We characterize the commuting Toeplitz operator and Hankel operator with quasihomogeneous symbols. Also, we use it to show the necessary and sufficient conditions for commuting Toeplitz operator and Hankel operator with ordinary functions.

1. Introduction

Let dA denote Lebesgue area measure on the unit disk 𝔻, normalized so that the measure of 𝔻 equals 1. For α > −1, we denote by dA_α the measure dA_α(z) = (α + 1)(1−|z|²) ^αdA(z). For 1 ≤ p < +∞, the space L^p(𝔻, dA_α) is a Banach space. The weighted Bergman space

is the closed subspace of analytic functions in the Hilbert space L²(𝔻, dA_α). For each z ∈ 𝔻, the application

is continuous and can be represented as

, where

()

This means that, if P_α is the orthogonal projection from L²(𝔻, dA_α) onto

, then P_α can be defined by

()

For a function f ∈ L^∞(𝔻, dA_α(z)), we define the Toeplitz operator

with symbol f by

()

It is well known that

()

Let U^(α) : L²(𝔻, dA_α(z)) → L²(𝔻, dA_α(z)) be the unitary operator defined by

, where f belongs to L²(𝔻, dA_α(z)). Let g be in L²(𝔻, dA_α(z)); we define a bounded linear operator M_g on L²(𝔻, dA_α(z)) as follows:

()

Then we can define the small Hankel operator as follows:

()

as H_g = P_αU^(α)M_g.

The study of commuting Toeplitz operators on the Bergman and Hardy spaces over various domains and related operator algebras has a long lasting history. On the Hardy space of the unit disk, Brown and Halmos [1] first showed that two Toeplitz operators are commuting if and only if either both symbols of these operators are analytic, or both symbols of these operators are coanalytic or a nontrivial linear combination of the symbols of these operators is constant. On the Bergman space, the situation is more complicated. Axler and Čučković obtained the analogous result for Toeplitz operators with bounded harmonic symbols on the Bergman space of the unit disk [2]. The problem of characterizing commuting Toeplitz operators with arbitrary bounded symbols seems quite challenging and is not fully understood until now. In [3], Čučković and Rao used the Mellin transform to characterize all Toeplitz operators on which commute with for (m, p) ∈ N × N. Later in [4] Louhichi and Zakariasy gave a partial characterization of commuting Toeplitz operators on with quasihomogeneous symbols. Recently, Lu and Zhang [5, 6] characterized the commuting Toeplitz operators and Hankel operators with quasihomogeneous symbols. There are also many other important results [7–13]. Motivated by those works, we study commuting Toeplitz operator and Hankel operator on the weighted Bergman space. In this paper, we obtain the necessary and sufficient conditions for commuting Toeplitz operator and Hankel operator.

An operator that will arise in our study of Toeplitz operators is the Mellin transform, defined for any function φ ∈ L¹([0,1], r dr); from the formula,

is the Mellin transform as follows:

()

which is a bounded holomorphic function in the half plane {z : Re z > 2}.

Let φ ∈ L¹(𝔻, dA_α) be a radial function; that is, suppose that φ(z) = φ(|z|), z ∈ 𝔻. In fact, if we define the function φ_r on [0,1] by φ_r(s) = φ(s), then a direct calculation shows that

()

so that if k ∈ ℕ,

()

Thus, T_φ is a diagonal operator on

with coefficient sequence as follows:

()

This makes it relatively simple to work with the product of two operators with such radial symbols.

Now, we define the “radialization” of a function f ∈ L¹(𝔻, dA_α) by the following:

()

It is clear that a function f is a radial if and only if rad(f) = f.

Let ℜ^α be the space of weighted square integrable radial functions on 𝔻. By using that, trigonometric polynomials are dense in L²(𝔻, dA_α) and that, for k₁ ≠ k₂,

is orthogonal to

, we see that is

()

Definition 1. Let φ be a function in L¹(𝔻, dA_α) which is of the form e^ikθf, where f is a radial function. Then one says that φ is a quasihomogeneous function of quasihomogeneous degree k.

A direct calculation gives the following lemmas which we will use often.

Lemma 2. Let p ∈ ℕ and let φ be an integrable radial function. Then,

()

Lemma 3. Let φ be an integrable radial function. Then, for p ∈ ℤ₊,

()

and for p ∈ ℕ,

()

2. Commuting of Toeplitz Operator and Hankel Operator

Theorem 4. Let e^ipθf be a bounded function of quasihomogeneous degree p ≥ 0 and g = ∑_k∈Z e^ikθg_k,α(r) ∈ L^∞(𝔻, dA_α). Then if and only if the following conditions holds

(1)
, if 0 ≤ k ≤ p − 1 and j ≥ 0;
(2)
, if k ≥ 0 and j ≥ 0.

Proof. For j ≥ 0,

()

, then we have

()

which is equivalent to

()

that is,

()

From the aforementioned we get the following.

Case 1. For 0 ≤ k ≤ p − 1 and j ≥ 0,

()

Case 2. For k ≥ 0 and j ≥ 0,

()

As a special case of Theorem 4, we can have the following corollary.

Corollary 5. Let f be a bounded radial function and g = ∑_k∈Z e^ikθg_k,α(r) ∈ L^∞(𝔻, dA_α). Then T_fH_g = H_gT_f if and only if

()

for k ≥ 0 and j ≥ 0.

Theorem 6. Let e^−ipθf be a bounded function of quasihomogeneous degree −p < 0 and g = ∑_k∈Z e^ikθg_k,α(r) ∈ L^∞(𝔻, dA_α). Then if and only if the following conditions holds

(1)
, if k ≥ 0 and p > j ≥ 0;
(2)
, if k ≥ 0 and j ≥ p.

Proof. For j ≥ 0, we have

()

Then one has the following.

Case 1. For p > j ≥ 0,

()

Case 2. For j ≥ p,

()

, then we have the following.

Case 1. For p > j ≥ 0,

()

Case 2. For j ≥ p,

()

That is one has the following.

Case 1. For p > j ≥ 0, k ≥ 0,

()

Case 2. For j ≥ p, k ≥ 0,

()

Theorem 7. Let e^−isθg be a bounded function of quasihomogeneous degree −s ≤ 0 and f = ∑_k∈Z e^ikθf_k,α(r) ∈ L^∞(𝔻, dA_α). Then if and only if or

(1)
, if s ≥ k ≥ 0 and j > s;
(2)
j + 2), if s ≥ k ≥ 0 and s ≥ j ≥ 0;
(3)
, if k > s and s ≥ j ≥ 0.

Proof. For j ≥ 0, we have the following.

Case 1. For j > s,

()

Case 2. For s ≥ j ≥ 0,

()

Applying

, we have the following.

Case 1. For j > s,

()

Case 2. For s ≥ j ≥ 0,

()

, then the equation holds.

Otherwise , we have the following.

Case 1. For j > s,

()

Case 2. For s ≥ j ≥ 0,

()

that is one has the following.

Case 1. For s ≥ k ≥ 0, j > s,

()

Case 2. For s ≥ j ≥ 0,

()

Then we get the following.

Case 1. For s ≥ k ≥ 0, j > s, .

Case 2. For s ≥ j ≥ 0,

()

Case 3. For k > s, s ≥ j ≥ 0,

Corollary 8. Let g be a bounded radial function and f = ∑_k∈Z e^ikθf_k,α(r) ∈ L^∞(𝔻, dA_α). Then T_fH_g = H_gT_f if and only if

(1)
;
(2)
and , for k > 0.

Finally, we will investigate the situation that both functions are ordinary functions.

Theorem 9. Let f = ∑_k∈Z e^ikθf_k,α(r) ∈ L^∞(𝔻, dA_α) and g = ∑_l∈Z e^ilθg_l,α(r) ∈ L^∞(𝔻, dA_α). Then T_fH_g = H_gT_f if and only if

()

for m ≥ 0 and n ≥ 0.

Proof. For j ≥ 0, we have

()

From the aforementioned we get, for m ≥ 0 and n ≥ 0,

()

If H_gT_f = T_fH_g, then we get

()

for m ≥ 0 and n ≥ 0.

The converse is easy to get.

Acknowledgments

The author would like to thank the referees for their valuable comments and suggestions, which helped to improve the paper. This work is supported by the National Natural Science Foundation of China (no. 11126061), Innovation Program of Shanghai Municipal Education Commission (no. 13YZ090) and the Science & Technology Program of Shanghai Maritime University (no. 20120098).

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Commuting Quasihomogeneous Toeplitz Operator and Hankel Operator on Weighted Bergman Space

Abstract

1. Introduction

2. Commuting of Toeplitz Operator and Hankel Operator

Acknowledgments

References

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Commuting Quasihomogeneous Toeplitz Operator and Hankel Operator on Weighted Bergman Space

Abstract

1. Introduction

2. Commuting of Toeplitz Operator and Hankel Operator

Acknowledgments

References

References

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