# Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols

Authors: Zhi Ling Sun; Yu Feng Lu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We construct an operatorR 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,μI. Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about
the operator, providing its spectral decomposition.


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## Body

## 1. Introduction

Let𝔹n 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=∑j=1nzjξj¯, 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=z1α1⋯znαn, where αk∈Z+=ℕ∪{0}, and |α|=α1+⋯+αn is its length, α!=α1!⋯αn!.The weighted pluriharmonic Bergman spacebμ2(𝔹n) is the subspace of the weighted space Lμ2(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. It is known that bμ2(𝔹n) is a closed subspace of Lμ2(𝔹n) and hence is a Hilbert space. Let Q𝔹nμ be the Hilbert space orthogonal projection from Lμ2(𝔹n) onto bμ2(𝔹n). For a function u∈Lμ2(𝔹n), the Toeplitz operator Tu:bμ2(𝔹n)→bμ2(𝔹n) with symbol u is the linear operator defined by
(1.1)Tuf=Q𝔹nμ(uf),f∈bμ2(𝔹n).Tu is densely defined and not bounded in general.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
(1.2)RR*=I:l2#→l2#,R*R=Q𝔹nμ:Lμ2(𝔹n)→bμ2(𝔹n);
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,μI 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 spaceLμ2(𝔹n). Consider a nonnegative measurable function μ(r),r∈(0,1), such that mes{r∈(0,1):μ(r)>0}=1, and
(2.1)∫𝔹nμ(|z|)dv(z)=|S2n-1|∫01μ(r)r2n-1dr<∞,
where |S2n-1|=2πn-(1/2)Γ-1(n-(1/2)) is the surface area of unit sphere S2n-1 and Γ(z) is the Gamma function.Introduce the weighted space(2.2)Lμ2(𝔹n)={f:∥f∥Lμ2(𝔹n)2=∫𝔹n|f(z)|2μ(|z|)dv(z)<∞},
where dv(z) is the usual Lebesgue volume measure and L2(S2n-1) is the space with the usual Lebesgue surface measure.The spaceL2(S2n-1) is the direct sum of mutually orthogonal spaces ℋk, that is,
(2.3)L2(S2n-1)=⨁k=0∞ℋk,
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]):
(2.4)ℋk=⨁p+q=kp,q∈ℤ+Hp,q,k∈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¯=(z¯1,…,z¯n). Thus, we can get
(2.5)L2(S2n-1)=⨁p,q∈ℤ+Hp,q.The Hardy spaceH2(𝔹n) in the unit ball 𝔹n is a closed subspace of L2(S2n-1). Denote by PS2n-1 the Szegö orthogonal projection of L2(S2n-1) onto the Hardy space H2(𝔹n). It is well known that H2(𝔹n)=⨁p=0∞Hp,0. The standard orthonormal base in H2(𝔹n) has the form
(2.6)eα(ω)=dn,αωα,dn,α=(n-1+|α|)!|S2n-1|(n-1)!α!for|α|=0,1,….Fix an orthonormal basis{eα,β(ω)}α,β, α,β∈ℤ+n, 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(2.7)Lμ2(𝔹n)=L2((0,1),μ(r)r2n-1dr)⊗L2(S2n-1).
For any function f(z)∈Lμ2(𝔹n) have the decomposition
(2.8)f(z)=∑|α|+|β|=0∞cα,β(r)eα,β(ω),r=|z|,ω=zr,
with the coefficients cα,β(r) satisfying the condition (2.9)∥f∥Lμ2(𝔹n)2=∑|α|+|β|=0∞∫01|cα,β(r)|2μ(r)r2n-1dr<∞.According to the decomposition (2.7), (2.8) together with Parseval’s equality, we can define the unitary operator (2.10)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)),by the ruleU1:f(z)→{cα,β(r)}, and
(2.11)∥f∥Lμ2(𝔹n)2=∥cα,β(r)∥l2(L2((0,1),μ(r)r2n-1dr))2=∑|α|+|β|=0∞∥cα,β(r)∥L2((0,1),μ(r)r2n-1dr)2.Letf(z) be a pluriharmonic in the unit ball 𝔹n and write f=g+h¯, where the functions g, h are holomorphic in 𝔹n. Suppose
(2.12)g(z)=∑|α|=0∞cαzα,h(z)=∑|β|=0∞cβzβare their power series representations ofg and h, respectively. We have
(2.13)f(z)=∑|α|=0∞cαzα+∑|β|=0∞cβzβ¯=∑|α|=0∞cα(r)eα(ω)+∑|β|=0∞cβ(r)eβ(ω)¯,
where cα(r)=cαdn,α-1r|α|, cβ(r)=cβdn,β-1r|β|, r=|z|, ω=(z/r).Letbμ2(𝔹n) be the pluriharmonic Bergman space in 𝔹n from Lμ2(𝔹n). Denote by Q𝔹nμ the pluriharmonic Bergman orthogonal projection of Lμ2(𝔹n) onto the Bergman space bμ2(𝔹n). From the above it follows that to characterize a function f(z)∈bμ2(𝔹n) and considering its decomposition according to (2.13), one can restrict to the function having the representation
(2.14)f(z)=g(z)+h(z)¯=∑|α|=0∞cα,0(r)eα,0(ω)+∑|β|=0∞c0,β(r)e0,β(ω).
Now let us take an arbitrary f(z) from bμ2(𝔹n) in the form (2.14). It will satisfy the Cauchy-Riemann equations, that is,
(2.15)∂∂z¯kg(z)≡12(∂∂xk+i∂∂yk)g(z)=0,k=1,…,n,z∈𝔹n,∂∂zkh¯(z)≡12(∂∂xk-i∂∂yk)h¯(z)=0,k=1,…,n,z∈𝔹n.Applying∂/∂z¯k, ∂/∂zk to g and h¯, respectively, we have(2.16)∂∂z¯k∑|α|=0∞cα,0(r)eα,0(ω)=zk2r∑|α|=0∞(ddrcα,0(r)-|α|rcα,0(r))eα,0(ω),∂∂zk∑|β|=0∞c0,β(r)e0,β(ω)=z¯k2r∑|β|=0∞(ddrc0,β(r)-|β|rc0,β(r))e0,β(ω),wherek=0,…,n, and we come to the infinite system of ordinary linear differential equations
(2.17)ddrcα,0(r)-|α|rcα,0(r)=0,|α|=0,1,…ddrc0,β(r)-|β|rc0,β(r)=0,|β|=0,1,….Their general solution has the formcα,0=bαr|α|=λ(n,|α|)cα,0r|α|, c0,β=bβr|β|=λ(n,|β|)c0,βr|β|, with λ(n,m)=(∫01t2m+2n-1μ(t)dt)-1/2. Hence, for any f(z)∈bμ2(𝔹n) we have
(2.18)f(z)=∑|α|=0∞cα,0λ(n,|α|)r|α|eα,0+∑|β|=0∞c0,βλ(n,|β|)r|β|e0,β.And, it is easy to verify∥f∥Lμ2(𝔹n)2=∑|α|=0∞|cα,0|2+∑|β|=0∞|c0,β|2. Thus the image b1,μ2(𝔹n)=U1(bμ2(𝔹n)) is characterized as the closed subspace of
(2.19)L2((0,1),μ(r)r2n-1dr)⊗l2=l2(L2((0,1),μ(r)r2n-1dr))
which consists of all sequences {cα,β(r)} of the form
(2.20)cα,β(r)={λ(n,|α|)cα,0r|α|,|β|=0λ(n,|β|)c0,βr|β|,|α|=00,|α|≠0,|β|≠0.For eachm∈ℤ+ introduce the function
(2.21)φm(ρ)=λ(n,m)1/n(∫0ρr2m+2n-1μ(r)dr)1/2n,ρ∈[0,1].
Obviously, there exists the inverse function for the function φm(ρ) on [0,1], which we will denote by ϕm(r). Introduce the operator
(2.22)(umf)(r)=2nλ(n,m)ϕm-m(r)f(ϕm(r)).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
(2.23)um(λ(n,m)rm)=2n,m∈ℤ+.Intoduce the unitary operator(2.24)U2:l2(L2((0,1),μ(r)r2n-1dr))→l2(L2((0,1),r2n-1dr)),
where
(2.25)U2:{cα,β(r)}→{(u|α|+|β|cα,β)(r)}.By (2.23), we can get  the space b2,μ2=U2(b1,μ2) coincides with the space of all sequences {kα,β} for which(2.26)kα,β={2ncα,0,|β|=02nc0,β,|α|=00,|α|≠0,|β|≠0.Letl0(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
(2.27)(P0f)(r)=〈f,l0〉l0=2n∫01f(ρ)2nρ2n-1dρ.
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,β)I, where χ+(α,β)=0, if |α∥β|>0 and χ+(α,β)=1, if |α∥β|=0.Observe thatb2,μ2=L0⊗l2# and the orthogonal projection B2 of
(2.28)l2(L2((0,1),r2n-1dr))≡L2((0,1)r2n-1dr)⊗l2ontob2,μ2 has the form B2=P0⊗p#. This leads to the following theorem.Theorem 2.1.
The unitary operatorU=U1U2 gives an isometric isomorphism of the space Lμ2(𝔹n) onto l2(L2((0,1),r2n-1dr))≡L2((0,1),r2n-1dr)⊗l2 such that
(1) the pluriharmonic Bergman spacebμ2(𝔹n) is mapped onto L0⊗l2#,(2.29)U:bμ2(𝔹n)→L0⊗l2#,
where L0 is the one-dimensional subspace of L2((0,1),r2n-1dr), generated by the function l0(r)=2n;
(2) the pluriharmonic Bergman projectionQ𝔹nμ is unitary equivalent to(2.30)UQ𝔹nμU-1=P0⊗p#,
where P0 is the one-dimensional projection (2.27) of L2((0,1),r2n-1dr) onto L0.Introduce the operator(2.31)R0:l2#→L2((0,1),r2n-1dr)⊗l2by the rule(2.32)R0:{dα,β}→l0(r){dα,β}.The mappingR0 is an isometric embedding, and the image of R0 coincides with the space b2,μ2. The adjoint operator
(2.33)R0*:L2((0,1),r2n-1dr)⊗l2→l2#is given by(2.34)R0*:{cα,β(r)}→{χ+(α,β)∫01cα,β(ρ)2nρ2n-1dρ},R0*R0=I:l2#→l2#,R0R0*=B2:L2((0,1),r2n-1dr)⊗l2→b2,μ2.Meanwhile the operatorR=R0*U maps the space Lμ2(𝔹n) onto l2#, and its restriction (2.35)R∣bμ2(𝔹n):bμ2(𝔹n)→l2#is an isometric isomorphism. The adjoint operator(2.36)R*=U*R0:l2#→bμ2(𝔹n)⊂Lμ2(𝔹n)is isometric isomorphism ofl2# onto the subspace bμ2(𝔹n) of Lμ2(𝔹n).Remark 2.2.
We have(2.37)RR*=I:l2#→l2#,R*R=Q𝔹nμ:Lμ2(𝔹n)→bμ2(𝔹n).Theorem 2.3.
The isometric isomorphismR*=U*R0:l2#→bμ2(𝔹n) is given by
(2.38)R*:{dα,β}↦∑|α|=0∞λ(n,|α|)cα,0r|α|eα,0(ω)+∑|β|=1∞λ(n,|β|)c0,βr|β|e0,β(ω).Proof.
Let{dα,β}∈l2#, we can get
(2.39)R*=U1*U2*R0:{dα,β}↦U1*U2*({2ndα,β})=U1*({λ(n,|α|)cα,0r|α|}+{λ(n,|β|)c0,βr|β|})=∑|α|=0∞λ(n,|α|)cα,0r|α|eα,0(ω)+∑|β|=1∞λ(n,|β|)c0,βr|β|e0,β(ω).Corollary 2.4.
The inverse isomorphismR:bμ2(𝔹n)→l2# is given by
(2.40)R:φ(z)↦{dα,β}={(2n)-1kα,β},
where cα,0=〈φ,ẽα,0μ〉=λ(n,|α|)dn,α∫𝔹nφ(z)z¯αdv(z), c0,β=〈φ,ẽ0,βμ〉=λ(n,|β|)dn,β∫𝔹nφ(z)zβdv(z), |α|,|β|∈ℤ+, and {ẽα,0μ}|α|=0∞∪{ẽ0,βμ}|β|=1∞ is the standard basis for the pluriharmonic Bergman space bμ2(𝔹n); that is,
(2.41)ẽα,0μ=dn,αλ(n,|α|)zα,ẽ0,βμ=dn,βλ(n,|β|)z¯β.


## 3. Toeplitz Operator with Radial Symbols onbμ2(𝔹n)

In this section we will study the Toeplitz operatorsTa=Q𝔹nμa:φ∈bμ2(𝔹n)↦Q𝔹nμaφ∈bμ2(𝔹n) with radial symbols a=a(r).Theorem 3.1.
Leta(r) 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,μI acting on l2#. The sequence γa,μ={χ+(α,β)γa,μ(|α|+|β|)} is given by
(3.1)γa,μ(m)=λ2(n,m)∫01a(r)r2m+2n-1μ(r)dr,m∈Z+.Proof.
By means of Remark2.2, the operator Ta is unitary equivalent to the operator
(3.2)RTaR*=RQ𝔹nμaQ𝔹nμR*=R(R*R)a(R*R)R*=(RR*)RaR*RR*=RaR*=R0*U2U1a(r)U1-1U2-1R0=R0*U2a(r)U2-1R0=R0*{χ+(α,β)a(ϕ|α|+|β|(r))}R0.
Further, let {dα,β} be a sequence from l2#. By (2.21), we have
(3.3)R0*{χ+(α,β)a(ϕ|α|+|β|(r))}R0{dα,β}=R0*{2ndα,βχ+(α,β)a(ϕ|α|+|β|(r))}={∫01χ+(α,β)a(ϕ|α|+|β|(r))2ndα,βr2n-1dr}={χ+(α,β)dα,β∫01a(y)dφ|α|+|β|2n(y)}={χ+(α,β)dα,βλ2(n,|α|+|β|)∫01a(y)y2(|α|+|β|)+2n-1μ(y)dy}={χ+(α,β)dα,βγa,μ(|α|+|β|)}.Corollary 3.2.
(i) The Toeplitz operatorTa with measurable radial symbol a(r) is bounded on bμ2(𝔹n) if and only if supm∈ℤ+|γa,μ(m)|<∞. Moreover,(3.4)∥Ta∥=supm∈ℤ+|γa,μ(m)|.
(ii) The Toeplitz operatorTa is compact if and only if limm→∞γa,μ(m)=0. The spectrum of the bounded operator Ta is given by(3.5)spTa={γa,μ(m):m∈ℤ+}¯,
and its essential spectrum ess-spTa coincides with the set of all limits points of the sequence {γa,μ(m)}m∈ℤ+.LetH 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,
(3.6)∥φ∥2=(φ,φ)=∫G|(φ,φg)|2dμ,or equivalently, if for allφ1,φ2∈H,
(3.7)(φ1,φ2)=∫G(φ1,φg)(φ2,φg)¯dμ.We define the isomorphic inclusionV:H→L2(G) by the rule (3.8)V:φ∈H↦f=f(g)=(φ,φg)∈L2(G).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〉=f(h). The operator (Pf)(g)=∫G(φt,φg)f(t)dμ(t) is the orthogonal projection of L2(G) onto H2(G).The functiona(g), g∈G, is called the anti-Wick (or contravariant) symbol of an operator T:H→H if
(3.9)VTV-1∣H2(G)=Pa(g)P=Pa(g)I∣H2(G):H2(G)→H2(G),or, in other terminology, the operatorVAV-1∣H2(G) is the Toeplitz operator
(3.10)Ta(g)=Pa(g)I∣H2(G):H2(G)→H2(G)
with the symbols a(g).Given an operatorT:H→H, introduce the (Wick) function
(3.11)ã(g,h)=(Tφh,φg)(φh,φg),g,h∈G.If the operatorT has an anti-Wick symbols, that is, VTV-1=Ta(g) for some function a=a(g), then
(3.12)ã(g,h)=〈Tafh,fg〉〈fh,fg〉,g,h∈G.And the operatorTa admits the following representation in terms of its Wick function:
(3.13)(Taf)(g)=∫Ga(t)f(t)ft(g)dμ(t)=∫Ga(t)ft(g)dμ(t)∫Gf(h)fh(t)dμ(h)=∫Gf(h)dμ(h)∫Ga(t)ft(g)fh(t)dμ(t)=∫Gf(h)dμ(h)fh(g)〈fh,fg〉∫Ga(t)fh(t)fg(t)¯dμ(t)=∫Gã(g,h)f(h)fh(g)dμ(h).
Interchanging the integrals above, we understand them in a weak sense.The restriction of the functionã(g,h) onto the diagonal (3.14)ã(g)=ã(g,g)=(Tφg,φg)(φg,φg),g∈G,is called the Wick (or covariant or Berezin) symbols of the operatorT:H→H.The Wick and anti-Wick symbols of an operatorT:H→H are connected by the Berezin transform
(3.15)ã(g)=ã(g,g)=(Tφg,φg)(φg,φg)=〈Tafg,fg〉〈fg,fg〉=∫Ga(t)|fg(t)|2dμ(t)∫G|fg(t)|2dμ(t).The pluriharmonic Bergman reproducing kernel in the spacebμ2(𝔹n) has the form (3.16)Rz(w)=Kz(w)+Kz(w)¯-dn,02λ2(n,0)=∑|α|=0∞ẽαμ(w)ẽαμ(z)¯+∑|α|=0∞ẽαμ(z)ẽαμ(w)¯-dn,02λ2(n,0),
where α=0=(0,…,0). For f∈bμ2(𝔹n), the reproducing property
(3.17)f(z)=(Q𝔹nμf)(z)=∫𝔹nf(w)Rz(w)¯μ(|w|)dv(w)shows that the system of functionsRz(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)=Lμ2(𝔹n), φg=fg=Rg, where g=z∈𝔹n.Lemma 3.3.
LetTa be the Toeplitz operator with a radial symbol a=a(r). Then the corresponding Wick function (3.11) has the form
(3.18)ã(z,w)=Rw-1(z)(∑|α|=0∞ẽαμ(w)¯ẽαμ(z)γa,μ(|α|)+∑|α|=0∞ẽαμ(z)¯ẽαμ(w)γa,μ(|α|)-dn,02λ2(n,0)γa,μ(0)).Proof.
By (3.11) and (3.16), we have
(3.19)ã(z,w)=〈TaRw,Rz〉〈Rw,Rz〉=Rw-1(z)〈aRw,Rz〉=Rw-1(z)(〈aKw,Kz〉+〈aKw¯,Kz¯〉-dn,04λ4(n,0)〈a,1〉)=Rw-1(z)(∑|α|=0∞ẽαμ(w)¯ẽαμ(z)〈aẽαμ,ẽαμ〉+∑|α|=0∞ẽαμ(z)¯ẽαμ(w)〈aẽαμ,ẽαμ〉-dn,04λ4(n,0)〈a,1〉)=Rw-1(z)(∑|α|=0∞ẽαμ(w)¯ẽαμ(z)γa,μ(|α|)+∑|α|=0∞ẽαμ(z)¯ẽαμ(w)γa,μ(|α|)-dn,02λ2(n,0)γa,μ(0)).Denote byLαμ the one-dimensional subspace of bμ2(𝔹n) generated by the base element ẽαμ(z), |α|∈ℤ+. Then the one-dimensional projection Pαμ of bμ2(𝔹n) onto Lαμ has obviously the form (3.20)Pαμf=〈f,ẽαμ〉ẽαμ=ẽαμ(z)∫𝔹nf(w)ẽαμ(w)¯μ(|w|)dv(w).
In the similar method, Lαμ¯ denote the one-dimensional subspace of bμ2(𝔹n) generated by the base element ẽαμ(z)¯. Let Pαμ¯ be the projection from bμ2(𝔹n) onto Lαμ¯, and the projection can be rewritten as (3.21)Pαμ¯f(z)=〈f,ẽαμ¯〉ẽαμ(z)¯=ẽαμ(z)¯∫𝔹nf(w)ẽαμ(w)μ(|w|)dv(w).Theorem 3.4.
LetTa be a bounded Toeplitz operator having radial symbol a(r). Then one can get the spectral decomposition of the operator Ta:
(3.22)Ta=∑|α|=0∞γa,μ(|α|)Pαμ+∑|α|=0∞γa,μ(|α|)Pαμ¯-γa,μ(0)P0μ.Proof.
According to (3.13), (3.20), (3.21), and Lemma 3.3, we get
(3.23)(Taf)(z)=∫𝔹nã(z,w)f(w)Rw(z)μ(|w|)dv(w).=∫𝔹n(∑|α|=0∞ẽαμ(w)¯ẽαμ(z)γa,μ(|α|)+∑|α|=0∞ẽαμ(z)¯ẽαμ(w)γa,μ(|α|)-dn,02λ2(n,0)γa,μ(0)∑|α|=0∞ẽαμ(w)¯ẽαμ)f(w)μ(|w|)dv(w).=∑|α|=0∞γa,μ(|α|)Pαμf(z)+∑|α|=0∞γa,μ(|α|)Pαμ¯f(z)-γa,μ(0)P0μf(z).The valueγa,μ(|α|) depends only on |α|. Collecting the terms with the same |α| and using the formula (3.24)(z·w¯)m=∑|α|=mm!α!zαw¯αwe obtain(3.25)ã(z,w)=Rw-1(z)[∑m=0∞l(m,n)γa,μ(m)((z·w¯)m+(w·z¯)m)-dn,02λ2(n,0)γa,μ(0)],
where (l(m,n)=(m+n-1)!/|S2n-1|m!(n-1)!)λ2(n,m). The orthogonal projection of bμ2(𝔹n) onto the subspace generated by all element ẽαμ with |α|=m, m∈ℤ+ can be written as (3.26)(P(m)μf)(z)=l(m,n)∫𝔹nf(w)(z·w¯)mμ(|w|)dv(w);similarly,(3.27)(P(m)μ¯f)(z)=l(m,n)∫𝔹nf(w)(w·z¯)mμ(|w|)dv(w)denotes the orthogonal projection frombμ2(𝔹n) onto the subspace generated by all elements ẽαμ¯ with |α|=m. Therefore, (3.22) has the form
(3.28)Ta=∑m=0∞γa,μ(m)P(m)μ+∑m=0∞γa,μ(m)P(m)μ¯-γa,μ(0)P0μ.In view of (3.25), we can get the following useful corollary.Corollary 3.5.
LetTa be a bounded Toeplitz operator having radial symbol a(r). Then the Wick symbol of the operator Ta is radial as well and is given by the formula
(3.29)ã(z)=ã(r)=Rz-1(z)(2∑m=0∞l(m,n)γa,μ(m)r2m-dn,02λ2(n,0)γa,μ(0)),
where Rz(z)=2∑m=0∞l(m,n)r2m-dn,02λ2(n,0).In terms of Wick function the composition formula for Toeplitz operators is quite transparent.Corollary 3.6.
LetTa, Tb be the Toeplitz operators with the Wick function
(3.30)ã(z,w)=Rw-1(z)[∑m=0∞l(m,n)γa,μ(m)((z·w¯)m+(w·z¯)m)-dn,02λ2(n,0)γa,μ(0)],b̃(z,w)=Rw-1(z)[∑m=0∞l(m,n)γb,μ(m)((z·w¯)m+(w·z¯)m)-dn,02λ2(n,0)γb,μ(0)],
respectively. Then the Wick function c̃(z,w) of the composition T=TaTb is given by
(3.31)c̃(z,w)=Rw-1(z)[∑m=0∞l(m,n)γb,μ(m)γa,μ(m)((z·w¯)m+(w·z¯)m)-dn,02λ2(n,0)γb,μ(0)γa,μ(0)].Proof.
According to Lemma3.3 and (3.25), we have
(3.32)c̃(z,w)=〈TaTbRw,Rz〉〈Rw,Rz〉=Rw-1(z)〈TbRw,aRz〉=Rw-1(z)∫𝔹n(TbRw)(u)Rz(u)¯a(|u|)μ(|u|)dv(u)=Rw-1(z)∫𝔹n〈TbRw,Ru〉Rz(u)¯a(|u|)μ(|u|)dv(u)=Rw-1(z)(∑|α|=0∞ẽαμ(w)¯eαμ(z)γb,μ(|α|)γa,μ(|α|)+∑|α|=0∞ẽαμ(z)¯eαμ(w)γb,μ(|α|)γa,μ(|α|)-dn,02λ2(n,0)γb,μ(0)γa,μ(0)∑|α|=0∞)=Rw-1(z)[∑m=0∞l(m,n)γb,μ(m)γa,μ(m)((z·w¯)m+(w·z¯)m)-dn,02λ2(n,0)γb,μ(0)γa,μ(0)].



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*Source: 210596-2011-09-20.xml*