Abstract
We found that in the polydisk there exist different classes of commutative -algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that the -algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutative -algebra of Toeplitz operators.
1. Introduction
In recent years, Vasilevski, Quiroga, and coauthors found a connection that the commutative -algebras generated by Toeplitz operators acting on the weighted Bergman space were described and classified for the case of the unit disk and the unit ball in ; see [1, 2] for further results and details. The classification result states that given a maximal Abelian subgroup of biholomorphisms of the unit ball or the unit disk (i.e., ball of dimension one), the -algebra generated by Toeplitz operators whose symbols are invariant under the action of is commutative on each weighted Bergman space.
For the unit ball of dimension there are models of maximal Abelian subgroups of biholomorphisms; all others are conjugated with one of them via biholomorphisms of the unit ball.
We apply the above method to the polydisk and we find that there exist different classes of commutative -algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. Moreover, we show that in each case the Toeplitz operator is unitary equivalent to a multiplication operator . The explicit form of is given.
The main result of this paper is that we relate to each Abelian subgroup (not necessary maximal) of biholomorphisms a -algebra generated by Toeplitz operators. We consider the symplectic manifold where is the usual symplectic form; using the fact that the action of biholomorphisms is Hamiltonian we give the moment map associated with each Abelian subgroup , where . Then we introduce the family of symbols which are given by functions of the form , where is an arbitrary bounded function defined on . Finally, we prove that the -algebra generated by Toeplitz operators with symbols in the family is commutative.
One important advantage of the point of view of the moment map is that if we consider the symbol set invariant under the action of a nonmaximal Abelian subgroup of biholomorphisms, then the -algebra generated by Toeplitz operators with symbols in this set may not be commutative. In a sense, this symbol set is too big for our purposes, so this method no longer works for nonmaximal Abelian subgroups.
The method to find commutative -algebras of Toeplitz operators using the moment map has not been used in other manifolds and we suppose that we can apply this technique to other symplectic manifolds.
2. Weighted Bergman Space and Bergman Projection on the Polydisk
Let be fixed. Throughout this paper we use the notation for a typical element and , . The unit polydisk is defined byLet ; the normalized -weighted volume measure is given bywhere and , .
The weighted Bergman space is the space of all holomorphic functions which belongs to . It is well known (see, e.g., [3]) that is a closed subspace of . The (weighted) Bergman projection has the formwhere the (weighted) Bergman kernel is given byFrom now on we denote by a vector of nonnegative integer numbers such that . Denote by the upper half-plane; that is,Let denote the following domain in :Consider the Hilbert space , where the normalized -weighted volume measure has the formDenote by the (weighted) Bergman subspace of . As always, the Bergman space is the subspace of the corresponding -space which consists of all analytic functions.
The Cayley transform is given byand the inverse Cayley transform acts by the ruleLet denote the identity map. The map given byis obviously a biholomorphism.
It is easy to check directly thatwhere .
The map generates the operatoracting by the ruleLet ; changing the variables and using (11) we haveThe operator is unitary; its inverse operatorhas the formIt is obvious that ; then the (weighted) Bergman space is a closed subspace of . Changing the variables and using (11), (12), and (13) we have that the (weighted) Bergman projection of onto is given bywhere the (weighted) Bergman kernel has the form
3. Representation of the Weighted Bergman Space on the Polydisk
We present a construction of the Bargmann type transform for the case of the polydisk, which maps isometrically the Bergman space onto an appropriate space and which allows us to factor the identity operator in the space and the Bergman projection of the polydisk. This scheme has been already successfully used, for example, in [2, 4–6]; we give here an analogous study for the polydisk.
We will use the multi-index notation. For an element , let denote the sumLet denote by the vector . Introduce in the coordinatesUnder the identificationwe havewhereLet , we define the measureWe haveThen we can writewhereConsider the following three unitary operators:(i)The discrete Fourier transform is given by the inverse operator has the form (ii)The Fourier integral transform is given by the inverse operator has the form(iii)The Mellin transform is given by the inverse operator has the formLet . Introduce the unitary operatoracting fromontowhere , , , andIn the coordinates the Cauchy-Riemann equations have the formLet . It straightforward to see thatwith and . Here denotes the vector whose th entry is and zero elsewhere. Thus the image consists of all sequences satisfying the equationsThese equations are easy to solve; the general solution has the formwith , . The function has to be in , which implies that its support on each variable , , has to be in and each integer , , has to be in . That is,where , , is the characteristic function of , and (see, e.g., formulas 3.251, 3.381, and 3.892 of [7])with , . Moreover, we have
Lemma 1. The unitary operator is an isometric isomorphism of the space ontounder which the Bergman space is mapped onto
Following [2] we introduce the isometric embedding acting from the spaceontoby the rulewhere is extended by , .
The adjoint operator acting fromonto has obviously the formIt is easy to check thatwhere is the orthogonal projection ofonto .
The operator maps the space ontoand the restrictionis an isometric isomorphism. The adjoint operatoris an isometric isomorphism.
Remark 2. We have
Theorem 3. The isometric isomorphism is given by
Proof. Let be a sequence of functions in ; then we have which proves the theorem.
Corollary 4. The inverse isomorphism is given by
4. Toeplitz Operators and Maximal Commutative Subgroups of Biholomorphisms
One of the main results of [8] gives a classification of the maximal Abelian subgroups of biholomorphisms of the unit ball . In particular for the unit disk or the upper half-plane , we have three different types of these subgroups.
The group of biholomorphisms can be described as follows. Denote by the special unitary group defined as the set of matrices of dimension such that and , whereWe have that is realized by the action of given by
Definition 5. Let be a Lie group. A maximal connected Abelian subgroup of is a Lie subgroup of that satisfies the following:(i) is connected Abelian.(ii)If is a connected Abelian Lie subgroup of such that , then .
Definition 6. Let and be connected complex manifolds. If and are subgroups of and , respectively, we will say that and are analytically equivalent if there is a biholomorphism such that .
It is proved in [8] that every maximal connected Abelian subgroup of is analytically equivalent to one of the following three groups, while neither two from the list are analytically equivalent.
The elliptic group of biholomorphisms of the unit disk is isomorphic to with the following group action:for each .
The parabolic group of biholomorphisms of the upper half-plane is isomorphic to with the following group action:for each .
The hyperbolic group of biholomorphisms of the upper half-plane is isomorphic to with the following group action:for each .
Now we turn our attention to the case of the polydisk. We have [3] that is realized by the action of extended by the th symmetric group. Consider the Lie groupacting on byfor each . It is evident that this group is analytically equivalent to a maximal Abelian subgroup of for each . Therefore we have different types of maximal Abelian subgroups of biholomorphisms of .
We have that a function , is invariant with respect to the action of if and only if . Hence making use of the results of Section 3 we can give a spectral representation of the Toeplitz operators with symbols invariant under the action of a maximal Abelian subgroup of biholomorphisms.
Theorem 7. Let be a bounded measurable function. Then the Toeplitz operator acting on is unitary equivalent to the multiplication operator . The sequence , where , , is given by
Proof. The operator defined in Section 3 factors the identity operator and the Bergman projection (Remark 2); then we haveLet be a sequence in . Theorem 3 and Corollary 4 give an integral representation of the operators and , respectively; then we have
5. The Moment Map
Let be a symplectic manifold, let be a Lie group, and let be the Lie algebra of . Denote by the set of all symplectomorphisms of . The actionis called a symplectic action if for all . In this case we say that acts by symplectomorphisms.
A symplectic action is a Hamiltonian action if there exists a mapsatisfying the following:(1)For each , let Let be the vector field on generated by the one-parameter subgroup ; then is a Hamiltonian function for the vector field ; that is,(2) is equivariant with respect to the action and the coadjoint action of on ; that is, for all .The vector is called a Hamiltonian -space and is a moment map.
In what follows we will suppose that is an Abelian Lie group. It is known that the Lie algebra of a connected Lie group is Abelian if and only if its Lie algebra is Abelian. Therefore if is Abelian we have that , with (see [9, Proposition ]). A direct calculation shows the following.
Lemma 8. Let be a connected Abelian Lie group. If , thenwhere are real numbers.
Moreover, making use of the linearity of the symplectic form we can state the following result.
Proposition 9. Let be a connected Abelian Lie group which acts by symplectomorphisms on the symplectic manifold . Let be a basis of . If is a map such that is a Hamiltonian function of , , thenfor each .
If is Abelian, then the conjugate action of on is the trivial action. This implies that the coadjoint action of on is the identity map for all . Consequently, we can restate condition as follows:(2) is invariant with respect to the action ; that is, for all .
6. The Moment Map of the Polydisk
Let be the symplectic manifold whereFor each consider the symplectic manifold where It is easy to check directly that the mapwhere is the Cayley transform defined by 1, is a symplectomorphism of onto the polydisk ; that is, .
Let denote the Lie group . The Lie algebra of this group is isomorphic to ; then . Let be the action of onto defined byWe note that for each the map is a diffeomorphism; moreover we havewhich implies that is a symplectic action.
We introduce the functions , , given by
Lemma 10. The map defined byis a moment map for the symplectic action of into ; that is, is a Hamiltonian -space.
Proof. Let be the standard basis of . The vector fields over generated by the the elements of this basis are given byWe observe thatwith . Hence Lemma 8 implies thatfor all . It is easy to check directly that each function , , is invariant under the action of ; consequently is invariant under this action too. Therefore is a moment map.
Our aim is to give a moment map for the action of each Abelian subgroup of biholomorphisms of the polydisk. Let be a Lie subgroup of dimension of ; then the Lie algebra of is a Lie subalgebra of , which implies that is a subspace of . Let be the natural projection and consider the symplectic action of into as the restricted action .
Theorem 11. The map is a moment map for the symplectic action of into ; that is, is a Hamiltonian -space.
Proof. Let be a basis of ; we can write , , as a linear combination of the standard basis . We observeUsing Proposition 9 we have that for every in . Since it is immediate that is invariant under the action, we conclude that it is a moment map.
7. Symbol Classes Generated by the Moment Map
In this section we introduce a class of bounded measurable functions over the polydisk associated with each connected Abelian Lie subgroup of biholomorphisms of which are a group of isometries over . Moreover, we have that the -algebra generated by Toeplitz operators with symbols in is commutative.
We consider a connected Abelian subgroup of . In the previous Section we calculate the moment map associated with the symplectic action of ; now using this function we define the following set of functions:We denote by the set of bounded measurable functions such that there exist satisfying and the set of bounded functions over invariant under the restricted action . Moreover, we know that is invariant under the action of , so we have that for every .
Definition 12. We denote by the algebra generated by symbols in where is an Abelian subgroup of isometries.
In particular, we take a maximal Abelian subgroup and consider the injective map defined by where , , and is the moment map associated with . Thus, we obtain that the image coincides with the image . On the other hand, we have that the inverse map is given bywith . Hence, we obtain that and ; thus it is clear that .
Therefore, if we consider a maximal Abelian subgroup , we have that the -algebra induced by the moment map is equal to the -algebra of Toeplitz operators generated by the set of symbols invariant under the action of . This affirmation is not true for a (nonmaximal) Abelian subgroup of isometries.
Example 13. Let and we consider the Abelian subgroup as follows: for each . A function , is invariant with respect to this action if and only if , where . It is obvious that Toeplitz operators with this class of symbols do not commute.
On the other hand, we observe that a moment map for is given by it follows that an element has the form . A straightforward calculus shows that the -algebra is commutative.
Example 14. Let be the connected Lie subgroup of -torus which acts as follows: We denote by and the Lie algebras corresponding to and , respectively. We know that the Lie algebra is generated by , where is standard basis of . Hence the moment map associated with is given by since we have that hence Lemma 10 implies that is a moment map. We have that every function belonging to has the form Thus it is clear that the -algebra is commutative.
Remark 15. Consider and as the above examples; its clear that , where is the identity in the group. In consequence of the above examples we have that , where is the operator identity on the Bergman space.
In summary, we have that every Abelian group of isometries is a subgroup of some maximal Abelian group . Hence, we have that . From this we obtain the following theorem.
Theorem 16. If is a connected Abelian Lie group of biholomorphisms in , then the -algebra is commutative.
Remark 17. Consider and Abelian Lie groups of biholomorphisms in such that is subgroup of ; under the considerations of the above theorem we have that is subalgebra of . Therefore a chain of commutative Abelian groups of isometries induce a chain of commutative -algebras.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was partially supported by National Council on Science and Technology of México under Grant no. 236109.