Toeplitz Operators on the Weighted Bergman Space over the Two-Dimensional Unit Ball

Abstract

We extend the known results on commutative Banach algebras generated by Toeplitz operators with radial quasi-homogeneous symbols on the two-dimensional unit ball. Spherical coordinates previously used hid a possibility to detect an essentially wider class of symbols that can generate commutative Banach Toeplitz operator algebras. We characterize these new algebras describing their properties and, under a certain extra condition, construct the corresponding Gelfand theory.

1. Introduction

After a detailed study of the commutative -algebras generated by Toeplitz operators acting on the weighted Bergman spaces on the unit ball [1, 2] it was quite unexpectedly observed [3] that, contrary to the one-dimensional case of the unit disk, there exist many other Banach (not !) algebras generated by Toeplitz operators that are commutative on each weighted Bergman space. They were generated by Toeplitz operators with radial and the so-called quasi-homogeneous symbols, and the commutativity of the corresponding algebras was established just by an observation that the generating operators commute among themselves. Then the problem of constructing the Gelfand theory for these algebras emerged, being a tool to describe the properties, in particular spectral ones, of the operators forming the algebra. For the two-dimensional ball, the case relevant to this paper, this step was done in [4].

Note that the Introduction of [4] stated that the paper studies the unique commutative Toeplitz operator Banach algebra on the two-dimensional ball. This was indeed completely true for the only known by that time generating radial quasi-homogeneous symbols. As it later turns out (and this is exactly what this paper is about) the spherical coordinates used in [4] hide the possibility to detect many other commutative Banach algebras on the two-dimensional ball generated by Toeplitz operators with symbols of a more general type.

In this paper we use another representation of the points of the two-dimensional ball, which permits us to extend essentially the previous class of quasi-homogeneous symbols. Instead of a very specific function of [4], we are dealing here with arbitrary . Each commutative Banach algebra , considered in the paper, is generated by Toeplitz operators with radial symbols and by the Toeplitz operator with symbol . That is, the whole variety of our algebras is parametrized by and .

All these algebras share many common properties. We discuss, in particular, the description of the invariant subspaces, the property of being not semisimple, radical elements description, and non-uniqueness of the representation of elements in a dense subalgebra.

The tools that we use for the explicit description of both the compact set of maximal ideals of our algebras and the Gelfand transform require the continuity of on the boundary of the unit ball. Thus here we impose an extra condition: and (which is obviously satisfied in the particular case of in [4]).

2. Preliminaries

2.1. Definitions and Basic Properties

Let be the open unit ball in and let denote the standard volume form on . For , we introduce the one-parameter family of the standard weighted (probability) measures The weighted Bergman space is the closed subspace of that consists of all analytic functions. The orthogonal Bergman projection of onto has the form The reproducing kernel of is defined by The standard orthonormal monomial basis of is of the form

Given a function , the Toeplitz operator with symbol and acting on is defined by

Recall also [4] that, given a radial function , the Toeplitz operator is diagonal with respect to the monomial basis (4) and its eigenvalue sequence is given by

In order to define our class of symbols we start with some notation.

We write a point as , . Given a multi-index , we use the standard notation .

We represent each coordinate of in the form , where Then we pass from the Cartesian coordinates , on the base of the unit ball to the polar coordinates , : , and , with and .

The Toeplitz operators with radial quasi-homogeneous symbols on the two-dimensional ball were studied in detail in [4]. The quasi-homogeneous part of the symbols consisted of the following functions: Here we extend this class of symbols to the functions of the form Note that the old functions (8) correspond to the very particular case of (9):

To start our analysis we consider the action of on monomials: where (making the change of variables , )

Taking into account that we come to the following result.

Lemma 1. Given , one has where

Corollary 2. The action of does not depend on the weight parameter .

Consider now some particular cases of with .(i)Let ; that is, we deal with the operator ; then (ii)Let ; that is, we deal with the operator ; then

This is a particular case of (8) which was studied in [4].

Take now any and , both from , and ; then Thus we calculate That is, where or

Note that the last formula implies the next result.

Lemma 3. Given any and any , both from , one has

In what follows, to simplify the notation, we introduce the pairs , , and . Then so that .

2.2. Comparison of the Toeplitz Operators with Symbols (8) and (9)

Let , , and , . Consider the symbols and , and calculate the action on the monomials of the products of the corresponding Toeplitz operators: That is, the Toeplitz operators and do not commute, in general.

Consider now a more specific case, close to (8): Then That is, commutes with if and only if The last equality holds if and only if

This brings us back to the case analyzed in [4], showing that in our more general case of we have fewer properties as compared with the specific case of [4]. In particular, we lose the commutativity of generating operators in [4], extending at the same time the class of generating symbols.

3. The Algebra

Denote by the -algebra generated by all Toeplitz operators ’s with radial symbols . We fix then a function and and denote by the unital Banach algebra generated by Toeplitz operator . Since the generators of both these algebras commute (Lemma 3), the Banach algebra generated by elements of and is commutative.

In this section we study the algebra as well its generating subalgebras. The analysis of the algebra is the same as in [4, Section 3.1]; thus we recall here just the most important facts.

3.1. Toeplitz Operators with Radial Symbols

Given a sequence , we denote by the diagonal operator defined on as follows: If is a Toeplitz operator with radial symbol we have obviously Recall [5, Section 5] that the sequence of a Toeplitz operator belongs to the -algebra SO, where SO consists of all bounded sequences that slowly oscillate in the sense of Schmidt [6]; that is, Moreover [5, Section 5], the -algebra is isomorphic and isometric to the algebra SO, via identification of a diagonal operator with its eigenvalue sequence.

Corollary 4. Let be a convergent sequence; then . In particular, for all the orthogonal projection of onto belongs to the algebra .

Recall [4] that the compact set of maximal ideals of the algebra has the form The fiber is the set of all multiplicative functionals such that whenever is a compact operator, or , where denotes the set of all sequences converging to zero. And can be considered a part of since each defines the multiplicative evaluation functional .

Moreover, by [7, Chapter I, Theorem 8.2], the set is densely and homeomorphically embedded into , and, by [8], the fiber is connected.

3.2. Toeplitz Operators with Symbol and Its Spectrum

Recall that the spectrum of the Toeplitz operator is independent of the weight parameter , and note that That is, the spectral radius of is at most . And (Lemma 1) since the operator is not invertible, we have that We denote by the set . Then the maximal ideal space of the Banach algebra coincides with the spectrum of its generator; that is,

3.3. Invariant Subspaces

Given , we denote by the finite dimensional subspace It is obvious that Observe now that each space is invariant for all operators from . Moreover, by Corollary 4, each orthogonal projection onto is a diagonal operator from .

Each diagonal operator restricted to is the scalar operator , while the operator acts on as a weighted shift operator. It is also nilpotent on each , since The last implies that

3.4. Radical

Lemma 5. The algebra is not semisimple. Its radical contains, in particular, the operators of the form , where .

Proof. Following the proof of [4, Lemma 3.7], we only need to prove that the operator is topologically nilpotent; that is,
Note that That is, since ,

3.5. Dense Subalgebra in

The set of all operators of the form where , constitutes the dense (nonclosed) subalgebra of the algebra . At the same time the representation of the operators from in the above form is not unique.

To describe the source of such non-uniqueness, we denote by the set of finite dimensional diagonal operators with eigenvalue sequences and for .

Lemma 6. One has if and only if , for each .

Proof. The proof follows almost literally the arguments of proof [4, Lemma 3.8], given for a particular case of and .

The next lemma is a special case of the previous one.

Lemma 7. Let where all are different. Then, for each at least one radial symbol is identically zero.

Proof. The previous lemma implies that if then each diagonal operator is finite dimensional. The rest of the proof follows the arguments of the proof of [4, Theorem 3.9].

As an application to the so-called zero-product problem we give the next corollary.

Corollary 8. For the operator the following statements are equivalent: (1),(2) is finite dimensional,(3)At least one radial symbol is identically zero.

3.6. A Special Case of

In what follows we will assume that a function satisfies the extra condition: We note that this is the (only) case when the homogeneous of order zero function (our generating symbol) is continuous on the boundary of the unit ball .

We note also that the quasi-homogeneous symbols (8) of [4] satisfy the above conditions.

Condition (50) permits us to make the statements of Section 3.2 more precise.

First of all (50), together with the results of [9], implies that Therefore as where is the disk of radius centered at the origin.

That is, now we have that

Theorem 9. The Banach algebra is isomorphic via the Gelfand transform to the algebra of all functions analytic in and continuous on .

Proof. It practically literally follows the proof of [4, Theorem 3.6].

4. Gelfand Theory of

4.1. Finitely Generated Subalgebras of

We present here relevant material from [4], slightly modified when the more general symbols (9) will be involved.

We start from a finite number of diagonal operators on from the algebra . They act on the standard monomial basis (4) as Denote by the unital -algebra generated by elements of .

Recall [10] that the joint spectrum of the operators can be identified with the maximal ideal space of the -algebra and it has the form where denotes the smallest ideal in the algebra containing the elements , for .

Lemma 10 (see [4, Corollary 4.11]). Let . Then either there is such that for all or there is a sequence such that for all

Let . Assuming the first option of the above lemma, we define a multiplicative functional on by for all . Note that , for all .

Alternatively, if the second option of Lemma 10 holds, then we define a multiplicative functional on by

Lemma 11 (see [4, Lemma 4.12]). The limit (60) exists for all . The functional is multiplicative with , for all .

Following [4] we give another formula defining the functional that permits an extension of to a larger subalgebra of .

We start with the set and define the Hilbert space The reproducing kernel of has the form

Given , let , be such that , and denote by the corresponding point on . Consider then a sequence defined by , and define the sequence of unit vectors in by

Then, by [4, Lemma 4.13], the functional (60) can be also defined as

In order to extend the multiplicative functional (64) to a larger algebra we proceed as follows. Let and let , be such that . Then we take a sequence convergent to and as above.

Lemma 12 (see [4, Lemma 4.14]). The functional extends to the functional on the algebra generated by elements of and via with and . Moreover, for elements of the form one has

4.2. Gelfand Theory

Recall that the commutative Banach algebra is generated by its two unital subalgebras and .

Recall also the following general fact: let be a unital commutative Banach algebra generated by its two subalgebras , sharing the same identity, and let , , and be their respective sets of maximal ideals. Then we have a natural continuous mapping defined by the restrictions and of the functional onto the subalgebras , .

The mapping is injective identifying thus its range with .

That is, by the results of Sections 3.1 and 3.2   identifies with a subset of .

Lemma 13. None of the points of the set belongs to .

Proof. Suppose that is an element of and let where is the orthogonal projection onto ; then and, on the other hand, the operator belongs to the radical of ; therefore and we have a contradiction.

Note that the result of this lemma is independent of an extra condition (50) and thus from the concrete form of the spectrum .

For the next result recall that the function satisfies condition (50), and thus .

Lemma 14. One has that (i)the set belongs to ,(ii)the set belongs to .

Proof. (i) Let . Let be a functional on a dense subalgebra of defined, for any where , by where is the multiplicative functional on given in (59).
The functional is well defined, since implies by Lemma 6; moreover, is continuous and extends to a multiplicative functional on since
(ii) Let and define a multiplicative functional on a dense subalgebra of as
The functional is well defined: implies that is compact for all and since , for all we have that .
The functional is continuous: let be a fixed element. Consider the unital -algebra generated by . The restriction of to defines obviously a multiplicative functional on . We note that Since maps compact operators to zero, by (64), it has the form where is a suitable sequence induced by as in Section 4.1 and is given by (63).
By Lemma 12 we have therefore, is continuous and thus extends to a multiplicative functional on .

Finally, Lemmas 13 and 14, Theorem 9, and properties of the injective tensor product imply the description of the set of maximal ideals and the Gelfand transform of Banach algebra .

Theorem 15. The compact set of maximal ideals of the algebra has the form (i)The Gelfand image of the algebra is isomorphic to and coincides with the algebra which is identified with the set of all pairs satisfying the following condition for all , where is identified with the value on the element . The symbol denotes the injective tensor product.(ii)The Gelfand transform is generated by the following mapping of elements of :

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work was partially supported by the CONACYT Project 180049, Mexico.