂ ∂yk ) g z 0, k 1, . . . , n, z ∈ B, ∂ ∂zk h z ≡ 1 2 ( ∂ ∂xk − i ∂ ∂yk ) h z 0, k 1, . . . , n, z ∈ B. 2.15 Applying ∂/∂zk, ∂/∂zk to g and h, respectively, we have ∂ ∂zk ∞ ∑ |α| 0 cα,0 r eα,0 ω zk 2r ∞ ∑ |α| 0 ( d dr cα,0 r − |α| r cα,0 r ) eα,0 ω , ∂ ∂zk ∞ ∑ |β| 0 c0,β r e0,β ω zk 2r ∞ ∑ |β| 0 ( d dr c0,β r − ∣ ∣β ∣ ∣ r c0,β r ) e0,β ω , 2.16 where k 0, . . . , n, andwe come to the infinite system of ordinary linear differential equations d dr cα,0 r − |α| r cα,0 r 0, |α| 0, 1, . . . d dr c0,β r − ∣ ∣β ∣ ∣ r c0,β r 0, ∣ ∣β ∣ ∣ 0, 1, . . . . 2.17 Their general solution has the form cα,0 bαr |α| λ n, |α| cα,0r |α|, c0,β bβr |β| λ n, |β| c0,βr |β|, with λ n,m ∫1 0 t 2m 2n−1μ t dt −1/2. Hence, for any f z ∈ b2 μ B we have f z ∞ ∑ |α| 0 cα,0λ n, |α| r |α|eα,0 ∞ ∑ |β| 0 c0,βλ ( n, ∣ ∣β ∣ ∣ ) r |β|e0,β. 2.18 And, it is easy to verify ‖f‖2 Lμ Bn ∑∞ |α| 0 |cα,0| ∑∞ |β| 0 |c0,β|. Thus the image b2 1,μ B U1 b2 μ B n is characterized as the closed subspace of L2 ( 0, 1 , μ r r2n−1dr ) ⊗ l2 l2 ( L2 ( 0, 1 , μ r r2n−1dr )) 2.19 which consists of all sequences {cα,β r } of the form cα,β r ⎧ ⎪ ⎪⎨ ⎪ ⎪⎩ λ n, |α| cα,0r |α|, ∣ ∣β ∣ ∣ 0 λ ( n, ∣ ∣β ∣ ∣ ) c0,βr |β|, |α| 0 0, |α|/ 0, ∣ ∣β ∣ ∣/ 0. 2.20 6 Abstract and Applied Analysis For each m ∈ Z introduce the function φm ( ρ ) λ n,m 1/n (∫ρ 0 r2m 2n−1μ r dr )1/2n , ρ ∈ 0, 1 . 2.21 Obviously, there exists the inverse function for the function φm ρ on 0, 1 , which we will denote by φm r . Introduce the operator ( umf ) r √ 2n λ n,m φ−m m r f ( φm r ) . 2.22 By Proposition 2.1 in 5 , the operator um maps unitary L2 0, 1 , μ r r2n−1 onto L2 0, 1 , r2n−1dr in such a way that um λ n,m r √ 2n, m ∈ Z . 2.23 Intoduce the unitary operator U2 : l2 ( L2 ( 0, 1 , μ r r2n−1dr )) −→ l2 ( L2 ( 0, 1 , r2n−1dr )) , 2.24 where U2 : { cα,β r } −→ u|α| |β|cα,β ) r } . 2.25 By 2.23 , we can get the space b2 2,μ U2 b 2 1,μ coincides with the space of all sequences {kα,β} for which kα,β ⎧ ⎪ ⎪⎨ ⎪ ⎪⎩ √ 2ncα,0, ∣ ∣β ∣ ∣ 0 √ 2nc0,β, |α| 0 0, |α|/ 0, ∣ ∣β ∣ ∣/ 0. 2.26 Let l0 r √ 2n. We have l0 r ∈ L2 0, 1 , r2n−1dr and ‖l0‖L2 0,1 ,r2n−1 1. Denote by L0 the one-dimensional subspace of L2 0, 1 , r2n−1dr generated by l0 r . The orthogonal projection P0 of L2 0, 1 , r2n−1dr onto L0 has the form ( P0f ) r 〈 f, l0 〉 l0 √ 2n ∫1 0 f ( ρ )√ 2nρ2n−1dρ. 2.27 Let dα,β kα,β √ 2n −1. Denote by l# 2 the subspace of l2 consisting of all sequences {dα,β}. And let p# be the orthogonal projections of l2 onto l# 2, then p # χ α, β I, where χ α, β 0, if |α‖β| > 0 and χ α, β 1, if |α‖β| 0. Abstract and Applied Analysis 7 Observe that b2 2,μ L0 ⊗ l# 2 and the orthogonal projection B2 of l2 ( L2 ( 0, 1 , r2n−1dr )) ≡ L2 ( 0, 1 r2n−1dr ) ⊗ l2 2.28and Applied Analysis 7 Observe that b2 2,μ L0 ⊗ l# 2 and the orthogonal projection B2 of l2 ( L2 ( 0, 1 , r2n−1dr )) ≡ L2 ( 0, 1 r2n−1dr ) ⊗ l2 2.28 onto b2 2,μ has the form B2 P0 ⊗ p#. This leads to the following theorem. Theorem 2.1. The unitary operator U U1U2 gives an isometric isomorphism of the space Lμ B n onto l2 L2 0, 1 , r2n−1dr ≡ L2 0, 1 , r2n−1dr ⊗ l2 such that (1) the pluriharmonic Bergman space b2 μ B n is mapped onto L0 ⊗ l# 2, U : b2 μ B n −→ L0 ⊗ l# 2, 2.29 where L0 is the one-dimensional subspace of L2 0, 1 , r2n−1dr , generated by the function l0 r √ 2n; (2) the pluriharmonic Bergman projection Q Bn is unitary equivalent to UQ μ Bn U−1 P0 ⊗ p#, 2.30 where P0 is the one-dimensional projection 2.27 of L2 0, 1 , r2n−1dr onto L0. Introduce the operator R0 : l# 2 −→ L2 ( 0, 1 , r2n−1dr ) ⊗ l2 2.31 by the rule R0 : { dα,β } −→ l0 r { dα,β } . 2.32 The mapping R0 is an isometric embedding, and the image of R0 coincides with the space b2 2,μ. The adjoint operator R0 : L2 ( 0, 1 , r2n−1dr ) ⊗ l2 −→ l# 2 2.33


Introduction
Let B n be the open unit ball in the complex vector space C n .For any z z 1 , . . ., z n and ξ ξ 1 , . . ., ξ n in C n , let z•ξ n j 1 z j ξ j , where ξ j is the complex conjugate of ξ j and |z| √ z • z.For a multi-index α α 1 , . . ., α n and z z 1 , . . ., z n ∈ C n , we write The weighted pluriharmonic Bergman space b 2 μ B n is the subspace of the weighted space L 2 μ B n consisting of all pluriharmonic functions on B n .A pluriharmonic function in the unit ball is the sum of a holomorphic function and the conjugate of a holomorphic functions.It is known that b 2 μ B n is a closed subspace of L 2 μ B n and hence is a Hilbert space.Let Q μ B n be the Hilbert space orthogonal projection from L 2 μ B n onto b 2 μ B n .For a function u ∈ L 2 μ B n , the Toeplitz operator T u : b 2 μ B n → b 2 μ B n with symbol u is the linear operator defined by T u is densely defined and not bounded in general.

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 C n .
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 b 2 μ B n is an isometric isomorphism between b 2 μ B n and l # 2 , and where l # 2 is the subspace of l 2 .Using the operator R we prove that each Toeplitz operator T a with radial symbols is unitary to the multication operator γ a,μ I acting on l # 2 .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.

Pluriharmonic Bergman Space and Orthogonal Projection
We start this section with a decomposition of the space L 2 μ B n .Consider a nonnegative measurable function μ r , r ∈ 0, 1 , such that mes{r ∈ 0, 1 : μ r > 0} 1, and where is the surface area of unit sphere S 2n−1 and Γ z is the Gamma function.
Introduce the weighted space where dv z is the usual Lebesgue volume measure and L 2 S 2n−1 is the space with the usual Lebesgue surface measure.The space L 2 S 2n−1 is the direct sum of mutually orthogonal spaces H k , that is, where H k denotes the space of spherical harmonics of order k.Meanwhile, each space H k is the direct sum under the identification C n R 2n of the mutually orthogonal spaces H p,q see, e.g., 7 : where H p,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 z 1 , . . ., z n .Thus, we can get The Hardy space H 2 B n in the unit ball B n is a closed subspace of L 2 S 2n−1 .Denote by P S 2n−1 the Szeg ö orthogonal projection of L 2 S 2n−1 onto the Hardy space H 2 B n .It is well H p,0 .The standard orthonormal base in H 2 B n has the form for |α| 0, 1, . . . .

2.11
Let f z be a pluriharmonic in the unit ball B n and write f g h, where the functions g, h are holomorphic in B n .Suppose are their power series representations of g and h, respectively.We have where the pluriharmonic Bergman orthogonal projection of L 2 μ B n onto the Bergman space b 2 μ B n .From the above it follows that to characterize a function f z ∈ b 2 μ B n and considering its decomposition according to 2.13 , one can restrict to the function having the representation

2.14
Now let us take an arbitrary f z from b 2 μ B n in the form 2.14 .It will satisfy the Cauchy-Riemann equations, that is,

2.15
Applying ∂/∂z k , ∂/∂z k to g and h, respectively, we have

2.16
where k 0, . . ., n, and we come to the infinite system of ordinary linear differential equations

2.18
And, it is easy to verify f 2 which consists of all sequences {c α,β r } of the form

6 Abstract and Applied Analysis
For each m ∈ Z introduce the function , ρ ∈ 0, 1 .

2.21
Obviously, there exists the inverse function for the function ϕ m ρ on 0, 1 , which we will denote by φ m r .Introduce the operator By Proposition 2.1 in 5 , the operator u m maps unitary L 2 0, 1 , μ r r 2n−1 onto L 2 0, 1 , r 2n−1 dr in such a way that Intoduce the unitary operator
Theorem 2.1.The unitary operator U U 1 U 2 gives an isometric isomorphism of the space where L 0 is the one-dimensional subspace of L 2 0, 1 , r 2n−1 dr , generated by the function l 0 r √ 2n; (2) the pluriharmonic Bergman projection where P 0 is the one-dimensional projection 2.27 of L 2 0, 1 , r 2n−1 dr onto L 0 .
Introduce the operator by the rule

2.32
The mapping R 0 is an isometric embedding, and the image of R 0 coincides with the space b 2 2,μ .The adjoint operator is given by

Toeplitz Operator with Radial Symbols on b 2 μ B n
In this section we will study the Toeplitz operators Proof.By means of Remark 2.2, the operator T a is unitary equivalent to the operator

3.2
Further, let {d α,β } be a sequence from l # 2 .By 2.21 , we have and its essential spectrum ess-spT a coincides with the set of all limits points of the sequence {γ a,μ m } m∈Z .
Let H be a Hilbert space and {ϕ g } g∈G a subset of elements of H parameterized by elements g of some set G with measure dμ.
Then {ϕ g } g∈G is called a system of coherent states, if for all ϕ ∈ H, We define the isomorphic inclusion V : H → L 2 G by the rule By 3.7 we have ϕ 1 , ϕ 2 f 1 , f 2 , where •, • and •, • are the scalar products on H and L 2 G , respectively, and The function a g , g ∈ G, is called the anti-Wick or contravariant symbol of an operator T : or, in other terminology, the operator with the symbols a g .Given an operator T : H → H, introduce the Wick function a g, h Tϕ h , ϕ g ϕ h , ϕ g , g,h ∈ G.

3.11
If the operator T has an anti-Wick symbols, that is, V TV −1 T a g for some function a a g , then

3.12
And the operator T a admits the following representation in terms of its Wick function:

3.13
Interchanging the integrals above, we understand them in a weak sense.The pluriharmonic Bergman reproducing kernel in the space b 2 μ B n has the form  Abstract and Applied Analysis shows that the system of functions R z w , w ∈ B n , forms a system of coherent states in the space b 2 μ B n .In our context, we have G

3.18
Proof.By 3.11 and 3.16 , we have

3.23
The value γ a,μ |α| depends only on |α|.Collecting the terms with the same |α| and using the formula
. ., 0 .For f ∈ b 2 μ B n , the reproducing property f z Q μ B n f z B nf w R z w μ |w| dv w 3.17 12

Lemma 3 . 3 .
where g z ∈ B n .Let T a be the Toeplitz operator with a radial symbol a a r .Then the corresponding Wick function 3.11 has the form a,μ |α| − d 2 n,0 λ 2 n, 0 γ a,μ 0 ⎞ ⎠ .

α 21 Theorem 3 . 4 .
the one-dimensional subspace of b 2 μ B n generated by the base element e μ α z , |α| ∈ Z .Then the one-dimensional projection P μ α of b 2 μ B n onto L |w| dv w .3.Let T a be a bounded Toeplitz operator having radial symbol a r .Then one can get the spectral decomposition of the operator T a : According to 3.13 , 3.20 , 3.21 , and Lemma 3.3, we get T a f z B n a z, w f w R w z μ |w| dv w .a,μ |α| −d 2 n,0 λ 2 n, 0 γ a,μ 0 ⎞ ⎠ f w μ |w| dv w .

Corollary 3 . 5 .Corollary 3 . 6 .
m .The orthogonal projection of b 2 μ B n onto the subspace generated by all element e μ α with |α| m, m ∈ Z can be written as • z m μ |w| dv w 3.27 denotes the orthogonal projection from b 2 μ B n onto the subspace generated by all elements e μ α with |α| m.Therefore, 3.22 has the form 3.25 , we can get the following useful corollary.Let T a be a bounded Toeplitz operator having radial symbol a r .Then the Wick symbol of the operator T a is radial as well and is given by the formula n γ a,μ m r 2m − d 2 n,0 λ 2 n, 0 γ a,μ 0 , n r 2m − d 2 n,0 λ 2 n, 0 .In terms of Wick function the composition formula for Toeplitz operators is quite transparent.Let T a , T b be the Toeplitz operators with the Wick functiona n γ a,μ m z • w m w • z m − d 2 n,0 λ 2 n, 0 γ a,μ 0 , n γ b,μ m z • w m w • z m − d 2 n,0 λ 2 n, 0 γ b,μ 0 ,3.30respectively.Then the Wick function c z, w of the composition T T a T b is given by c z, w R −1 w z ∞ m 0 l m, n γ b,μ m γ a,μ m z • w m w • z m − d 2 n,0 λ 2 n, 0 γ b,μ 0 γ a,μ 0 .
The Toeplitz operator T a with measurable radial symbol a r is bounded on b 2 μ B n if and only if sup m∈Z |γ a,μ m | < ∞.Moreover, The Toeplitz operator T a is compact if and only if lim m → ∞ γ a,μ m 0. The spectrum of the bounded operator T a is given by