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

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.


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 twodimensional 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 () = cos  sin  of [4], we are dealing here with arbitrary () ∈  ∞ [0, /2].Each commutative Banach algebra T rad,, , considered in the paper, is generated by Toeplitz operators with  ∞ radial symbols and by the Toeplitz operator with symbol ()( 1  2 )  .That is, the whole variety of our algebras is parametrized by () ∈  ∞ [0, /2] and  ∈ N.
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 T rad,, and the Gelfand transform require the continuity of ()( 1  2 )  on the boundary of the unit ball.Thus here we impose an extra condition: () ∈ [0, /2] and (0) = (/2) = 0 (which is obviously satisfied in the particular case of () = cos  sin  in [4]).

Preliminaries
The weighted Bergman space A 2  (B 2 ) is the closed subspace of  2 (B 2 , ]  ) that consists of all analytic functions.The orthogonal Bergman projection ) has the form The reproducing kernel of A 2  (B 2 ) is defined by The standard orthonormal monomial basis Given a function  ∈  ∞ (B 2 ), the Toeplitz operator   with symbol  and acting on A 2  (B 2 ) is defined by Recall also [4] that, given a radial function  = () ∈  ∞ (0, 1), 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  ∈ We represent each coordinate of  ∈ B 2 in the form   =     , where Then we pass from the Cartesian coordinates  1 ,  2 on the base (B 2 ) of the unit ball B 2 to the polar coordinates , :  1 =  cos , and  2 =  sin , with  ∈ [0, 1) and  ∈ [0, /2].
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  ()( 1  2 )  on monomials: where (making the change of variables Taking into account that we come to the following result. where Corollary 2. The action of  ( 1  2 )  does not depend on the weight parameter .
Consider now some particular cases of  with  = 1.
(i) Let  ≡ 1; that is, we deal with the operator   1  2 ; then (ii) Let  = cos  sin ; that is, we deal with the operator  cos  sin  1  2 ; then This is a particular case of ( 8) which was studied in [4].Take now any  = () and  = (), both from  ∞ , and  ∈ N; then Thus we calculate That is, where or Note that the last formula implies the next result.

Comparison of the Toeplitz Operators with Symbols (
. Consider the symbols  1   1 and  2   2 , and calculate the action on the monomials of the products of the corresponding Toeplitz operators: That is, the Toeplitz operators   1   1 and   2   2 do not commute, in general. Consider now a more specific case, close to (8): Then That is,   1   1 commutes with   2   2 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  ∈  ∞ [0, /2] 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.

The Algebra T rad,𝑞,𝑘
Denote by T rad the  * -algebra generated by all Toeplitz operators   's with radial symbols  = () ∈  ∞ [0, 1).We fix then a function  = () ∈  ∞ [0, /2] and  ∈ N and denote by T , the unital Banach algebra generated by Toeplitz operator    .Since the generators of both these algebras commute (Lemma 3), the Banach algebra T rad,, generated by elements of T rad and T , is commutative.
In this section we study the algebra T rad,, as well its generating subalgebras.The analysis of the algebra T rad is the same as in [4, Section 3.1]; thus we recall here just the most important facts.

Toeplitz Operators with Radial Symbols.
Given a sequence  = {(||)} ||∈Z + ∈ ℓ ∞ , we denote by   the diagonal operator defined on A 2  (B 2 ) 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(Z + ), where SO(Z + ) consists of all bounded sequences that slowly oscillate in the sense of Schmidt [6]; that is, Moreover [5, Section 5], the  * -algebra T rad is isomorphic and isometric to the algebra SO(Z + ), via identification of a diagonal operator with its eigenvalue sequence.

Corollary 4.
Let  be a convergent sequence; then   ∈ T rad .
In particular, for all  ∈ Z + the orthogonal projection   of A 2  (B 2 ) onto span{  : || = } belongs to the algebra T rad .
Recall [4] that the compact set (T rad ) of maximal ideals of the algebra T rad has the form The fiber  ∞ is the set of all multiplicative functionals  such that (  ) = 0 whenever   is a compact operator, or  ∈  0 , where  0 denotes the set of all sequences converging to zero.And Z + can be considered a part of (T rad ) since each ℓ ∈ Z + defines the multiplicative evaluation functional  ℓ :    → (ℓ).Moreover, by [7, Chapter I, Theorem 8.2], the set Z + is densely and homeomorphically embedded into (T rad ), and, by [8], the fiber  ∞ is connected.

Toeplitz Operators with
We denote by Δ(, ) the set sp   .Then the maximal ideal space (T , ) of the Banach algebra T , coincides with the spectrum of its generator; that is,  (T , ) = Δ (, ) . (36)

Invariant Subspaces.
Given  ∈ Z + , we denote by   the finite dimensional subspace It is obvious that Observe now that each space   is invariant for all operators from T rad,, .Moreover, by Corollary 4, each orthogonal projection   onto   is a diagonal operator from T rad .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 Note that That is, since  ∈  0 , (43) 3.5.Dense Subalgebra in T rad,, .The set of all operators of the form where    ∈ T rad , constitutes the dense (nonclosed) subalgebra D = D rad,, of the algebra T rad,, .At the same time the representation of the operators from D in the above form is not unique.
Following [4] we give another formula defining the functional  { ℓ } that permits an extension of  { ℓ } to a larger subalgebra of T rad,, .
Lemma 12 (see [4,Lemma 4.14] Recall also the following general fact: let A be a unital commutative Banach algebra generated by its two subalgebras A 1 , A 2 sharing the same identity, and let (A), (A 1 ), and (A 2 ) be their respective sets of maximal ideals.Then we have a natural continuous mapping  :  ∈  (A)  → ( 1 ,  2 ) ∈  (A 1 ) ×  (A 2 ) , (67) defined by the restrictions  1 and  2 of the functional  onto the subalgebras A 1 , A 2 .
The mapping  is injective identifying thus its range with (A).
Note that the result of this lemma is independent of an extra condition (50) and thus from the concrete form of the spectrum Δ(, ).
where  () is the multiplicative functional on T rad given in (59).
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 T rad,, .(78)
Symbol ()( 1  2 )  and Its Spectrum.Recall that the spectrum of the Toeplitz operator    is independent of the weight parameter , and note that              ≤      ()( 1  2 )      ∞ (34)That is, the spectral radius of    is at most ‖‖ ∞ .And (Lemma 1) since the operator    is not invertible, we have that 0 ∈ sp   ⊂  (0,         ∞ ) .
The algebra T rad,, is not semisimple.Its radical RadT rad,, contains, in particular, the operators of the form      , where  ∈  0 .Proof.Following the proof of [4, Lemma 3.7], we only need to prove that the operator      is topologically nilpotent; ).The functional  { ℓ } extends to the functional  = ( { ℓ } , ) on the algebra generated by elements of A * D and    via Recall that the commutative Banach algebra T rad,, is generated by its two unital subalgebras T rad and T , .