# Derivatives of the Berezin Transform

Authors: Hélène Bommier-Hato
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

For a rotation invariant domainΩ, we consider A2(Ω,μ) the Bergman
space and we investigate some properties of the rank one projection A(z):=〈·,kz〉kz. We prove that the trace of all the strong derivatives of A(z) is zero. We also focus on the
generalized Fock space A2(μm), where μm is the measure with weight e-|z|m, m>0,
with respect to the Lebesgue measure on ℂn and establish estimations of derivatives of
the Berezin transform of a bounded operator T on A2(μm).


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## Body

## 1. Introduction and Statement of the Main Results

We consider a rotation invariant open setΩ in ℂn and μ a positive rotation invariant measure on Ω; we suppose that μ has moments of every order. Let L2(Ω,μ) be the Hilbert space of square integrable complex-valued functions on Ω and 𝒜2(Ω,μ) its subspace consisting of holomorphic elements. We assume that for each compact set K⊂Ω there exists C=C(K) such that for all f∈𝒜2(Ω,μ)(1.1)sup⁡z∈K|f(z)|≤C‖f‖L2(Ω,μ).
It is known that 𝒜2(Ω,μ) is a closed space of L2(Ω,μ) and possesses a reproducing kernel K(ω,z):=Kz(ω): we have(1.2)f(z)=∫Ωf(ω)K(z,ω)dμ(ω)=〈f,Kz〉,
for all f∈𝒜2(Ω,μ) and z∈Ω.For a bounded linear operatorT on 𝒜2(Ω,μ), the Berezin transform of T is the function T̃ defined on Ω by(1.3)T̃(z):=〈Tkz,kz〉,
where kz is the normalized reproducing kernel(1.4)kz(ω):=K(ω,z)K(z,z)1/2.
The case of the Fock space, where Ω=ℂn and μ2 is the Gaussian measure, was considered by Coburn, Englis, and Zhang. Coburn [1] has shown that T̃ is a Lipschitz function. Namely, for x,y∈ℂn,(1.5)|T̃(x)-T̃(y)|≤2‖T‖|x-y|,
the constant 2 being sharp (see [2]).Englis and Zhang [3] have shown that T̃ has bounded derivatives of all orders. Namely, for any multi-indices β, γ, there exists a constant cβ,γ, depending on β, γ, and n only, such that(1.6)‖∂β∂¯γT̃‖∞≤cβ,γ‖T‖,
where ∂β will stand for(1.7)∂|β|∂z1β1⋯∂znβn,where∂∂zj=12(∂∂xj-i∂∂yj)forzj=xj+iyj
and similarly for ∂¯. Recently, the author extended Coburn’s result to weighted Fock spaces, corresponding to Ω=ℂn endowed with the measure(1.8)dμm(z)=e-|z|mdV(z),
where m is a positive parameter and dV is the normalized Lebesgue measure on ℂn, such that the volume of the unit ball in ℂn is equal to 2n. We have showed that T̃ satisfies a local Lipschitz condition. There exist positive constants A, C, depending on n and m only, such that, for any a∈ℂn with |a|>A, there is a neighbourhood U of a that satisfies(1.9)|T̃(a)-T̃(x)|≤C‖T‖|a|(m/2)-1|x-a|,∀xinU.
In this paper we will investigate some properties of the derivatives of the Berezin transform on 𝒜2(Ω,μ). For z in Ω, if A(z) is the rank one projection(1.10)A(z):=〈⋅,kz〉kz,
we have(1.11)T̃(z)=tr⁡(TA(z)).
We first fix some notations. Let ℕ0n denote the set of all n-tuples with components in the set ℕ0 of all nonnegative integers. If β=(β1,…,βn)∈ℕ0n, we let |β|:=β1+⋯+βn denote the length of β and β! stands for ∏j=1nβj!. If γ=(γ1,…,γn)∈ℕ0n satisfies γj≤βj for all j=1,…,n, then we write γ≤β. Our first main result is about the strong derivatives of A(z).Theorem 1.1.
LetΩ be a rotation invariant open set in ℂn and μ a rotation invariant positive measure on Ω that satisfies (1.1) and has moments of every order. Moreover one assumes that, for any multi-index α and any compact set K of Ω, there exists Gα,K∈L2(Ω,μ) such that
(1.12)|w¯α∂αK(z,w)|≤Gα,K(w),∀z∈K.
Then for all β, γ multi-indices, the operators ∂β∂¯γA(z) and ∂¯β∂γA(z) are adjoint to each other; their rank is smaller or equal to the infimum of #{δ∈ℕ0n,δ≤γ} and #{δ∈ℕ0n,δ≤β}.
Moreover, if at leastβ or γ is different from 0, one has
(1.13)tr⁡[∂β∂¯γA(z)]=0.Our second main result generalizes the estimates of Englis and Zhang for the strong derivatives of the Berezin transform on weighted Fock spaces.Theorem 1.2.
For a bounded linear operatorT on 𝒜2(ℂn,μm), the Berezin transform T̃ has derivatives of all orders. In addition to any multi-indices β, γ, there exist positive constants cβ,γ and A, depending on β, γ, and n only such that
(1.14)|∂β∂¯γT̃(z)|≤cβ,γ|z|((m/2)-1)(|β+γ|)for|z|>A.


## 2. Preliminaries

We recall some properties of the Bergman kernelK, when μ is a positive rotation invariant measure on Ω. The kernel K is given by(2.1)K(z,ω)=∑β∈N0n1cβzβω¯β,
for z, ω in Ω, with the usual convention that, for z=(z1,…,zn) and β=(β1,…,βn), zβ stands for z1β1⋯znβn and where(2.2)cβ:=∫Ωxβx¯βdμ(x).
By Lemma  2.1 in [4], we know that(2.3)cβ=(n-1)!β!m|β|(n-1+|β|)!,wheremk=∫Ω|z|2kdμ(z)fork∈N0.
Thus we can write(2.4)K(z,ω)=F(〈z,ω〉),
where(2.5)F(t):=1(n-1)!∑d∈N0(n-1+d)!mdtd
is a holomorphic function of one complex variable.For a bounded operatorT on 𝒜2(Ω,μ), the Berezin transform can be written in the form(2.6)T̃(z)=tr⁡(TA(z)),forz∈Ω,
where A(z) is the rank one projection A(z)=〈·,kz〉kz.Recall [3] that a mapping h from a domain in ℂn into a Banach space possesses a strong holomorphic derivative ∂h/∂z1(z) at a point z=(z1,…,zn)∈ℂn if(2.7)lim⁡t→0‖h(z1+t,z2,…,zn)-h(z1,…,zn)t-∂h∂z1(z1,…,zn)‖=0;
similarly one can define the antiholomorphic derivative ∂¯. Englis and Zhang showed that the mapping z↦A(z) has strong derivatives.Lemma 2.1.
LetΩ be a domain in ℂn, μ a measure on Ω that satisfies (1.1). Then the function z↦kz has strong derivatives of all orders.Lemma 2.2.
Under the same hypothesis, the trace-class-operator-valued function(2.8)z⟼〈⋅,kz〉kz
from Ω into the space of trace-class operators on 𝒜2(Ω,μ) has strong derivatives of all orders.The mappingz↦A(z) and its derivatives can be expressed in terms of the function F. It is easy tosee that, for z, u in Ω,(2.9)A(z)⋅f(u)=〈f,F(〈⋅,z〉)F(〈u,z〉)F(〈z,z〉)〉.


## 3. Proof of Theorem1.1

We write, forf∈𝒜2(Ω,μ) and z, u in Ω,(3.1)A(z)⋅f(u)=[∫Ωf(ω)F(〈z,ω〉)dμ(ω)]F(〈u,z〉)F(〈z,z〉).
Since it is possible to differentiate under the integral sign at any order with respect to z, for β in ℕ0n, we have(3.2)(∂βA(z)⋅f)(u)=∫Ωf(ω)∂β[F(〈z,ω〉)F(〈z,z〉)]dμ(ω)F(〈u,z〉).
The Leibnitz rule leads to(3.3)(∂βA(z)⋅f)(u)=∑ν≤β(βν)∫Ωf(ω)ω¯νF(|ν|)(〈z,ω〉)z¯β-ν(1F)(|β-ν|)(〈z,z〉)dμ(ω)F(〈u,z〉).
Setting K̇z(β)(ω)=F(〈z,z〉)1/2∑ν⊂β(βν)ωνzβ-ν(1/F)(|β-ν|)(〈z,z〉)F(|ν|)(〈ω,z〉), we get(3.4)∂βA(z)⋅f=〈f,K̇z(β)〉kz
and then(3.5)∂¯γ∂βA(z)⋅f=∑δ≤γ〈f,Lz,δ〉Mz,δ,
where Lz,δ(ω)=(∂δ/∂zδ)(F(〈z,z〉)-1/2K̇z(β)(ω)) and Mz,δ(u)=(γδ)uγ-δF(|γ-δ|)(〈u,z〉). Notice that the rank of the operator ∂¯γ∂βA(z) is smaller than or equal to #{δ∈ℕ0n,δ≤γ}.Setting nowS:=∂¯γ∂βA(z), Lδ:=Lz,δ, and Mδ:=Mz,δ for a fixed element z in Ω, we obtain, for f,g in 𝒜2(Ω,μ),(3.6)〈Sf,g〉=〈∑δ≤γ〈f,Lδ〉Mδ,g〉=〈f,∑δ≤γ〈g,Mδ〉Lδ〉.
Consequently,(3.7)S*g=∑δ≤γ〈g,Mδ〉Lδ,
that is,(3.8)(S*g)(u)=∑δ≤γ(γδ)∫Ωg(ω)ω¯γ-δF(|γ-δ|)(〈z,ω〉)dμ(ω)∂δ∂zδ(1F(〈z,z〉)K̇z(β)(u))=∂γ∂zγ[∫Ωg(ω)F(〈z,ω〉)dμ(ω)(1F(〈z,z〉)K̇z(β)(u))]=∂γ∂zγ[〈g,kz〉K̇z(β)(u)]=∂γ∂zγ∂β∂z¯β[〈g,kz〉kz(u)]=(∂γ∂β¯A(z)⋅g)(u).
It follows that the rank of the operator ∂¯γ∂βA(z) is also smaller than or equal to #{δ∈ℕ0n,δ≤β}. Thus(3.9)(∂γ¯∂βA(z))*=∂γ∂β¯A(z),forz∈Ω.
Now let β and γ be multi-indices. Due to Lemma 2.2 and the continuity of the linear form Tr(X), we have(3.10)Tr⁡(∂γ¯∂βA(z))=∂γ¯∂βTr⁡A(z)=∂γ¯∂β1=δβ0δγ0.


## 4. Proof of Theorem1.2

WhenΩ=ℂn and dμm(z)=e-|z|mdv(z), for brevity we set 𝒜2(μm) for 𝒜2(ℂn,μm). The Bergman kernel Km of 𝒜2(μm) can be expressed in terms of the Mittag-Leffler function (see [5]). Putting α=2/m, we have(4.1)Km(z,ω)=F(〈z,ω〉),whereF(t)=CEα,α(n-1)(t),C=m(n-1)!,Eα,α(n-1) being the n-1-th derivative of the Mittag-Leffler function Eα,α, entire on ℂ and defined by(4.2)Eα,α(t):=∑d=0+∞tdΓ(αd+α),fort∈C.
In what follows, we will use some asymptotic properties of this function (see [6]) near the positive real axis.Lemma 4.1.
Letp a nonnegative integer. There exists a constant ϵ>0, such that, for any complex number z with |argz|<ϵ, one has
(4.3)F(p)(z)=F0(p)(z)+ϵp(z),
where
(4.4)F0(z)=CeZ0(z)Z0(z)-nαPn(Z0(z))
and Zs(z)=|z|1/αexp[(i/α)(argz+2πs)], for s integer.
The polynomials(Pp)p∈ℕ are defined by (dp/dup)[eu1/α]=(eu1/α/up)Pp(u1/α), Pp is of degree p, and when |z|→+∞, ϵp(r)(z)=o(F0(q)(Z0)) for any nonnegative integers r and q.We also need asymptotic estimates for some auxiliaries functions.Lemma 4.2.
Letp and d be nonnegative integers. When t is real and tends to +∞,
(4.5)(1F)(d)(t)~(1F0)(d)(t),(F(p)F)(d)(t)~(F0(p)F0)(d)(t).ForT a bounded linear operator on 𝒜2(μm), the Berezin transform is given by (see [3])(4.6)T̃(z)=tr⁡(TA(z)).
Let β and γ be some multi-indices. By differentiation, Lemma 2.2 gives(4.7)∂β∂¯γT̃(z)=tr⁡(T∂β∂¯γA(z)).
Due to the continuity of the bilinear form (X,Y)↦tr(XY), we have(4.8)|tr⁡(XY)|≤‖X‖‖Y‖tr⁡,
for all bounded operators X and trace class operators Y. Therefore, we obtain(4.9)|∂β∂¯γT̃(z)|≤‖T‖‖∂β∂¯γA(z)‖tr⁡.
Like in Section 3, we set S:=∂β∂¯γA(z). Next we shall estimate tr((SS*)1/2).Fixf in 𝒜2(μm) and z in ℂn. We recall that(4.10)Sf=∑δ≤γ〈f,Lz,δ〉Mz,δ,
where Lz,δ(ω)=(∂δ/∂zδ)(F(〈z,z〉)-1/2K̇z(β)(ω)) and Mz,δ(u)=(γδ)uγ-δF(|γ-δ|)(〈u,z〉). Then(4.11)SS*f=∑δ≤γ[∑δ′≤γ〈f,Mz,δ′〉〈Lz,δ′,Lz,δ〉]Mz,δ.
To estimate the eigenvalues of the finite rank positive operator SS*, we compute its trace:(4.12)tr⁡(SS*)=∑δ≤γaδ,withaδ=∑δ′≤γ〈Mz,δ,Mz,δ′〉〈Lz,δ′,Lz,δ〉.
With the operator being of finite rank, we observe that Tr(SS*)=Tr(S*S). We now compute and then estimate each diagonal term aδ. Fix some multi-indices δ and δ′ such that δ≤γ and δ′≤γ. To simplify the notation we set Lδ and Mδ instead of Lz,δ and Mz,δ. We obtain(4.13)〈Mδ,Mδ′〉=C2(γδ)(γδ′)∫Cnωγ-δF(|γ-δ|)(〈ω,z〉)ω¯γ-δ′F(|γ-δ′|)(〈z,ω〉)dμ(ω)=C2(γδ)(γδ′)1C∂γ-δ∂z¯γ-δ∂γ-δ′∂zγ-δ′F(〈z,z〉)=C(γδ)(γδ′)∂γ-δ∂z¯γ-δ[z¯γ-δ′F(|γ-δ′|)(〈z,z〉)]=C(γδ)(γδ′){∑κ≤γ-δ,κ≤γ-δ′(γ-δκ)(γ-δ′κ)κ!×z¯γ-δ′-κzγ-δ-κF(|γ-δ′+γ-δ-κ|)(〈z,z〉)∑κ≤γ-δ,κ≤γ-δ′(γ-δκ)(γ-δ′κ)κ!}.
For the second term, since(4.14)Lδ(ω)=∂δ∂zδ{∑ν≤β(βν)ωβ-νF(|β-ν|)(〈ω,z〉)zν(1F)(|ν|)(〈z,z〉)}=∑ν≤β(βν)ωβ-νF(|β-ν|)(〈ω,z〉)∂δ∂zδ[zν(1F)(|ν|)(〈z,z〉)],
we get(4.15)〈Lδ′,Lδ〉=∑ν,ν′≤β(βν)(βν′)∂δ′∂zδ′[zν′(1F)(|ν′|)(〈z,z〉)]∂δ∂z¯δ×[z¯ν(1F)(|ν|)(〈z,z〉)]∂β-ν′∂z¯β-ν′∂β-ν∂zβ-ν[F(〈z,z〉)C],(4.16)〈Lδ′,Lδ〉=1C∑ν,ν′≤β(βν)(βν′)∂δ′∂zδ′[zβ-ν′(1F)(|β-ν′|)(〈z,z〉)]∂δ∂z¯δ×[z¯β-ν(1F)(|β-ν|)(〈z,z〉)]∂ν′∂z¯ν′[z¯νF(|ν|)(〈z,z〉)].
By the Leibnitz rule, we see that(4.17)∂δ∂z¯δ[z¯β-ν(1F)(|β-ν|)(〈z,z〉)]=∑λ≤δ,λ≤β-ν(δλ)(β-νλ)λ!z¯β-ν-λzδ-λ(1F)(|β-ν+δ-λ|)(〈z,z〉),∂δ′∂zδ′[zβ-ν′(1F)(|β-ν′|)(〈z,z〉)]=∑λ′≤δ′,λ′≤β-ν′(δ′λ′)(β-ν′λ′)λ′!zβ-ν′-λ′z¯δ′-λ′×(1F)(|β-ν′+δ′-λ′|)(〈z,z〉),∂ν′∂z¯ν′[z¯νF(|ν|)(〈z,z〉)]=∑σ≤ν,σ≤ν′(ν′σ)(νσ)σ!z¯ν-σzν′-σF(|ν+ν′-σ|)(〈z,z〉).
Then
(4.18)〈Mδ,Mδ′〉〈Lδ′,Lδ〉=(γδ)(γδ′)∑ν≤β,ν′≤β(βν)(βν′)∑λ≤δ,β-νλ′≤δ′,β-ν′σ≤ν,ν′κ≤γ-δ,γ-δ′(ν′σ)(νσ)σ!(δ′λ′)×(β-ν′λ′)λ′!(δλ)(β-νλ)λ!(γ-δκ)×(γ-δ′κ)κ!zβ+γ-λ-λ′-σ-κz¯β+γ-λ-λ′-σ-κ×F(|γ-δ+γ-δ′-κ|)(〈z,z〉)F(|ν+ν′-σ|)(〈z,z〉)(1F)(|β-ν′+δ′-λ′|)×(〈z,z〉)(1F)(|β-ν+δ-λ|)(〈z,z〉).
Thus
(4.19)aδ=(γδ)∑δ′≤γ(γδ′)∑ν≤βν′≤β(βν)(βν′)∑λ≤δ,β-νλ′≤δ′,β-ν′β-ν,β-ν′≤ρκ≤γ-δ,γ-δ′(ν′β-ρ)(νβ-ρ)(β-ρ)!(δ′λ′)(β-ν′λ′)×λ′!(δλ)(β-νλ)λ!(γ-δκ)(γ-δ′κ)κ!zγ-λ-λ′-κ+ρz¯γ-λ-λ′-κ+ρ×F(|γ-δ+γ-δ′-κ|)(〈z,z〉)F(|ν+ν′-β+ρ|)(〈z,z〉)(1F)(|β-ν′+δ′-λ′|)(〈z,z〉)×(1F)(|β-ν+δ-λ|)(〈z,z〉),(4.20)aδ=(γδ)∑δ′≤γ(γδ′)∑κ∑λ,λ′β!(β-λ)!β!(β-λ′)!∑ν,ν′ρ(β-λρ-λ)(β-λ′ρ-λ′)(β-ρ)!(ρ-λβ-ν-λ)×(ρ-λ′β-ν′-λ′)(δλ)(δ′λ′)(γ-δκ)(γ-δ′κ)κ!zγ-λ-λ′-κ+ρz¯γ-λ-λ′-κ+ρ×F(|γ-δ+γ-δ′-κ|)(〈z,z〉)F(|ν+ν′-β+ρ|)(〈z,z〉)×(1F)(|β-ν′+δ′-λ′|)(〈z,z〉)(1F)(|β-ν+δ-λ|)(〈z,z〉),(4.21)aδ=(γδ)∑δ′≤γ(γδ′)∑κ(γ-δκ)(γ-δ′κ)κ!∑ν,ν′,λ,λ′,ρβ!(β-λ)!β!(β-λ′)!(β-λρ-λ)(β-λ′ρ-λ′)×(β-ρ)!(δλ)(δ′λ′)(ρ-λβ-ν-λ)(ρ-λ′β-ν′-λ′)zγ-λ-λ′-κ+ρz¯γ-λ-λ′-κ+ρ×F(|γ-δ+γ-δ′-κ|)(〈z,z〉)F(|ν+ν′-β+ρ|)(〈z,z〉)×(1F)(|β-ν′+δ′-λ′|)(〈z,z〉)(1F)(|β-ν+δ-λ|)(〈z,z〉).
Setting σ=β-ν-λ and σ′=β-ν′-λ′, we have(4.22)aδ=(γδ)∑δ′≤γ(γδ′)∑κ(γ-δκ)(γ-δ′κ)κ!∑λ≤δ,λ′≤δ′β!(β-λ)!β!(β-λ′)!(δλ)(δ′λ′)×∑λ≤ρ,λ′≤ρ(β-ρ)!(β-λρ-λ)(β-λ′ρ-λ′)∑σ≤ρ-λ,σ′≤ρ-λ′(ρ-λσ)(ρ-λ′σ′)×F(|γ-δ+γ-δ′-κ|)(〈z,z〉)F(|β+ρ-λ-λ′-σ-σ′|)(〈z,z〉)×(1F)(|σ+δ|)(〈z,z〉)(1F)(|σ′+δ′|)(〈z,z〉)zγ-λ-λ′-κ+ρz¯γ-λ-λ′-κ+ρ,(4.23)aδ=(γδ)∑δ′≤γ(γδ′)∑κ(γ-δκ)(γ-δ′κ)κ!∑λ≤δ,λ′≤δ′β!(β-λ)!β!(β-λ′)!(δλ)(δ′λ′)×∑λ,λ′≤ρ≤β(β-ρ)!(β-λρ-λ)(β-λ′ρ-λ′)zγ-λ-λ′-κ+ρz¯γ-λ-λ′-κ+ρSρ(〈z,z〉),
where(4.24)Sρ=∑σ≤ρ-λσ′≤ρ-λ′(ρ-λσ)(ρ-λ′σ′)F(|γ-δ+γ-δ′-κ|)F(|β+ρ-λ-λ′-σ-σ′|)(1F)(|σ+δ|)(1F)(|σ′+δ′|).
Setting τ̆=(τ1,…,τn-1) for a multiindex τ=(τ1,…,τn), we see that(4.25)Sρ=F(|γ-δ+γ-δ′-κ|)∑σ̆≤ρ̆-λ̆σ′̆≤ρ̆-λ′̆(ρ̆-λ̆σ̆)(ρ̆-λ′̆σ′̆)∑σn′≤ρn-λn′(ρn-λn′σn′)(1F)(|σ′+δ′|)×∑σn≤ρn-λn(ρn-λnσn)((1F)(|δ|+σ1+⋯+σn-1))(σn)((F)(|β-λ′-σ′|+|ρ̆-λ̆-σ̆|))(ρn-λn-σn).
Using the Leibnitz rule(4.26)Sρ=F(|γ-δ+γ-δ′-κ|)∑σ̆≤ρ̆-λ̆σ′̆≤ρ̆-λ′̆(ρ̆-λ̆σ̆)(ρ̆-λ′̆σ′̆)∑σn′≤ρn-λn′(ρn-λn′σn′)(1F)(|σ′+δ′|)×[(1F)(|δ|+σ1+⋯+σn-1)F(|β-λ′-σ′|+|ρ̆-λ̆-σ̆|)](ρn-λn).
An induction process shows that(4.27)Sρ=F(|γ-δ+γ-δ′-κ|)∑σ′≤ρ-λ′(ρ-λ′σ′)(1F)(|σ′+δ′|)[F(|β-λ′-σ′|)(1F)(|δ|)](|ρ-λ|).
Therefore(4.28)aδ=(γδ)∑δ′≤γ(γδ′)∑κ(γ-δκ)(γ-δ′κ)κ!∑λ≤δ,λ′≤δ′β!(β-λ)!β!(β-λ′)!(δλ)(δ′λ′)×∑λ,λ′≤ρ≤β(β-ρ)!(β-λρ-λ)(β-λ′ρ-λ′)F(|γ-δ+γ-δ′-κ|)(〈z,z〉)zγ-λ-λ′-κ+ρz¯γ-λ-λ′-κ+ρ×∑σ′≤ρ-λ′(ρ-λ′σ′)(1F)(|σ′+δ′|)(〈z,z〉)[F(|β-λ′-σ′|)(1F)(|δ|)](|ρ-λ|)(〈z,z〉).
Now let τ=γ-λ′-κ. It follows that(4.29)aδ=(γδ)∑ρ≤β∑τ≤γ∑λ,λ′≤ρ∑λ′≤δ′≤τ+λ′(γδ′)(γ-δγ-λ′-τ)(γ-δ′γ-λ′-τ)(γ-λ′-τ)!β!(β-λ)!β!(β-λ′)!×(δλ)(δ′λ′)(β-ρ)!(β-λρ-λ)(β-λ′ρ-λ′)F(|γ-δ-δ′+λ′+τ|)(〈z,z〉)∑σ′≤ρ-λ′(ρ-λ′σ′)×(1F)(|σ′+δ′|)(〈z,z〉)[F(|β-λ′-σ′|)(1F)(|δ|)](|ρ-λ|)(〈z,z〉)zτz¯τzρ-λz¯ρ-λ.
On the other hand(4.30)(γδ′)(γ-δ′γ-λ′-τ)(γ-λ′-τ)!(δ′λ′)=γ!(-δ′+λ′+τ)!λ′!(δ′-λ′)!=γ!τ!λ′!(τδ′-λ′).
Thus, if we set σ=δ′-λ′, then(4.31)aδ=(γδ)∑ρ≤β∑τ≤γ∑λ,λ′≤ρσ′≤ρ-λ′∑σ≤τ(ρ-λ′σ′)(β-ρ)!(β-λρ-λ)(β-λ′ρ-λ′)(δλ)γ!zτ+ρ-λz¯τ+ρ-λ×(γ-δγ-λ′-τ)1τ!λ′!(τσ)F(|γ-δ+τ-σ|)(〈z,z〉)×(1F)(|σ+σ′+λ′|)(〈z,z〉)[F(|β-λ′-σ′|)(1F)(|δ|)](|ρ-λ|)(〈z,z〉).
Using the identity(4.32)∑σ≤τ(τσ)(F(|γ-δ|))(|τ-σ|)((1F)(|σ′+λ′|))(|σ|)=[(1F)(|γ-δ|)F(|σ′+λ′|)](|τ|),
we see that(4.33)aδ=(γδ)γ!∑ρ≤βτ≤γ∑λ,λ′≤ρσ′≤ρ-λ′(ρ-λ′σ′)(β-ρ)!(β-λρ-λ)(β-λ′ρ-λ′)(δλ)(γ-δγ-λ′-τ)1τ!λ′!zτ+ρ-λz¯τ+ρ-λ×[F(|β-λ′-σ′|)(1F)(|δ|)](|ρ-λ|)(〈z,z〉)[(1F)(|γ-δ|)F(|σ′+λ′|)](|τ|)(〈z,z〉).
Notice that, for p and q integers, we have, when t is real and t→+∞,(4.34)F(p)(t)F(t)~t((1/α)-1)p,(1F)(q)(t)~t((1/α)-1)pF(t).
Hence by Lemmas 4.1 and 4.2 we have the following estimates:(4.35)|[F(|β-λ′-σ′|)(1F)(|δ|)](|ρ-λ|)(t)|≤Ct((1/α)-1)(|β+δ-λ′-σ′|)-|ρ-λ|,|[(1F)(|γ-δ|)F(|σ′+λ′|)](|τ|)(t)|≤Ct((1/α)-1)(|γ-δ+σ′+λ′|)-|τ|
for some constant C when the real t tends to +∞. Thus for any multiindex δ≤γ, there exists a constant C′ such that aδ≤C′|z|2((1/α)-1)(|β+γ|). Then taking the sum over δ we get(4.36)tr⁡(SS*)≤C′′|z|2((1/α)-1)(|β+γ|).
Since Tr(SS*)=Tr(S*S) and the operator S*S is positive, its eigenvalues are bounded above by C′′|z|2((1/α)-1)(|β+γ|). Thus the eigenvalues of (SS*)1/2 are bounded above by C′′′|z|((1/α)-1)(|β+γ|). Since (S*S)1/2 is of finite rank, the proof is complete.



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*Source: 160808-2012-02-29.xml*