Hyponormality of Toeplitz operators with polynomial symbols on the weighted Bergman space

This paper gives the complete proof of the Conjecture given by Hazarika and this author jointly which deals with a necessary and sufficient condition for the hyponormality of Toeplitz operator, $$T_\varphi $$Tφ on the weighted Bergman space with certain polynomial symbols under some assumptions about the Fourier coefficients of the symbol $$\varphi $$φ.


Introduction
Let D be the open unit disc in the complex plane C. For −1 < α < ∞, let L 2 (D, d A α ) denote the Hilbert space consisting of functions on D which are square integrable with respect to the measure d A α (z) = (α + 1)(1 − |z| 2 ) α d A(z), where d A denotes the normalized Lebesgue area measure on D. The inner product on is then defined as the closed subspace of L 2 (D, d A α ) consisting of analytic functions on D. For any nonnegative integer n, if e n (z) = Γ (n+α+2) Γ (n+1)Γ (α+2) z n , (z ∈ D) with the usual Gamma function Γ (s), then the set {e n } forms an orthonormal basis for A 2 α (D) [12,13]. The reproducing kernel in A 2 α (D) is defined as K (α) A bounded linear operator T on a Hilbert space is said to be hyponormal if its self-commutator [T * , T ] := T * T − T T * is positive semi-definite. The hyponormality of Toeplitz operators T ϕ on the Hardy space H 2 (T) where T = ∂D was first characterised by Cowen [2] which was simplified by Nakazi and Takahashi [9]. For ϕ ∈ L ∞ (T), we write Then T ϕ is hyponormal if and only if E(ϕ) is nonempty. The solution was given on the basis of a dilation theorem of Sarason [11]. As for the Bergman space, no such dilation theorem exists [3]; so, the characterization for the hyponormality of Toeplitz operators on the Bergman space is still remained an open question. For ϕ = f +ḡ where f, g are bounded analytic functions, Sadraoui [10] gave a necessary and sufficient condition for the hyponormality of Toeplitz operators on the Bergman space which is equivalently true on the weighted Bergman space too. His theorem is stated below: Theorem 1.1 [10] Let f, g be bounded and analytic in L 2 (D, d A). The following statements are equivalent: where C is of norm less than or equal to one.
Recently, Hwang and Lee [5], Hwang, Lee and Park [6], Lu and Liu [7], Lu and Shi [8] and Hazarika along with this author in [4] gave some necessary and sufficient conditions for the hyponormality of Toeplitz operators on weighted Bergman space with the class of functions ϕ = f +ḡ where f, g ∈ L 2 (D, d A α ). In [4], the authors gave a Conjecture in which it was made an attempt to give a complete criteria for the hyponormality of T ϕ for this class of functions in the weighted Bergman space. The Conjecture is: In this paper, we give the proof of this Conjecture for all α > −1. Since, the hyponormality of operators is translation invariant, we may assume that f (0) = g(0) = 0. We have the following properties of Toeplitz operators: where s and t are nonnegative integers. Again, for γ k = z k α , we have

The proof of the Conjecture
To prove the Conjecture we need some specific lemmas.
If we denote the coefficients of k (n) 's of the polynomial Q n (k) by C n 's, then we have C 2m = 0, C 2m−1 = 0. Since,

That is, if and only if
Hf That is, if and only if Hḡk i (z), Hḡk j (z) α ≥ 0.
Thus, using Lemma 2.4 in (2), we have that T ϕ is hyponormal if and only if We have, Therefore, using the assumption a māN = a −mā−N , we have Also, we have And, Therefore, using the assumption a māN = a −mā−N and Lemma 2.2, we have Thus by applying (4), (5) in (3) and simplifying, we have that T ϕ is hyponormal if and only if Let for all k ≥ 1 Now, if |a N | ≥ |a −N | then, For all k = 1, 2, . . . , N − 1, we have Hence from (8) and (9), T ϕ is hyponormal if and only if By Lemma 2.3, we get If |a −N | ≥ |a N |, then |a −N | 2 − |a N | 2 |a m | 2 − |a −m | 2 ≤ ξ(k) for all k ≥ 1.
Hence in this case T ϕ is hyponormal if and only if Thus the results follow from (10) and (13).

Conclusion
This theorem may give a clue to the readers to think about the generalised form of the hyponormality of Toeplitz operators with a class of polynomial symbols relaxing some of the restrictions to the Fourier coefficients.
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