# Toeplitz Operators, Pseudo-Homogeneous Symbols, and Moment Maps on the Complex Projective Space

Authors: Miguel Antonio Morales-Ramos; Raul Quiroga-Barranco; Armando Sanchez-Nungaray
Journal: Journal of Function Spaces
 (2017)

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## Abstract

Following previous works for the unit ball due to Nikolai Vasilevski, we define quasi-radial pseudo-homogeneous symbols on the projective space and obtain the corresponding commutativity results for Toeplitz operators. A geometric interpretation of these symbols in terms of moment maps is developed. This leads us to the introduction of a new family of symbols, extended pseudo-homogeneous, that provide larger commutative Banach algebras generated by Toeplitz operators. This family of symbols provides new commutative Banach algebras generated by Toeplitz operators on the unit ball.


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## Body

## 1. Introduction

The study of commutative algebras generated by Toeplitz operators has been a very interesting subject in the last decade. The first works studied commutativeC∗-algebras generated by Toeplitz operators on the unit disk and the unit ball, where an important connection with the geometry of the underlying space was found (see [1–3]). In particular, the commutativity of the C∗-algebras generated by Toeplitz operator with symbols invariant under the action of maximal Abelian groups was proved. A dual study was carried out for the complex projective space Pn(C) in [4]. It was proved that results similar to those of the unit ball hold for projective spaces, but in this case the symbols that yield commutative C∗-algebras are those that depend only on the radial part of the homogeneous coordinates.On the other hand, Vasilevski [5] introduced a new family of commutative algebras generated by Toeplitz operators which are Banach algebras but not C∗-algebras. These Banach algebras are those generated by Toeplitz operators whose symbols are the so-called quasi-radial quasi-homogeneous symbols on the unit ball. This was again brought into the setup of the projective space in [6]. In this latter work, it was proved that the quasi-radial quasi-homogeneous symbols yield commutative Banach algebras generated by Toeplitz operators on each weighted Bergman space of the complex projective space. A remarkable fact found in [6] is that a geometric description of the symbols was also given by proving that the quasi-homogeneous symbols can be associated with an Abelian subgroup of holomorphic isometries and that one can also construct Lagrangian foliations over principal bundles over such Abelian subgroups. Furthermore, this was proved for both the projective space and the unit ball.On the other hand, García and Vasilevski considered in [7] new families of commutative Banach algebras generated by Toeplitz operators over the Bergman space of the two-dimensional unit ball. These are obtained by a generalization of the quasi-homogeneous symbols over the two-dimensional unit ball.Recently Vasilevski [8] introduced two further and more general commutative Banach algebras over the unit ball. These algebras are generated by Toeplitz operator where the symbols are called quasi-radial pseudo-homogeneous symbols, with a variation to consider a product of spheres or a single sphere. Moreover, Vasilevski showed that the quasi-radial quasi-homogeneous symbols of both types have interesting properties. More precisely, if a and b are symbols from the product of spheres type, then Tab=TaTb. However, when the symbols a and b are from the case of a single sphere, then one still has TbTa=TaTb but Tab≠TaTb.Our goal is to extend the existence of such type of commutative Banach algebras generated by Toeplitz operators from the unit ball to the setup of the complex projective space. Thus we introduce the quasi-radial pseudo-homogeneous symbols for projective spaces, both the multisphere and single-sphere cases. Furthermore, we prove that the pseudo-homogeneous symbols have a geometric background which is supplied by the moment maps associated with Hamiltonian actions. The latter are given in our case by toral actions on projective spaces that are restricted to the usual toral actions on complex spaces. In particular, our geometric interpretation for the pseudo-homogeneous symbols applies to the case of the unit ball as well. This allows us to raise some question about the existence of larger families of symbols that properly contain the pseudo-homogeneous ones. This motivates us to consider what we called extended pseudo-homogeneous symbols that embrace all the possibilities discussed so far for both the unit ball and the projective space.The paper is organized as follows. After some preliminary remarks presented in Section2, in Sections 3 and 4 we prove the existence of projective versions (dual to the unit ball) of commutative Banach algebras generated by Toeplitz operators whose symbols are quasi-radial pseudo-homogeneous, first for the multisphere case and then for the single-sphere case. This is achieved on each weighted Bergman space. In Section 5 we consider some aspects of symplectic geometry as Hamiltonian actions, moment maps, symplectic reductions, and Delzant polytopes. Using these tools, we give a geometric description of quasi-radial pseudo-homogeneous symbols and we indicate how these techniques may be used to extend these families of symbols. Thus, we introduce in Section 6 the extended pseudo-homogeneous symbols, which contain many families contained up to this point. In particular, we exhibit commutative Banach algebras generated by Toeplitz operator with extended quasi-radial pseudo-homogeneous symbols. It is important to note that for this kind of symbols we have that Tab≠TaTb in general. Finally, we show in Section 7 how to obtain commutative Banach algebras in the case of the unit ball for our extended pseudo-homogeneous symbols.


## 2. Geometry and Analysis onPn(C)

On then-dimensional complex projective space Pn(C) and for every j=1,…,n we consider the open set(1)Uj=w∈PnC:wj≠0,where the natural homogeneous coordinates are given by the holomorphic chart φj:Uj→Cn defined by(2)φjw=1wjw0,…,w^j,…,wn=z1,…,zn.These yield the holomorphic atlas of Pn(C).The canonical Kähler structure onPn(C) is given by the closed (1,1)-form(3)ω=i∂∂¯log⁡fjon Uj for every j=0,…,n, where(4)fjw=∑k=0nwkw¯kwjw¯j=1+∑k=1nzkz¯k.The corresponding Riemannian metric is the well-known Fubini-Study metric. The volume element of Pn(C) with respect to the Fubini-Study metric is defined by(5)Ω=12πnωn.On the other hand, the hyperplane line bundle onPn(C) is given by(6)H=w,λ∈PnC×Cn+1:λ∈Cw∗,which assigns to every point in Pn(C) the dual of the line in Cn+1 that such point represents. The bundle H has a natural Hermitian metric h inherited from the usual Hermitian inner product on Cn+1. As usual, let us denote by Hm the mth tensor power of H and by h(m) its corresponding Hermitian inner product. Hence, we denote by L2(Pn(C),Hm) the L2-completion of the space of smooth sections Γ(Pn(C),Hm) with respect to the Hermitian inner product:(7)ζ,ξm=∫PnChmζ,ξΩ,where ζ,ξ∈Γ(Pn(C),Hm). Since Pn(C) is compact, the space of global holomorphic sections of Hm, denoted by Am2(Pn(C)), is finite dimensional and so closed in L2(Pn(C),Hm). This yields the so-called weighted Bergman space on Pn(C) with weight m.We note that there is a trivialization ofHm on U0 given by (8)Cn×C⟶HmU0z,c⟼1,z,λ1,z⟼cλ.This map together with φ0 yields an isometry L2(Pn(C),Hm)→L2(Cn,νm), where(9)νm=n+m!πnm!dVz1+z12+⋯+zn2n+m+1,and dV(z) denotes the Lebesgue measure on Cn (see [4] for further details). With respect to this isometry, the Bergman space Am2(Pn(C)) is identified with the space of polynomials of degree at most m on Cn which is denoted by Pm(Cn).In the rest of this work we will use the multi-index notation without further notice. In particular, the set of monomialszα on z=(z1,…,zn)∈Cn of degree at most m is a basis for Am2(Pn(C)). This basis is indexed by the set Jn(m)={α∈Nn:|α|≤m}. And it is easy to compute that(10)m!α!m-α!1/2zα:α∈Jnmis an orthonormal basis of Am2(Pn(C)).On the other hand, the Bergman projection forAm2(Pn(C)) is given by(11)Bmψz=n+m!πnm!∫Cnψw1+z1w¯1+⋯+znw¯nm1+w1w¯1+⋯+wnw¯nn+m+1dVw.Finally, the Toeplitz operator Ta on Am2(Pn(C)) with bounded symbol a∈L∞(Pn(C)) is given by(12)Taf=Bmaf,for every f∈Am2(Pn(C)).


## 3. Quasi-Radial Pseudo-Homogeneous Symbols

The symbols introduced in this section follow those considered in [8] as well as the notions developed in [6].Letk=(k0,…,kl)∈Nl+1 be a multi-index so that |k|=n+1. This partition provides a decomposition of the coordinates of w∈Cn+1 as w=(w(0),…,w(l)), where(13)wj=wk0+⋯+kj-1+1,…,wk0+⋯+kj=wj,1,…,wj,kj,for every j=0,…,l, and the empty sum is 0 by convention. We further decompose w∈Cn+1 as follows. For every j=0,…,l we define rj=|w(j)|. And for any such j we write(14)wj,l=rjsj,ltj,l,where(15)sj,l=wj,lrj,tj,l=wj,lwj,l.Correspondingly, we write s(j)=(sj,1,…,sj,kj) and t(j)=(tj,1,…,tj,kj)∈Tkj. We observe that for every j we have s(j)∈S+kj-1=Skj-1∩R+kj, the subset of the real (kj-1)-sphere with nonnegative coordinates.Definition 1.
A symbolψ∈L∞(Pn(C)) is called k-pseudo-homogeneous if it has the form(16)ψw=bs0,…,sltp=bs0,…,sl∏j=0ltjpj,where b(s0,…,sl)∈L∞(S+k0-1×⋯×S+kl-1) and p=(p0,…,pn)∈Zn+1 with |p|=0.A particularly useful case is that whereb(s(0),…,s(l))=∏j=0l‍bj(s(j)) with bj∈L∞(S+kj-1), j=0,…,l.It is easily seen from the definition of the variables involved that the value of a pseudo-homogeneous symbol is well defined in terms of homogeneous coordinates. In fact, we observe that the condition|p|=0 is used to have a well-defined expression in Definition 1. This can be checked, for example, following the arguments from Section 3 in [6]. Furthermore, the proof of Lemma 3.6 from the previous reference ensures that we can take k0=1 without any loss of generality. Hence, from now on we will consider partitions of the form (1,k) where k∈Nl is a partition of n and we will also call the corresponding symbols k-pseudo-homogeneous. With such restriction, the general form of a pseudo-homogeneous symbol is given by(17)ψw=bs1,…,sltp=bs1,…,sl∏j=1ltjpj,since s(0)=t(0)=1.This convention allows us to consider a natural corresponding expression for thek-pseudo-homogeneous symbols on Cn through the canonical embedding Cn↪Pn(C) given by the chart φ0. More precisely, the k-pseudo-homogeneous symbol ψ associated with p and b on Cn is given by(18)z⟼ψ1,z=bs1,…,sl∏j=1ltlpj.Next, we consider the quasi-radial symbols onPn(C) as first introduced in [4].Definition 2.
Letk=(k0,…,kl)∈Nl+1 be a partition of n+1. A k-quasi-radial symbol is a function a∈L∞(Pn(C)) that satisfies(19)aw=a~w0,…,wlfor some function a~:[0,+∞)l+1→C which is homogeneous of degree 0.A similar discussion as the one provided above applies in this case (see [4]). Hence, we will consider k0=1 and k-quasi-radial symbols on Cn↪Pn(C) for k∈Nl. Their corresponding expression is given by(20)z⟼a1,z=a~z1,…,zldefined for z∈Cn and a~:Rl→C some function.Collecting the previous two types of symbols we obtain the following.Definition 3.
Letk∈Nl be a partition of n. A k-quasi-radial k-pseudo-homogeneous symbol is a function Pn(C)→C of the form aψ where a is k-quasi-radial symbol and ψ is a k-pseudo-homogeneous symbol.By the previous discussion, for a partitionk∈Nl of n we will consider symbols defined on Cn of the form(21)ψz=az1,…,zl∏j=1lbjsjtjpj,where p∈Zl satisfies |p|=0.It follows from Lemma3.8 in [6] that for a given k-quasi-radial symbol a∈L∞(Cn) the Toeplitz operator Ta acting on Am2(Pn(C))≃Pm(Cn) satisfies(22)Tazα=γa,k,mαzαfor every α∈Nl, where γa,k,m:Nl→C is given by(23)γa,k,mα=n+m!m-α!∏j=1lkj-1+αj!∫R+lar1,…,rl1+rn+m+1∏j=1lrjαj+kj-1drj.We now extend to quasi-radial pseudo-homogeneous symbols the previous result and the corresponding one for quasi-homogeneous symbols found in [6]. Note that the following computation is dual to the one found in [8]. In what follows we will denote bj(s(j))=bj(sj,1,…,sj,kj). We will also consider the following sets:(24)B+kj-1=x∈R+kj-1∣x12+⋯+xkj-12≤1Δkj-1=x∈R+kj-1∣x1+⋯+xkj-1≤1.Theorem 4.
Letψ∈L∞(Cn) be a k-quasi-radial k-pseudo-homogeneous symbol whose expression is given as in (21). Then, the Toeplitz operator Tψ acting on the weighted Bergman space Am2(Pn(C))≃Pm(Cn) satisfies(25)Tψzα=γψ,mαzα+pifα+p∈Jnm0ifα+p∉Jnmfor every α∈Jn(m), where(26)γψ,mα=n+m!m-α!∏j=1lkj-1+αj+1/2pj!×∫R+lar1,…,rl∏j=1lrjαj+1/2pj+kj-1drj1+r1+⋯+rnn+m+1×∏j=1lkj-1+αj+1/2pj!∏l=1kjαj,l+pj,l!∫Δkj-1bjsj1-∑l=1kj-1sj,lαj,kj+1/2pj,kj∏l=1kj-1sj,lαj,l+1/2pj,ldsj,l.Proof.
It is enough to compute the following inner product in the Bergman spaceAm2(Pn(C))≃Pm(Cn).(27)Tψzα,zβ=ψzα,zβ=n+m!πnm!∫Cnψzzαz¯β1+z12+⋯+zn2n+m+1dVz=n+m!πnm!∫R+nar1,…,rl∏j=1lbjsj∏j=1nzjαj+βj+11+z12+⋯+zn2n+m+1∏j=1ndzj×∏u=1n∫Ttuαu+pu-βudtuitu.The last product of integrals is nonzero if and only if β=α+p, and in this case it equals (2π)n. Hence, we will assume that β=α+p for which we have(28)=2nn+m!m!∫R+nar1,…,rl∏j=1lbjsj∏j=1nzj2αj+pj+11+z12+⋯+zn2n+m+1∏j=1ndzj.If for every j=1,…,l we denote by dSj the volume element on Skj-1, apply spherical coordinates on R+kj, and use the identity rj,l=rjsj,l, then we now have(29)=2nn+m!m!∫R+l×∏j=1lSkj-1ar1,…,rl1+r12+⋯+rl2n+m+1×∏j=1lbjsj∏l=1kjrjsj,l2αj,l+pj,l+1∏j=1lrjkj-1drjdSj=2nn+m!m!∫R+lar1,…,rl∏j=1lrj2αj+pj+2kj-11+r12+⋯+rl2n+m+1∏j=1ldrj×∏j=1l∫Skj-1bjsj∏l=1kjsj,l2αj,l+pj,l+1dSj=2n-ln+m!m!∫R+lar1,…,rl∏j=1lrjαj+1/2pj+kj-11+r1+⋯+rln+m+1∏j=1ldrj×∏j=1l∫Skj-1bjsj∏l=1kjsj,l2αj,l+pj,l+1dSj=n+m!m!∫R+lar1,…,rl∏j=1lrjαj+1/2pj+kj-11+r1+⋯+rln+m+1∏j=1ldrj×∏j=1l2kj-1∫Skj-1bjsj∏l=1kjsj,l2αj,l+pj,l+1dSj.Applying the identity sj,kj=1-(sj,12+⋯+sj,kj-12) together with the corresponding expression of dSj in terms of the coordinates sj,1,…,sj,kj-1 on B+kj-1 we have(30)=n+m!m!∫R+lar1,…,rl∏j=1lrjαj+1/2pj+kj-11+r1+⋯+rln+m+1∏j=1ldrj×∏j=1l2kj-1∫B+kj-1bjsj∏l=1kj-1sj,l2αj,l+pj,l+11-∑l=1kj-1sj,l2αj,kj+1/2pj,kj∏l=1kj-1dsj,l=n+m!m!∫R+lar1,…,rl∏j=1lrjαj+1/2pj+kj-11+r1+⋯+rln+m+1∏j=1ldrj×∏j=1l∫Δkj-1bjsj∏l=1kj-1sj,lαj,l+1/2pj,l1-∑l=1kj-1sj,lαj,kj+1/2pj,kj∏l=1kj-1dsj,l.On the other hand, we have(31)zα+p,zα+pm=α+p!m-α!m!=m-α!∏j=1l∏l=1kjαj,l+pj,l!m!,because we assumed that |p|=0. The result now follows from this.The following is a particular case.Corollary 5.
Forψ as above, one assumes that |p(j)|=0 for every j=1,…,l. Then, the function γψ,m:Jn(m)→C from Theorem 4 satisfies the following:(32)γψ,mα=n+m!m-α!∏j=1lkj-1+αj!∫R+lar1,…,rl∏j=1lrjαj+kj-1drj1+r1+⋯+rnn+m+1×∏j=1lkj-1+αj!∏l=1kjαj,l+pj,l!∫Δkj-1bjsj1-∑l=1kj-1sj,lαj,kj+1/2pj,kj∏l=1kj-1sj,lαj,l+1/2pj,ldsj,l,for every α∈Jn(m) such that α+p∈Jn(m), and it is zero otherwise.We observe that Corollary5 reduces to formula (23) for quasi-radial symbols. Furthermore, for a symbol of the form(33)ψw=bjsjtjpj,where j is fixed and p∈Zn satisfies |p(j)|=0 and p(l)=0 for every l≠j, Corollary 5 gives the following expression:(34)γbj,k,pjα=kj-1+αj!∏l=1kjαj,l+pj,l!∫Δkj-1bjsj1-∑l=1kj-1sj,lαj,kj+1/2pj,kj∏l=1kj-1sj,lαj,l+1/2pj,ldsj,l.In particular, the Toeplitz operator with this latter symbol ψ is independent of the weight m.As a consequence of the previous discussion we have the following result.Corollary 6.
Letψ be a symbol as in Corollary 5. Then, one has(35)γψ=γa,k,m∏j=1lγbj,k,pj.Furthermore, the Toeplitz operators Ta, Tbjt(j)p(j), for j=1,…,m, pairwise commute and(36)Ta∏j=1lbjtjpj=Ta∏j=1lTbjtjpj.In particular, the Toeplitz operators for these symbols generate a commutative Banach algebra.


## 4. Single-Sphere Pseudo-Homogeneous Symbols

For a givenz∈Cn we consider the decomposition(37)zu=ρutu,where ρu=|zu|∈R+ and tu∈T, for every u=1,…,n. We also introduce the coordinates(38)σu=ρur,for every u=1,…,n, where r=|z|. Hence, we have σ=(σ1,…,σn)∈S+n-1. And with this notation we can write(39)z=rσt=rσ1t1,…,σntn.Definition 7.
For a partitionk∈Nl of n, a symbol ψ∈L∞(Pn(C)) is called single-sphere k-pseudo-homogeneous if it has the form(40)ψw=bσ∏j=1ltjpj,where σ∈S+n-1 and t∈Tn are computed for z=(w1,…,wn)/w0, the function b∈L∞(S+n-1), and p∈Zn satisfies |p(j)|=0 for every j=1,…,n.We observe that with this definition the partb(σ) of the symbols is well defined since it is given in terms of homogeneous coordinates. And as before the condition on p ensures that the part depending on t is well defined as well; this has to be considered since the symbol is a function of the class [w] and not just the corresponding homogeneous coordinate z. On the other hand, note that ψ is given on the complement of a hyperplane of Pn(C) and so it is well defined as an L∞ function.The difference between the single-spherek-pseudo-homogeneous symbols and the k-pseudo-homogeneous symbols from Definition 1 is that for the latter we partition the coordinates into several spheres according to the partition k. In particular, we can refer to the symbols from Definition 1 as multisphere k-pseudo-homogeneous symbols.As before, a particularly interesting case is the one for which we have(41)bσ=∏j=1lbjσj,where bj∈L∞(S+n-1∩Rkj), where Rkj is the subspace of Rn corresponding to the coordinates (kj-1+1,…,kj-1+kj). For simplicity, we will denote by χj this latter set. Also note that if σ∈S+n-1 then we have for every j that σ(j)∈S+n-1∩Rkj essentially by definition.We now fix a partitionk as above and we will now consider a symbol of the form(42)ψjz=bjσjtjpj,where pj∈Zkj. As before, we can assume our symbol defined for z∈Cn. Also note that for ψj the multi-index pj needs to be given only on the coordinates in the set χj, but it still has to satisfy |pj|=0. Since we need to consider multi-indexes for all the coordinates of Cn, we introduce the following notation. Given pj∈Zkj, we will denote by p~j∈Zn the multi-index such that(43)p~jv=pj,ifv=j,0,ifv≠j.In other words, p~j vanishes outside the coordinates in χj and coincides with pj in these coordinates.Theorem 8.
Letψj∈L∞(Cn) be a single-sphere k-pseudo-homogeneous symbol whose expression is given as in (42). Then, the Toeplitz operator Tψj acting on the weighted Bergman space Am2(Pn(C))≃Pm(Cn) satisfies(44)Tψjzα=γψj,mαzα+p~j,ifα+p~j∈Jnm,0,ifα+p~j∉Jnm,for every α∈Jn(m), where(45)γψj,mα=α+n-1!α-αj+n-kj-1!∏u∈χjαu+pu!×∫Δkjbjσj1-σj,1+⋯+σj,kjα-αj+n-kj-1×∏u∈χjσuαu+1/2p~judσu.Proof.
As before we proceed to calculate as follows:(46)Tψjzα,zβm=ψjzα,zβm=n+m!πnm!∫Cnbjsjtjpjzαz¯βdVz1+z12+⋯+zn2n+m+1=n+m!πnm!∫R+nbjσj∏u=1nρuαu+βu+11+ρ12+⋯+ρn2n+m+1dρ×∏u=1n∫Ttuαu+p~ju-βudtuitu.The last product of integrals is nonzero precisely when β=α+p~j, in which case its value is (2π)n. So we now assume that β=α+p~j(47)=2nn+m!m!∫R+nbjσj∏u=1nρu2αu+p~ju+11+ρ12+⋯+ρn2n+m+1dρ.We now replace ρu=rσu for every u=1,…,n and apply spherical coordinates in R+n to obtain the following where dσ is the volume element of S+n-1:(48)=2nn+m!m!∫R+×S+n-1bjσj∏u=1nrσu2αu+p~ju+11+r2n+m+1rn-1drdσ=2nn+m!m!∫R+r2α+2n-11+r2n+m+1dr∫S+n-1bjσj∏u=1nσu2αu+p~ju+1dσ=2n-1n+m!m!∫R+rα+n-11+rn+m+1dr∫S+n-1bjσj∏u=1nσu2αu+p~ju+1dσ=α+n-1!m-α!m!2n-1∫S+n-1bjσj∏u=1nσu2αu+p~ju+1dσ.Now we parametrize S+n-1 by n-1 coordinates as before. To simplify our computations we assume that n∉χj so that (p~j)n=0(49)=α+n-1!m-α!m!×2n-1∫B+n-1bjσj∏u=1n-1σu2αu+p~ju+11-σ12+⋯+σn-12αn∏u=1n-1dσu=α+n-1!m-α!m!×∫Δn-1bjσj∏u=1n-1σuαu+1/2p~ju1-σ1+⋯+σn-1αn∏u=1n-1dσu.We now apply the change of coordinates σu=τu for u∈χj and σu=(1-(τj,1+⋯+τ(j),kj))τu for u∉χj(50)=α+n-1!m-α!m!×∫Δkjbjσj1-σj,1+⋯+σj,kjα-αj+n-kj-1∏u∈χjσuαu+1/2p~judσu×∫Δn-kj-11-∑u∉χj∪nτuαn∏u∉χj∪nτuαudτu=α+n-1!m-α!m!∏u∉χjαu!α-αj+n-kj-1!×∫Δkjbjσj1-σj,1+⋯+σj,kjα-αj+n-kj-1×∏u∈χjσuαu+1/2p~judσu.Since we can write(51)zα+p~j,zα+p~jm=m-α!∏u∈χjαu+pu!∏u∉χjαu!m!,note that the action of the operator Tψj does not depend on the parameter m. From the previous results we also obtain the following.Corollary 9.
For a partitionk∈Nl of n, let a∈L∞(Pn(C)) be a k-quasi-radial symbol and for every j=1,…,l let ψj∈L∞(Pn(C)) be a symbol of form (42). Then the Toeplitz operators Ta, Tψj pairwise commute on every weighted Bergman space Am2(Pn(C)).Remark 10.
It is easy to check thatTaψj≠TaTψj for suitably chosen symbols as above. This should be compared with Corollary 6.


## 5. Geometric Description of Symbols through Moment Maps

After introducing some of the basic notions associated with Hamiltonian actions we will use some of their basic properties. We refer to [9, 10] for further details and proofs of known facts.We will consider a compact groupG acting on a manifold M, which in our case will be compact as well. For simplicity and without loss of generality we will assume that the G-action is free: the stabilizer of every point is trivial. We also note that any action of a compact Lie group is necessarily proper.For everyX∈g we will denote by X♯ the vector field on M whose flow is exp⁡(tX). More precisely, we have(52)Xx♯=ddtt=0exp⁡tXx.We further assume that the manifold carries a symplectic form ω and that the G-action preserves ω. In particular, the 1-form given by(53)ιX♯ω=ωX♯,·is closed for every X∈g. The symplectic G-action on (M,ω) is called Hamiltonian if there exists a smooth map(54)μ:M⟶g∗,which is G-equivariant for the coadjoint G-action on g∗ (where g∗ denotes the dual vector space of the Lie algebra g) and if we have(55)dμX=ιX♯ω,for every X∈g. Here, the function μX∈C∞(M) is defined by (56)μX:M⟶RμXx=μx,X.The map μ is called the moment map of the Hamiltonian G-action on M.Marsden-Weinstein theorem allows performing what is known as the symplectic point reduction of a Hamiltonian action. More precisely, for the above setup, ifv∈g∗ is a regular point of the moment map μ, then the quotient(57)M0=μ-1vGadmits a manifold structure such that the quotient map(58)π:μ-1v⟶M0defines a principal bundle with structure group G. Furthermore, there exists a unique symplectic form ω0 on M0 that satisfies(59)π∗ω0=ωμ-1v. The symplectic manifold (M0,ω0) is called a symplectic point reduction of (M,ω).This provides a well-known alternative definition of the projective space that we now describe (see [10] for further details). On Cn+1 we consider the canonical symplectic form(60)ω′=12Im∑j=0ndwj∧dw¯j and the symplectic action (61)T×Cn+1⟶Cn+1,t,w⟼tw0,…,twn.This T-action is Hamiltonian with moment map (62)μ:Cn+1⟶R,w⟼12w2.We recall that R is the Lie algebra of T, and since the latter is Abelian the adjoint and coadjoint actions are trivial. The symplectic point reduction in this case reduces to the well-known fact that(63)μ-112=S2n-1/T≃PnC.It is also well known that the corresponding symplectic form is precisely the one introduced in Section 2.Our second fundamental example is given by the action(64)Tn×PnC⟶PnC,t,w⟼tw=t0w0,…,tnwn,where we have identified(65)Tn≃t0,…,tn∣tj∈T∀j=0,…,n,t0,…,tn=1.A straightforward computation shows that for the natural embedding Cn↪Pn(C) used before the subset Cn is Tn-invariant with restricted action given by(66)t·z=t1z1,…,tnzn,where Tn is now realized as the usual product of n circles T. This Tn-action is Hamiltonian. In fact, if we realize the Lie algebra of Tn as(67)R^n=x∈Rn+1∣x0+⋯+xn=12,then the moment map of the Tn-action on Pn(C) is given by (68)μn:PnC⟶R^n,μnw=12w2w02,…,wn2.Note that R^n is an n-dimensional affine space, but it carries a natural structure of vector space by choosing the point(69)12n+11,…,1as its origin.This last example can be generalized to the notion of toric manifolds (see [9, 10]).Definition 11.
A toric manifold is a compact symplectic manifoldM with dimension 2n together with an effective Hamiltonian Tn-action.A fundamental result due to Delzant establishes a one-to-one correspondence between toric manifolds and a certain family of polytopes.Theorem 12 (Delzant).
IfM is 2n-dimensional toric manifold with moment map μ, then the image μ(M) is a polytope that satisfies the following properties.
Simple. There are n edges meeting at each vertex.
Rational. The edges meeting at a vertex p are of the form p+tvj for t∈R+, where v1,…,vn∈Zn.
Smooth. The vectors v1,…,vn can be chosen to be a basis for Zn.
A convex polytope satisfying these properties is called a Delzant polytope. Furthermore, there is a one-to-one correspondence(70)toricmanifolds⟶DelzantpolytopesM,ω,μ⟼μM.Delzant’s theorem applied to the aboveTn-action on Pn(C) yields the following Delzant polytope given by the following expression and their corresponding realizations: (71)μnPnC=x∈Rn+1∣x0+⋯+xn=12≃x∈Rn+1∣x0+⋯+xn=1≃x∈R+n∣x1+⋯+xn≤1,and the latter is precisely the set Δn introduced in (24). As a consequence, we obtain the following remark.Remark 13.
The expressions found in Sections3 and 4 that describe the action on monomials of the Toeplitz operators with pseudo-homogeneous symbols (both single and multisphere) are all given as integrals on the Delzant polytopes of projective spaces. Furthermore, the same remark applies to the corresponding results for the unit ball: Lemmas 3.2 and 5.2 from [8]. These facts allow us to raise the following questions. (i)
Is there a relationship between the quasi-radial pseudo-homogeneous symbols and Delzant polytopes?(ii)
With the introduction of the notion of Delzant polytopes, is it possible to infer the existence of further symbols whose associated Toeplitz operators can be explicitly described on monomials?We now proceed to answer the first question in Remark13. Hence, we fix a partition k∈Zl of n as before.


### 5.1. Geometric Description ofk-Quasi-Radial Symbols

Let us consider the action (64) restricted to the torus Tl(72)Tl×PnC⟶PnC,t,w⟼t0w0,t1w1,…,tlwl.As before we have made the identification(73)Tl≃t0,…,tl∣tj∈T∀j=0,…,l,t0,…,tl=1.Since the previous action is the restriction of a Hamiltonian action it is Hamiltonian as well. Furthermore, the corresponding moment map is given by (74)μl:PnC⟶R^l,w⟼12w2w02,w12,…,wl2.This follows from the fact that the Tl-action is a restriction of the action given by (64), for which the moment map is already known, and the results in Section 4.5 are from [9].The following result describes thek-quasi-radial symbols as the pullback under the moment map μl of bounded functions on the polytope of such map.Theorem 14.
A bounded symbola∈L∞(Pn(C)) is k-quasi-radial if and only if it belongs to μl∗(L∞(Pl)), where Pl=μl(Pn(C)), the polytope associated with the moment map μl. More precisely, a is k-quasi-radial if and only if there exists a^∈L∞(Pl) such that a=a^∘μl.Proof.
Suppose thata=a^∘μl for some a^∈L∞(Pl). Then, we can write(75)aw=a^w022w2,w122w2,…,wl22w2,which is clearly the composition of the map w↦(|w0|,|w(1)|,…,|w(l)|) with a homogeneous function of degree 0. Hence, a is k-quasi-radial by Definition 2.
Conversely, suppose thata is a bounded k-quasi-radial symbol. By definition we can write(76)aw=a~w0,w1,…,wl,for some function a~ which is homogeneous of degree 0. In particular, we have(77)aw=a~w0w,w1w,…,wlw,and then we see that a belongs to μl∗(L∞(Pl)) by a trivial change of coordinates.


### 5.2. Geometric Description of Symbols of the Formtp

Besides the partitionk we now fix p∈Zn such that |p(j)|=0 for every j=1,…,l. In particular, and according to the notation from Section 3, the symbol (78)PnC⟶C,w⟼tp=∏j=1ltjpjis k-pseudo-homogeneous.Corresponding to theTl-action on Pn(C) given by (72) we have an induced quotient map (79)πk:PnC⟶∏j=1kjPkj-1C,w⟼w1,…,wl.On the other hand, the condition|p(j)|=0 for every j=1,…,l implies that the symbol tp is Tl-invariant. As a consequence we now have the following.Theorem 15.
Every symbol of the typetp belongs to πk∗(L∞(∏j=1kj‍Pkj-1(C))). More precisely, one has(80)tp∣p∈Zn,suchthatpj=0foreveryj=1,…,l⊂f∘πk∣f∈L∞∏j=1kjPkj-1C.In other words, any such symbol can be considered as a function on the product of projective spaces ∏j=1kj‍Pkj-1(C).


### 5.3. Geometric Description of the Pseudo-Homogeneous Symbols of the Formb(s(1),…,s(l))

We will now consider a bounded functionb∈L∞(S+k1-1×⋯×S+kl-1) and the corresponding k-pseudo-homogeneous symbol (81)PnC⟶C,w⟼bs1,…,sl,where(82)sj=1wjwj,1,…,wj,kj,for all j=1,…,l.From this description, it is easy to see that the symbol considered above is invariant under theTn-action given by (64). We now conclude the following result.Theorem 16.
Every symbol of the typeb(s(1),…,s(l)) belongs to μn∗(L∞(Dn)) where Dn=μn(Pn(C)) is the Delzant polytope of the projective space Pn(C). More precisely, one has(83)bs1,…,sl∣b∈L∞S+k1-1×⋯×S+kl-1⊂f∘μn∣f∈L∞Dn.In other words, any such symbol can be considered as a function on the Delzant polytope of the projective space Pn(C).


### 5.4. Geometric Description of the Single-Sphere Pseudo-Homogeneous Symbols

By the previous subsections, to describe these symbols it is enough to consider those of the form(84)PnC⟶C,w⟼bσ,for b∈L∞(S+n-1), where we have followed the notation from Section 4. With the definition of the variables involved, these symbols are clearly invariant under the Tn-action given by (64). Hence, we obtain the following result.Theorem 17.
Every symbol of the typeb(σ) belongs to μn∗(L∞(Dn)) where Dn=μn(Pn(C)) is the Delzant polytope of the projective space Pn(C). More precisely, one has(85)bσ∣b∈L∞S+n-1⊂f∘μn∣f∈L∞Dn.In other words, any such symbol can be considered as a function on the Delzant polytope of the projective space Pn(C).


## 5.1. Geometric Description ofk-Quasi-Radial Symbols

Let us consider the action (64) restricted to the torus Tl(72)Tl×PnC⟶PnC,t,w⟼t0w0,t1w1,…,tlwl.As before we have made the identification(73)Tl≃t0,…,tl∣tj∈T∀j=0,…,l,t0,…,tl=1.Since the previous action is the restriction of a Hamiltonian action it is Hamiltonian as well. Furthermore, the corresponding moment map is given by (74)μl:PnC⟶R^l,w⟼12w2w02,w12,…,wl2.This follows from the fact that the Tl-action is a restriction of the action given by (64), for which the moment map is already known, and the results in Section 4.5 are from [9].The following result describes thek-quasi-radial symbols as the pullback under the moment map μl of bounded functions on the polytope of such map.Theorem 14.
A bounded symbola∈L∞(Pn(C)) is k-quasi-radial if and only if it belongs to μl∗(L∞(Pl)), where Pl=μl(Pn(C)), the polytope associated with the moment map μl. More precisely, a is k-quasi-radial if and only if there exists a^∈L∞(Pl) such that a=a^∘μl.Proof.
Suppose thata=a^∘μl for some a^∈L∞(Pl). Then, we can write(75)aw=a^w022w2,w122w2,…,wl22w2,which is clearly the composition of the map w↦(|w0|,|w(1)|,…,|w(l)|) with a homogeneous function of degree 0. Hence, a is k-quasi-radial by Definition 2.
Conversely, suppose thata is a bounded k-quasi-radial symbol. By definition we can write(76)aw=a~w0,w1,…,wl,for some function a~ which is homogeneous of degree 0. In particular, we have(77)aw=a~w0w,w1w,…,wlw,and then we see that a belongs to μl∗(L∞(Pl)) by a trivial change of coordinates.


## 5.2. Geometric Description of Symbols of the Formtp

Besides the partitionk we now fix p∈Zn such that |p(j)|=0 for every j=1,…,l. In particular, and according to the notation from Section 3, the symbol (78)PnC⟶C,w⟼tp=∏j=1ltjpjis k-pseudo-homogeneous.Corresponding to theTl-action on Pn(C) given by (72) we have an induced quotient map (79)πk:PnC⟶∏j=1kjPkj-1C,w⟼w1,…,wl.On the other hand, the condition|p(j)|=0 for every j=1,…,l implies that the symbol tp is Tl-invariant. As a consequence we now have the following.Theorem 15.
Every symbol of the typetp belongs to πk∗(L∞(∏j=1kj‍Pkj-1(C))). More precisely, one has(80)tp∣p∈Zn,suchthatpj=0foreveryj=1,…,l⊂f∘πk∣f∈L∞∏j=1kjPkj-1C.In other words, any such symbol can be considered as a function on the product of projective spaces ∏j=1kj‍Pkj-1(C).


## 5.3. Geometric Description of the Pseudo-Homogeneous Symbols of the Formb(s(1),…,s(l))

We will now consider a bounded functionb∈L∞(S+k1-1×⋯×S+kl-1) and the corresponding k-pseudo-homogeneous symbol (81)PnC⟶C,w⟼bs1,…,sl,where(82)sj=1wjwj,1,…,wj,kj,for all j=1,…,l.From this description, it is easy to see that the symbol considered above is invariant under theTn-action given by (64). We now conclude the following result.Theorem 16.
Every symbol of the typeb(s(1),…,s(l)) belongs to μn∗(L∞(Dn)) where Dn=μn(Pn(C)) is the Delzant polytope of the projective space Pn(C). More precisely, one has(83)bs1,…,sl∣b∈L∞S+k1-1×⋯×S+kl-1⊂f∘μn∣f∈L∞Dn.In other words, any such symbol can be considered as a function on the Delzant polytope of the projective space Pn(C).


## 5.4. Geometric Description of the Single-Sphere Pseudo-Homogeneous Symbols

By the previous subsections, to describe these symbols it is enough to consider those of the form(84)PnC⟶C,w⟼bσ,for b∈L∞(S+n-1), where we have followed the notation from Section 4. With the definition of the variables involved, these symbols are clearly invariant under the Tn-action given by (64). Hence, we obtain the following result.Theorem 17.
Every symbol of the typeb(σ) belongs to μn∗(L∞(Dn)) where Dn=μn(Pn(C)) is the Delzant polytope of the projective space Pn(C). More precisely, one has(85)bσ∣b∈L∞S+n-1⊂f∘μn∣f∈L∞Dn.In other words, any such symbol can be considered as a function on the Delzant polytope of the projective space Pn(C).


## 6. Extended Pseudo-Homogeneous Symbols

In this section we bring upfront the second question from Remark13.In the geometric description of symbols considered in Section5 we have observed some proper inclusions in Theorems 15, 16, and 17. This certainly suggests that there could be some room to consider additional symbols for which the description of their Toeplitz operators is possible and useful. In particular, further study of these symbols and their geometric description seems to be needed. We now present a new set of symbols that support these claims. As before, k∈Nl is a fixed partition of n.Definition 18.
A symbolψ∈L∞(Pn(C)) is called extended k-pseudo-homogeneous if it has the form(86)ψw=br1,…,rl,s1,…,sltp=br1,…,rl,s1,…,sl∏j=1ltjpj,where b∈L∞([0,∞)l×S+k1-1×⋯×S+kl-1) and p∈Zn satisfies |p|=0.We observe that the set of quasi-radial pseudo-homogeneous symbols considered in the previous sections is a proper subset of the set of extended pseudo-homogeneous symbols. Furthermore, in the notation of Section4 we have(87)σ=1r12+⋯+rl2r1s1,…,rlsl,from which it follows that every single-sphere quasi-radial pseudo-homogeneous symbol is also an extended pseudo-homogeneous symbol. In other words, all the symbols considered in previous sections are particular cases of the extended pseudo-homogeneous symbols.A particular case that will be interesting to consider is given by symbols of the form(88)ψw=ar1,…,rl∏j=1lbjr1,…,rl,sjtjpj,where we continue to assume conditions as in Definition18. Following the previous remarks, we observe that the symbols in (88) have as particular case those considered by expressions (21) and (42).Theorem 19.
Letψ∈L∞(Cn) be an extended k-pseudo-homogeneous symbol whose expression is given as in (88). Then, the Toeplitz operator Tψ acting on the weighted Bergman space Am2(Pn(C))≃Pm(Cn) satisfies(89)Tψzα=γψ,mαzα+p,ifα+p∈Jnm,0,ifα+p∉Jnm,for every α∈Jn(m), where (90)γψ,mα=n+m!m-α!∏j=1lkj-1+αj+1/2pj!∫R+lar1,…,rl1+r1+⋯+rln+m+1×∫∏j=1lΔkj-1∏j=1lkj-1+αj+1/2pj!∏l=1kjαj,l+pj,l!1-∑l=1kj-1sj,lαj,kj+1/2pj,kj×bjr1,…,rl,sj∏l=1kj-1sj,lαj,l+1/2pj,ldsj,l∏j=1lrjαj+1/2pj+kj-1drj.Proof.
We compute the inner product(91)Tψzα,ζβ=ψzα,zβ=n+m!πnm!∫Cnψzzαz¯β1+z12+⋯+zn2n+m+1dVz=n+m!πnm!∫R+nar1,…,rl∏j=1lbjr1,…,rl,sj1+ρ12+⋯+ρn2n+m+1∏u=1nρuαu+βu+1dρu×∏u=1n∫Ttuαu+pu-βudtuitu.This vanishes unless β=α+p, and in this case the computation continues by introducing spherical coordinates in each variable z(j)(92)=2nn+m!m!∫R+l×∏j=1lSkj-1ar1,…,rl1+r12+⋯+rl2n+m+1×∏j=1lbjr1,…,rl,sj∏l=1kjrjsj,l2αj,l+pj,l+1∏j=1lrjkj-1drjdSj=2nn+m!m!∫R+lar1,…,rl∏j=1lrj2αj+pj+kj-11+r12+⋯+rl2n+m+1×∫∏j=1lSkj-1∏j=1lbjr1,…,rl,sj∏l=1kjsj,l2αj,l+pj,l+1dSjdr=2n-ln+m!m!∫R+lar1,…,rl∏j=1lrjαj+1/2pj+kj-11+r1+⋯+rln+m+1×∫∏j=1lSkj-1∏j=1lbjr1,…,rl,sj∏l=1kjsj,l2αj,l+pj,l+1dSjdr.We now parametrize spheres by balls to obtain(93)=2n-ln+m!m!∫R+lar1,…,rl∏j=1lrjαj+1/2pj+kj-11+r1+⋯+rln+m+1×∫∏j=1lB+kj-11-∑l=1kj-1sj,l2αj,l+1/2pj,l×∏j=1lbjr1,…,rl,sj∏l=1kj-1sj,l2αj,l+pj,l+1dsj,ldr.And now balls by simplices(94)=n+m!m!∫R+lar1,…,rl∏j=1lrjαj+1/2pj+kj-11+r1+⋯+rln+m+1×∫∏j=1lΔkj-11-∑l=1kj-1sj,lαj,l+1/2pj,l×∏j=1lbjr1,…,rl,sj∏l=1kj-1sj,lαj,l+1/2pj,ldsj,ldr.Finally, we use the fact that(95)zα+p,zα+pm=α+p!m-α!m!=m-α!∏j=1l∏l=1kjαj,l+pj,l!m!,because |p|=0, to obtain the result.The following is a particular case.Corollary 20.
Forψ as above, one assumes that pj=0 for every j=1,…,l. Then, the function γψ,m:Jn(m)→C from Theorem 19 satisfies the following.(96)γψ,mα=n+m!m-α!∏j=1lkj-1+αj!∫R+lar1,…,rl1+r1+⋯+rln+m+1×∫∏j=1lΔkj-1∏j=1lkj-1+αj!∏l=1kjαj,l+pj,l!1-∑l=1kj-1sj,lαj,kj+1/2pj,kj×bjr1,…,rl,sj∏l=1kj-1sj,lαj,l+1/2pj,ldsj,l∏j=1lrjαj+kj-1drj.As a consequence we also obtain the following result.Corollary 21.
Letψ=a∏j=1l‍bjt(j)p(j) be a symbol as in Corollary 20. Then, the Toeplitz operators Ta, Tbjt(j)p(j) pairwise commute on every weighted Bergman space Am2(Pn(C)).Remark 22.
As in the case of Remark10 it is not difficult to prove that in the previous result and for suitable symbols we have Taψj≠TaTψj.


## 7. Extended Pseudo-Homogeneous Symbols on the Unit Ball

The extended pseudo-homogeneous symbols introduced in Section6 can be easily considered on the unit ball Bn. As before, we fix a partition k∈Nn. The notation for coordinates is the same as in the previous sections.Definition 23.
A symbolψ∈L∞(Bn) is called extended k-pseudo-homogeneous if it has the form(97)ψz=br1,…,rl,s1,…,sltp=br1,…,rl,s1,…,sl∏j=1ltjpj,where b∈L∞(B+l×S+k1-1×⋯×S+kl-1) and p∈Zn satisfies |p|=0.As before, a particular case that is important to consider is given by(98)ψz=ar1,…,rl∏j=1lbjr1,…,rl,sjtjpj.Using similar computations to those from Theorem 19 we obtain the following result. We leave the details to the reader. For the standard definitions of weighted Bergman spaces and Toeplitz operators on the unit ball Bn we refer to [8].Theorem 24.
Letψ∈L∞(Bn) be an extended k-pseudo-homogeneous symbol whose expression is given as in (98). Then, the Toeplitz operators Tψ action on the weighted Bergman space Aλ2(Bn) satisfies(99)Tψzα=γψ,λαzα+p,ifα+p≥0,0,ifα+p≱0,for every α≥0, where(100)γψ,λα=Γn+α+λ+1Γλ+1∏j=1lkj-1+αj+1/2pj!∫Δlar1,…,rl1-∑j=1lrjλ×∫∏j=1lΔkj-1∏j=1lkj-1+αj+1/2pj!∏l=1kjαj,l+pj,l!1-∑l=1kj-1sj,lαj,kj+1/2pj,kj×bjr1,…,rl,sj∏l=1kj-1sj,lαj,l+1/2pj,ldsj,l∏j=1lrjαj+1/2pj+kj-1drj.As a consequence, we have the following commutativity results.Corollary 25.
Forψ as above, one assumes that |p(j)|=0 for every j=1,…,l. Then, the function γψ,λ:Nn→C from Theorem 24 satisfies (101)γψ,λα=Γn+α+λ+1Γλ+1∏j=1lkj-1+αj!∫Δlar1,…,rl1-∑j=1lrjλ×∫∏j=1lΔkj-1∏j=1lkj-1+αj!∏l=1kjαj,l+pj,l!1-∑l=1kj-1sj,lαj,kj+1/2pj,kj×bjr1,…,rl,sj∏l=1kj-1sj,lαj,l+1/2pj,ldsj,l∏j=1lrjαj+kj-1drj.Corollary 26.
Letψ=a∏j=1l‍bjt(j)p(j) be a symbol as in Corollary 25. Then, the Toeplitz operators Ta,Tbjt(j)p(j) pairwise commute on every weighted Bergman space Aλ2(Bn).Remark 27.
In this case we have in general thatTaψj≠TaTψj as in the previous corresponding remarks.



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