constant) of polynomial growth such that [Tfm,δ, TΨjeijθ] = 0. As an application we show that in casem 6= 0the condition[Tfm,0, TΨ] = 0is equivalent toΨbeing radial. Section 2.5 deals with the case j <0, and we prove that there is no non-zero functionΨj of polynomial growth satisfying [Tfm,δ, TΨjeijθ] = 0 forj < −2[δ4]. By using similar techniques, Section 2.6 treats the case 0 ≤ j ≤ δ. In Section 2.7, some applications of our results are indicated. We show that the radial componentsΨj can be extended to a complex analytic function on the right half plane. Moreover, by our method we recover the examples in [17] of radial functions u such thatTu commutes with another Toeplitz operator having a non-radial symbol. Motivated by a conjecture of Louhichi and Rao in [129] and using our results, we give a triple (Tf, Tg, Th)of Toeplitz operators with Tf 6= Idsuch that[Tf, Tg] = [Tf, Th] = 0but[Tg, Th]6= 0. Finally, we ask if for everyj ∈Zthere is exactly one functionΨj such that[Tfm,δ, TΨjeijθ] = 0.
2.2 Preliminaries
Let us start by defining the space of measurable functions of at most polynomialgrowth at infinity to be
S :=
n
g :C−→C| ∃C, c >0s.t.|g(z)|6C(1 +|z|)co . For a givenc > 0, we also define a function space:
Dc:=
n
Ψ :C−→C|Ψmeasurable and∃d >0s.t.|Ψ(z)|6dec|z|2 o
equipped with the norm kΨkDc := kΨe−c|·|2k∞. We write Fc := (Dc ∩H2,k·kDc) for the intersection withH2and we note thatE :=S
c<14 Dc.
Throughout the chapter,r and θ denote the polar coordinates ofz ∈ C i.e. z = reiθ. We fix a symbol fm,δ(reiθ) = rmeiδθ, where m ∈ R+, δ ∈ Z, and we write [·] for the greatest integer function. Note that each function Ψ ∈ S can be expanded to an L2-convergent series Ψ(reiθ) =P∞
j=−∞Ψj(r)eijθ(c.f. Lemma 2.2.1).
For a fixed symbolfm,δ we aim to characterizeΨ∈Ssuch that the commutator[Tfm,δ, TΨ] vanishes as an operator on the space of holomorphic polynomials P[z]. It turns out that it is sufficient to characterize the coefficient functions {Ψj}j∈Z in the series expansion above. We express{Ψj}j∈Z in terms of the inverse Mellin transform of a function formed up of Gamma functions and a trigonometric polynomial. Moreover, iffm,δ is a monomial (i.e.m±δ∈2N0), then for eachj ∈Zsuch thatj >−2[|δ|4]we find a collection of quasi-homogeneous functions {ϕjeijθ} ⊂ E such that[Tfm,δ, Tϕjeεijθ] = 0onP[z]whereε := sign(δ).
First it is essential to make clear that the commutator[Tfm,δ, TΨ]is well defined onP[z], and that it is sufficient to solve the above problem whenδ∈N0.
44 CHAPTER 2. COMMUTING TOEPLITZ OPERATORS Letf ∈ S, Ψ ∈ E and pick up0 < < 14 such thatΨ ∈ D. We define a dense subspace H ⊂ H2 containingP[z]such that the operator productsTfTΨ andTΨTf are well defined on H. Moreover, we prove the equivalence: [Tf, TΨ] = 0onHif and only if[Tf, TΨ] = 0onH. In order to construct the spaceHwe need to introduce a scale of subspaces (c.f. [14] ):
D ⊂Dc1 ⊂Dc2 ⊂ · · · ⊂ [
j∈N
Dcj ⊂L2,
where (cj)j∈N is the increasing sequence of real numbers cj := 1/2−1/(2j + 2) such that cj ∈[1/4,1/2)for allj ∈Nand
cj+1 = 1 4(1−cj).
It is known that the orthogonal projectionP is continuous fromDcj toDcj+1 for allj ∈N(c.f.
Lemma 10 in [14]). Using this fact, we can prove the following:
Proposition 2.2.1. Let f ∈ S and fix < 14. Then there exists a densely embedded Hilbert spaceH ,→H2 containingP[z]as well as the linear space
L=span n
kz(w) :=K(w, z)e−12|z|2 : z ∈C o
,
such that for all Ψ ∈ D ⊂ E the operator products TΨTf and TfTΨ are well defined and continuous fromH toH2.
Proof. The first step in the proof is to find a positive numberς = ς()such that the operator productsTΨTf andTfTΨare well defined and continuous from Fς toH2 for allΨ∈ D. Pick a real numberγwith0< γ <1− 2(1−2)1 . Then there existsj0 ∈Nsuch that 4(1−γ)1 + < cj0, and one can find j1 > j0 with ς := cj1+1−cj1 < γ. LetMf andMΨ denote the operators of multiplication byf andΨ, respectively. The assertion then follows by noting thatTΨTf is the composition of the following well defined and continuous operators:
Fς −−→Mf Dγ −→ FP 1
4(1−γ)
MΨ
−−→Dcj
0
−→ FP cj
0+1 ,→H2.
Sinceς + < cj0 +cj1+1−cj1 6 cj1+1, the operator product TfTΨ is the composition of the continuous operators:
Fς −−→MΨ Dcj
1+1
−→ FP cj
1+2
Mf
−−→Dcj
1+3
−→ FP cj
1+4 ,→H2.
Next, we modifyFςto obtain a densely embedded Hilbert spaceH,→ Fςsuch thatL ∪P[z]⊂ H and the inclusion L ⊂ H is dense. In particular, the restriction [Tf, TΨ] : H −→ H2 defines a continuous operator.
In order to construct the spaceHwe fix a positive number λ =λ()such that λ2 < ς, and we denote byµλ the probability Gaussian measure onCgiven by
dµλ(z) := λ
πe−λ|z|2dv(z).
2.2. PRELIMINARIES 45 Moreover, leth·,·iλ be the usual inner product onL2(C, dµλ)and define
H :=H2(C, dµλ)
to be the subspace of all entire µλ-square integrable functions on C equipped with the usual topology inherited from L2(C, dµλ) (with the notation used in Section 1.3, H is the Segal-Bargmann space H(21
4λ) over C). Then it is easy to check that H has the above mentioned properties.
Remark 2.2.1. In comparison to the case of the Bergman space over the unit disc considered in [66] and using the notation of Theorem A0 the Toeplitz operators Tψ and Tϕ are bounded onA(D). HenceTψ andTϕ commute onA(D)if and only if they commute on the holomorphic polynomials defined onD.
Remark 2.2.2. According to the above proposition, for a fixed < 14 the Toeplitz operatorTΨ is continuous onH for anyΨ ∈ D ⊂ E. SinceL ⊂ H, by Proposition 1.1.4 it follows that the Berezin transform is one-to-one on the space of Toeplitz operators with symbols in D for any < 14. Moreover, since D1 ⊂ D2 for1 < 2 andE =S
<14 Dthe Berezin transform is one-to-one on the space of Toeplitz operators with symbols inE.
Another consequence of Proposition 1.1.4 we have:
Corollary 2.2.1. Letf ∈ Sand < 14 such that Ψ∈ D ⊂ E. Then[Tf, TΨ]and[Tf, TΨ]are well defined and continuous onHand satisfies
[Tf, TΨ] = 0 ⇐⇒[Tf, TΨ] = 0.
Proof. By a direct calculation, we obtain
[T^f, TΨ] =[T^f, TΨ], (2.2.1) whereg[·,·]denotes the Berezin transform of the corresponding commutator. Since the operator [Tf, TΨ] :H −→H2 is continuous, it follows from (2.2.1) and Proposition 1.1.4 that
[Tf, TΨ] = 0⇐⇒[T^f, TΨ] = 0 ⇐⇒[T^f, TΨ] = 0⇐⇒[Tf, TΨ] = 0.
The followingL2-series expansion is essential in our proofs:
Lemma 2.2.1. Let < 12 andΨ∈D, thenΨhas the series expansion:
Ψ(reiθ) =
∞
X
j=−∞
Ψj(r)eijθ z
with|Ψj(r)| 6 Cer2 for some constantC > 0and almost allr > 0. The above convergence holds inL2(C, dµ). Moreover, ifΨ ∈S thenΨj(r)is defined a.e. onR+ and is of polynomial growth at infinity for allj ∈Z.
46 CHAPTER 2. COMMUTING TOEPLITZ OPERATORS Proof. IfΨ∈S, then by Lemma 4.2 in [17],Ψhas anL2-convergent expansionzwhere each Ψj(r) is of polynomial growth at infinity. Now, ifΨ ∈ D \S thenz is obtained from the expansion of the bounded functionΨ(z)e−|z|2.
Remark 2.2.3. Let Ψ ∈ E and fm,δ = rmeiδθ be the function defined earlier. Assume that [Tfm,δ, TΨ] = 0 on P[z]. Then by Corollary 2.2.1 and due to the dense inclusion P[z] ⊂ H it follows that [Tf
m,δ, TΨ] = [Tfm,−δ, TΨ] = 0on P[z]. Since (Ψ)j = Ψ−j for allj ∈ Z, it is sufficient to consider only the caseδ∈N0 in the following.
Remark 2.2.4. In [66], it has been shown that any ψ ∈ L2(D, dv) can be written as an L2(D,1πdv)-series expansion ψ(reiθ) = P∞
j=−∞ψj(r)eijθ. Indeed, if R denotes the space of complex valued functions defined on[0,1)which are integrable with respect to therdr mea-sure (r varies in [0,1)) then the spaces {eijθR}j∈Z are mutually orthogonal in L2(D,1πdv).
Moreover, every polynomial can be written as a finite sum in ⊕jeijθR. But the polynomials are dense inL2(D, dv)hence L2(D, dv) = ⊕j∈ZeijθR. As in the case of the Segal-Bargmann space considered in the above remark, Theorem A0 treats also the case δ < 0. In fact, if ψ and ϕ(reiθ) = rmeiδθ are the bounded functions considered in Theorem A0 then Tϕ and Tψ commute if and only if the adjoint operatorsTϕ? =Trme−iδθandTψ? =Tψ commute.
Now we recall some properties of the Mellin transform, which forms an important tool throughout this chapter. Let g be a (suitable) complex valued function defined on the interval (0,∞). We writeM[g]for the Mellin transform ofg:
M[g](z) = Z ∞
0
xz−1g(x)dx.
Recall thatM[g]is complex analytic on a strip in the complex plane parallel to the imaginary axis. For a suitable function ϕ(z) which is complex analytic in a strip a < Re(z) < b, the inverse Mellin transformM−1[ϕ]is the function on(0,∞)given by:
M−1[ϕ](x) = 1 2πi
Z c+i∞
c−i∞
x−zϕ(z)dz,
wherecis any number betweenaandb. LetΓ(z)denote the Gamma function, and write
H(x) =
0 ifx <0;
1 ifx >0;
1
2 ifx= 0;
for the Heaviside step function. Then one has the following well known identities:
(1) ForRe(z)>0,Γ(z) =M[e−x](z), and 12Γ(z2) = M[e−x2](z).
(2) ForRe(z)>0,1
z =M[H(1−x)](z).
2.2. PRELIMINARIES 47 The relation (2) can be generalized as follows: Let j ∈ Z, n ∈ N with n > j, then for Re(z)>−j we have:
Qjn(x) :=M−1[
n
Y
l=j
(z+l)−1](x) = 1
(n−j)!xj(1−x)n−jH(1−x). (2.2.2) Indeed, letB(z, w)be the Beta function defined forRe(z),Re(w)>0by:
B(z, w) :=
Z 1 0
xz−1(1−x)w−1dx = Γ(z)Γ(w) Γ(z+w). Note that
M
xj(1−x)n−jH(1−x) (z) =
Z 1 0
xz+j−1(1−x)n−jdx=B(z+j, n−j+ 1)
= Γ(z+j)Γ(n−j+ 1) Γ(z+n+ 1)
= Γ(z)Γ(n−j+ 1)Qj−1
l=0(z+l) Γ(z)Qn
l=0(z+l)
= (n−j)!
Qn
l=j(z+l).
We shortly writeQn(x) := Q1n(x). Ifφ ∈ Dc is a radial function onC, i.e. φ(z) = φ(|z|) = φ(r), withc <1thenM[φ(r)e−r2](z)exists for allz ∈Csuch thatRe(z)>0.
In order to relate the Mellin transform of a function to Toeplitz operators, we need the following simple observation:
Proposition 2.2.2. Letφ ∈ E be a radial function onC, and letk∈N0. Then:
(1) Forl∈N0, we have:
P(φrkeilθ) = 2M[φe−r2](l+k+ 2)zl l!. (2) Forl∈ {· · · ,−2,−1}, it holdsP(φrkeilθ) = 0.
Proof. (1) : Forl ∈N0
P(φrkeilθ) =hφrkeilθ, K(·, z)i
=X
n>0
φrkeilθ,znrneinθ n!
= 1 π
X
n>0
Z 2π 0
ei(l−n)θdθ Z ∞
0
rk+n+1φ(r)e−r2drzn n!
= 2M[φe−r2](l+k+ 2)zl l!. (2) : The second assertion follows by a similar calculation.
48 CHAPTER 2. COMMUTING TOEPLITZ OPERATORS