Hyponormal Toeplitz operators with non-harmonic symbols on the weighted Bergman spaces

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The space L 2 ( , dA ) is a Hilbert space with the above inner product. For any nonnegative integer n and z ∈ , let where Γ(⋅) is the usual Gamma function. Then {e n } is an orthonormal basis for the weighted Bergman spaces (cf. [4]). If f , g ∈ A 2 ( ) of the form then, the inner product of f and g is Given a bounded measurable function ∈ L ∞ ( ), the Toeplitz operator T with symbol on A 2 ( ) is defined by where P denote the orthogonal projection from L 2 ( , dA ) onto A 2 ( ).
For bounded linear operators A, B on a Hilbert space, we let [A, B] ∶= AB − BA. A bounded linear operator T is said to be hyponormal if its selfcommutator [T * , T] is positive (semidefinite). Hyponormal operators theory is a broad and highly advanced field, that has contributed a lot of problems in operator theory, functional analysis, and mathematical physics. On the Hardy space H 2 ( ), the hyponormal Toeplitz operators has been researched in [1,2,5]. In [1], the author characterized the hyponormality of T on H 2 ( ) by using the properties of the symbol ∈ L ∞ ( ) as follows.
The main idea of the proof of Cowen's Theorem is a dilation theorem as in [14]. However, the Cowen's theorem cannot be utilized to A 2 ( ). So, for the weighted Bergman space, determining the hyponormality of Toeplitz operators is very difficult. In fact, there seems to be very little study of hyponormal Toeplitz operators on A 2 ( ) in the literature. In [7,8,10,11,13], the authors characterized the hyponormality of Toeplitz operators T , in terms of the coefficient of the symbol under certain assumptions on A 2 ( ). Moreover, since the hyponormality of Toeplitz operators on A 2 ( ) is translation invariant, we can assume that the constant term is zero. By the definition and properties of Toeplitz operators, we have that for g, h ∈ L ∞ ( ) and , ∈ ℂ, we have that T g+ h = T g + T h , T * g = T g , and T g T h = T gh if g or h is analytic. We briefly summarize the results relating to the hyponormality of Toeplitz operator with non-harmonic symbols on the Bergman space, which have been recently obtained in [3,9,15]. In [15], the author provide a necessary condition on the complex constant C for the operator T z n +C|z| s to be hyponormal on the Bergman space, and after that, in [12], hyponormality of Toeplitz operator T z n +C|z| s is characterized on the weighted bergman spaces. In [3], the authors consider the sufficient conditions of hyponormality of T f +g on the Bergman space where f (z) = a m,n z m z n (m > n) and Recently, the authors as in [9] characterized the necessary conditions for the hyponormality of Toeplitz operators T on the Bergman space where (z) = az m z n + bz s z t (m ≥ n, t ≥ s) with m ≠ t and In this paper, we consider the hyponormality of Toeplitz operators when is bivariate polynomials in z and z acting on the weighted Bergman space A 2 ( ). First, we briefly recall some basic consequences of the hyponormality of Toeplitz operators on A 2 ( ). Next, we study either necessary or sufficient condition for the hyponormality of Toeplitz operators T with non-harmonic symbols (z) = az m z n + bz s z t on the weighted Bergman space.

Main results
Firstly, we need the following lemmas for our program. We recall the properties of projection and norm on A 2 ( ).
Lemma 2.1 [7] For any nonnegative integers s, t, Notation 2.2 For our convenience, we shall use the following notations.
. By using the Lemma 2.1 and the relation (1.1), we have the results.

Lemma 2.3 [7]
For any nonnegative m, we deduce that Next, we deal with the hyponormality of Toeplitz operators with non-harmonic symbols (z) on A 2 ( ). In particular, (z) be a bivariate polynomials in z and z of the form (z) = az m z n + bz s z t . Many researchers have used the Proposition 1.1 as in [6] to consider the hyponormality of T . However, in the case of Toeplitz operators T with non-harmonic symbols , we cannot apply the well-known consequences of Proposition 1.1 as in [6], since the symbol cannot separated into analytic and coanalytic parts. So, we have to find the self-commutator of T directly. First, we will consider some of the relation introduced in Notation 2.2.

Sufficient condition for hyponormal Toeplitz operators
Now, we consider the sufficient condition for hyponormal Toeplitz operators T on A 2 ( ) where the symbols is of the form (z) = az m z n + bz s z t under certain assumptions about the coefficients and degree of (z).

Theorem 2.7 For any a, b ∈ ℂ and nonnegative integers m, n, s, t, if
Proof For (z) = az m z n + bz s z t , T is hyponormal if and only if for any c i ∈ ℂ. We get that We let a notation P(a, n, m) ∶= P(az n ∑ ∞ i=0 c i z m+i ) and assume that c k = 0 when k ≤ 0. Then (2.8) Using the condition m − n = s − t then we have By Lemma 2.4(ii), we have that for i ≥ m − n, In [3], showed that the sufficient conditions for the hyponormality of Toeplitz operators with nonharmonic symbols of fixed related degrees on the Bergman space. The following Example generalizes the result of [3,Theorem 12] to the space A 2 ( ).
Example Let (z) = r 1 e i 1 z m z n + r 2 e i 2 z s z t with m − n = s − t ≥ 0. If | 1 − 2 | < 2 then T on A 2 ( ) is hyponormal.
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