1. Introduction
In recent years, an important subject of study has been the relationship between the
algebras generated by Toeplitz operators and geometric tools such as Lie groups, representation theory, etc. Several authors found commutative
algebras, which are generated by Toeplitz operators with invariant symbols under the action of some maximal abelian subgroup on different domains such as the unit disk, unit ball, Siegel domain, and projective spaces. For more details, see [
1,
2,
3,
4,
5,
6,
7].
Another step in this direction was to find commutative
algebras using geometry tools such as symplectic geometry and moment mapping; for more details, see [
8,
9,
10]. Moreover, using these techniques, several authors found commutative Banach algebras generated by Toeplitz operators, which are not
; for example, see [
11,
12,
13,
14,
15,
16,
17,
18]. More precisely, they used quasi-homogeneous, quasi-radial, and some generalizations of type symbols, which are invariant under the action of some group
G or are associated with the moment map of
G on different domains.
On the other hand, several authors found representations of the spaces of the analytic functions in terms of analytic functions in domains of the lower dimension. For example, in [
6,
19] the Bergman space over the Siegel domain is decomposed as a direct integral of weighted Fock spaces. Following this approach, we found a representation of the Fock space of
in the function of the weighted Bergman spaces of the projective spaces
; i.e., every function in the Fock space can be written in terms of elements in the weighted Bergman space on
. In other words, we show that the Fock space on
is unitarily equivalent to the direct sum of the weighted Bergman spaces on
; i.e.,
where
U is unitary.
In this paper we take the action of
on
, defined by
to study the
algebras of the Toeplitz operators, considering the following cases:
- (a).
The symbols depend of the moment map associated to the action; i.e., every symbol has the form where is the moment map associated to the above action.
- (b).
The symbols are invariant under this action; i.e., every symbol c satisfies the relation , for each and .
Using the above actions and their moment map, we introduce a coordinate system on , which is given by these geometrical objects; this coordinate system is very useful to find the representation of the Fock space in terms of the weighted Bergman spaces on . Hence, the algebras generated by Toeplitz operators where the symbols are as in (a) and (b) are denoted by and , respectively.
We show that each element of the algebra
is a direct sum of the multiples of the identity operator on each component of
. Similarly, we show that each element of the algebra
can be written as a direct sum of the Toeplitz operators on the weighted Bergman spaces of the projective space
. Using this result, we show the following relationship
where
and
. Note that in [
6,
19] the authors presented a similar result for the Siegel domain.
Moreover, using the above relation between the algebras and , we introduced a commutative Banach algebra of Toeplitz operators on the Fock space, which is obtained from and the sub-algebra of generated by Toeplitz operators with quasi-homogeneous symbols on the projective space (these symbols can be considered as a function of ).
We have organized the rest of this paper in the following way: In
Section 2, we present some known facts regarding the Bergman spaces and Toeplitz operators on projective space
. In
Section 3, we present the action of
on
, along with the moment map and a coordinates system associated with this action. In
Section 4, we present some known results about algebras generated by Toeplitz operators with quasi-radial and quasi-homogeneous symbols over the complex projective space. In
Section 5, we present a connection between the space de Fock of
and the direct sum of the weighted Bergman spaces of
. Finally, in
Section 6, we introduce some commutative Banach algebras of Toeplitz operators on the Fock space using the commutation relations between the algebras
and
, presented in
Section 4.
2. Preliminaries
As usual, the complex projective space
is the complex
dimensional manifold that consists of all elements
, where
. For every
we have an open set
and a holomorphic chart
given by
where the notation
means that
is omitted. The numbers
are known as the homogeneous coordinates with respect to the map
. Note that the collection of all such maps yields a holomorphic atlas of
.
From [
2], we know that the volume element on
induced by the Fubini–Study metric has the following form
where
is the canonical Lebesgue measure on
. Moreover, in polar coordinates we have that
where
,
and
. Also, the volume form for
is given by
where
.
With respect to the coordinates induced by
, given
, the
m-weighted measure on
is defined by
To simplify the notation, we use the same symbol to denote the weighted measures for both and , respectively.
Definition 1. The weighted Bergman space on , with weight , is defined bywhere is the tensor product of m copies of and T is the tautological or universal line bundle of . It is also known that for every , the Bergman space satisfies the following properties:
- (i).
With respect to the homogeneous coordinates of , the Bergman space can be identified with the space of all homogeneous polynomials of degree m over .
- (ii).
The map
defined by
is an isometry and it is well known that
where
denotes the space of all polynomials on
of degree less than or equal to
m.
Recall the following usual notation for multi-index: given
and
we have that
Considering the identification of the Bergman space
with the space
, we have that the monomial functions
form an orthogonal basis. Thus, the set of functions
is an orthonormal basis for
, where the inner product is defined by
for all
. Furthermore, in local coordinates, the Bergman projection from
onto
is defined by
where
and this function is called the Bergman kernel for
.
Definition 2. If then the Toeplitz operator with symbol a is the bounded operator on defined by , for each .
Note that the Toeplitz operator with symbol
can be represented as a matrix
A where the entries are given by
where
. This fact is clear since the Bergman space
is finite dimensional. For more detail, see [
2,
20].
3. Some Properties of the Action of on
Given a manifold
N and a Lie group
G with Lie algebra
associated to
G. If
G acts on
N then for every
we have a family of diffeomorphisms
, which is an one parameter group and the vector field
associated to this group
; that is,
It therefore makes sense to define
. Unfortunately the map
is an anti-Lie algebra map:
Recall the following definition:
An action of G on a symplectic manifold is symplectic if for all .
An action of a Lie group G on a manifold M is proper if the map defined by is proper. Recall that a continuous map between two topological spaces is proper if the preimage under f of a compact set is compact.
An action of a group G on a set X is free if for any , the equation implies that
Definition 3. Consider a Hamiltonian action of a Lie group G on a symplectic manifold . Let be a corresponding anti-Lie algebra map. The moment map corresponding to the action is defined byfor and , where is the canonical pairing. For more details about the above definition, we can see [
21].
We recall that the standard action of the unit circle
over
is given by
Also recall that the canonical symplectic form on
is defined by
and we have that the action (
6) is symplectic, proper, and free.
We denote by
the Lie algebra associated to
which is given by scalar multiples of the identity matrix in
; note that this algebra is generated by
and the dual algebra
is generated by
. Now we consider an arbitrary element
where
; thus we have a collection of diffeomorphisms over
defined by
and the corresponding vector field to
X in polar coordinates is given by
where
. We calculate the contraction of the symplectic form
with respect to
Therefore, we have that the moment map associated with the group
in the symplectic manifold
is given by
On the other hand, we apply the theorem of Marsden–Weinstein–Meyer to obtain a reduced space which is also a symplectic manifold. As a first step, we take the value 1, which is a regular value of the moment map
, and then
is a submanifold of
and
where
. In the second step, note that the action of
on
is free, thus
is a smooth manifold, where the action of
on
is given by
for all
and
. It is immediate to see that
and the orbit map
define a principal
bundle over
. Moreover, there exists a symplectic form
on
such that
.
Now, consider the local coordinates of the reduced manifold
given by
where
. We consider the embedding map
from
to
defined as follows:
where
The image
is a submanifold of
and we have that the symplectic form in local coordinates is given by
where
is the canonical symplectic form in
.
The image
is a dense open set of
, and the action of
is free, and we have that the map
is injective, since
is a local system of coordinates which is dense in
(which is given by (
2) with
) and the quotient of
with
is called the Hopf fibration, which is isomorphic to
. Therefore
is a dense local system of coordinates for
, in similar way we can introduce
associated to (
2). Theses
, for
, provide a global system coordinates of
. The volume element
on
can be expressed in the coordinates
as follows:
where
is the volume element of
given in local coordinates by (
3) and
is the invariant volume of
.
In summary, we have a system of coordinates of
given by
where
is given by (
7).
4. Commutative Algebras Generated by Toeplitz Operators with Symbols in the Projective Space
In this section some basic definitions and concepts related to Toeplitz operators with symbols on the complex projective space are presented. For a more detailed description, we can see [
14,
15].
Let
be a multi-index so that
. We will call such multi-index
k a partition of
n. For the sake of definiteness, we will always assume that
. This partition provides a decomposition of the coordinates of
as
, where
for every
, and the empty sum is 0 by convention. Each element
has a decomposition as follows: For every
we define
. And for any
j we write
Correspondingly, we write and . Note that , whenever and where denote the r-dimensional torus.
Definition 4. Let be a partition of n.
- (i).
A k-pseudo-homogeneous symbol is a function that can be written in the formwhere and with . - (ii).
A k-quasi-radial symbol is a function that can be written in the formfor some function which is homogeneous of degree 0. - (iii).
A k-quasi-radial-pseudo-homogeneous symbol is a function in of the form where is k-quasi-radial symbol and is a k-pseudo-homogeneous symbol.
From Lemma 3.8 in [
15], we know that for each
k-quasi-radial symbol
, the Toeplitz operator
acting on
satisfies
for every
, where
Considering that a
k-quasi-radial-pseudo-homogeneous symbol has the form
where
and
are
k-radial and
k-pseudo homogeneous symbols, respectively. The previous result implies that the Toeplitz operator
, acting on
, satisfies the following relation:
for every
, where
Moreover, if
is a
k-quasi-radial
k-pseudo-homogeneous symbol with
for
, then the function
presented in (
11) has the following form:
for every
such that
and is zero otherwise. Also, we have the following decomposition
So, the Toeplitz operators
,
,
, pairwise commute and
And therefore, the Banach algebra generated by this kind of Toeplitz operator is commutative. For more detail, see ([
14]
Section 3).
5. A Connection between the Space de Fock of and the Direct Sum of the Weighted Bergman Spaces of
The Fock or Segal–Bargmann space
is defined by the set of all holomorphic functions on
satisfying the condition
where
denotes the Lebesgue measure on
. In this space, one can define an inner product as follows:
It is known that the Fock space
is a Hilbert space with orthogonal basis
and
, where
. Then, the functions
form an orthonormal basis for
.
Considering the above orthonormal basis, we define the following operator:
where
, and the functions
and
are defined by (
15) and (
4), respectively.
Note that in
we have that
, where
is an homogeneous polynomial of degree
m. Thus,
The corresponding adjoint operator
has the form:
where
and
.
Example 1. For , we have that and , thusIn consequence,In particular, if and , then In summary, we have the following result, which is very important for the development of this work; i.e., we present a relation between the Fock space and the weighted Bergman spaces of .
Theorem 1. The operator U maps ontoMoreover, U is an isometric isomorphism. Proof. By straightforward calculation, we have
The result follows from the previous equation and the fact that
is an orthonormal basis on the Fock space
. □
6. Toeplitz Operators on the Fock Space over with Invariant Symbols under the Action of
The aim of this section is to decompose a Toeplitz operator on the Fock space of as a direct sum of Toeplitz operators on the weighted Bergman spaces of space using the unitary operator U, defined in the previous section.
Recall that a
-invariant symbol
c on
is a function
which is invariant under the action of
given in Equation (
6). In other words,
We denote by
the set of all
-invariant functions. Each element
has the form
, where
are the local coordinates of
presented in (
9).
In [
6], the authors studied three types of invariant symbols under the action of a commutative subgroup. These symbols were considered as three types: The family
,
, and
, respectively. The authors obtained that the Toeplitz operators with symbols in the family
are direct integrals of multiplication operators. In the family
, the Toeplitz operators with symbols in this family are direct integrals of Toeplitz operators with the same symbol. And finally, for the family
, the authors showed that the Toeplitz operators are a direct integral of Toeplitz operators, where the symbol depend of the base spaces of direct integral.
In [
6], the authors studied three types of invariant symbols under the action of a commutative subgroup over the Siegel domain
. These symbols over the Bergman space of the Siegel domain were considered as three types:
In the family A, a Toeplitz operator over the Bergman space of the Siegel domain is a direct integral of multiplication operators over the weighted Fock spaces.
The family B, the Toeplitz operators over the Bergman space of with symbols in this family can be written as direct integrals of Toeplitz operators over the weighted Fock space, where the symbol is constant on each element of the direct integral.
For the family C, the authors showed that the Toeplitz operators over the Bergman space of Siegel domain can be written as a direct integral of Toeplitz operators over the weighted Fock space, where the symbol varies on each element of the direct integral.
Furthermore, the authors decompose the weighted Bergman space of the Siegel domain as a direct integral of the weighted Fock space.
Inspired by the families presented in the previous paragraph, we will now consider three families of symbols over , which will be used to study Toeplitz operators on the Fock space .
Definition 5. We introduce the following families of functions:
;
;
;
where are the local coordinates given by (9); recall that w is connected to the local coordinates of the projective space . Remark 1. Note that the symbols of the family depend on the moment map associated with the action of the unit circle. In addition, the symbols of the family are invariant under the action of the unit circle. Finally, the symbols of the family are a special case of the symbols of the family .
The main result of this work is presented below, which connects the Toeplitz operators on the Fock space of
with the Toeplitz operators on the weighted Bergman space of the projective space
, since every Toeplitz operator on
can be decomposed as a direct sum of Toeplitz operators on
. Note that this result is analogous to Theorem 3.3 in [
6].
Theorem 2. Let c be an element in ; the Toeplitz operator acting on the Fock space is the unitary equivalent to the direct sum of Toeplitz operators ; that is,where U is given by (16) and is a Toeplitz operator acting on Bergman space with weight m over , with symbol Proof. Given
, we have:
We use the coordinates
, which are associated with the moment map and symplectic reduction of the unit circle and defined by (
9). Moreover, since the function
c is
-invariant, we have the following relations:
where
is given by (
3).
On the other hand, for two elements
and
belonging to the space
, we have
And so, we define the following function:
for all
.
On the other hand, we obtain the following relations:
where
, and
are defined in Equation (
4). Therefore, the result follows from Equation (
5). □
The following result describes a Toeplitz operator in Fock space, where the symbol depends on the moment map in terms of operators in complex projective space.
Corollary 1. Given , the Toeplitz operator acting on the Fock space is unitarily equivalent to the direct sum of multiplication operators ; that is,where is a function given by Proof. From Equation (
18), we have that
From the above relation, we have that
is a constant. In consequence,
. Thus, the result follows from Equation (
17). □
Corollary 2. Given , the Toeplitz operator acting in the Fock space is unitarily equivalent to the direct sum of Toeplitz operators ; that is,where is a Toeplitz operator acting on the weighted Bergman space over . Proof. We just need to calculate
has the following form:
In consequence, . □
For the symbols presented in the above statements, we can conclude the following results:
Corollary 3. For any symbol and , we have that , and Corollary 4. For any pair of symbols and , we have that , and Remark 2. It is straightforward to check that, contrary to the case of Corollary 4, for the symbols of the previous corollary, we have that , in general.
The results of this section used the system coordinated (
9). Note that every function in
is equivalent to a function in
since it just depends on the variable
w in (
9). Now, we consider functions
and
, where
is a
k-quasi-radial symbol and
is a
k-quasi-homogeneous symbol given in Definition 4.
From Corollary 3 and Equation (
14), the Toeplitz operators
,
pairwise commute and
7. Conclusions
In the present investigation, firstly we obtained the moment map
for the action of
on
in a similar way as Sánchez and Quiroga in [
10] obtained the moment map for the action on the unit ball of any maximal abelian subgroup of biholomorphisms of the unit ball. In consequence, Equations (
8) and (
9) provide a coordinate system, and this system uses the action of
on
together with the function
and the symplectic reduction of
. These coordinates relate the space
with the moment map, the action of
, and the projective space
.
On the other hand, a crucial point of this paper was the introduction of the operator
U defined in (
16), and we show that
U is unitary. This operator allows us to connect the Fock space of
with the weighted Bergman spaces of the complex projective space
.
Moreover, in
Section 6, the operator
U was used to decompose a Toeplitz operator with symbols into one of the three families
,
, and
(presented in Definition 5 and contained in
) as direct sums of Toeplitz operators on the weighted Bergman spaces of the projective space
. And so, these decompositions are used to find commutativity relations between the algebras
,
, and
generated by the Toeplitz operators with symbols in the families
,
, and
, respectively.
On the other hand, as a future direction of this work, we will explore the use of the techniques used in this paper in several symmetric domains such as the unit ball, the Siegel domain, and Cartan domains, among others. In particular, we will use a Hamiltonian action and its moment map in these domains. Moreover, we will try to provide a characterization of the Bergman spaces in the mentioned domains. We will study the algebras generated by Toeplitz operators in these domains, where the symbol depends on the moment map or is invariant under the action of the group.
Author Contributions
Conceptualization, C.G.-F.; Investigation, C.G.-F., L.A.D.-G., R.R.L.-M., and F.G.H.-Z.; Writing—review and editing, C.G.-F., L.A.D.-G., R.R.L.-M., and F.G.H.-Z. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Acknowledgments
The authors are appreciative of the reviewers who provided insightful criticism, suggestions, and counsel that helped them to modify and enhance the paper’s final version.
Conflicts of Interest
On behalf of all authors, the corresponding author states that there are no conflict of interest.
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