Toeplitz and Hankel Operators on Weighted Bergman Spaces

2015

In this paper, we derive certain algebraic properties of Toeplitz and Hankel operators defined on the vector-valued Bergman spaces $L_a^{2, mathbb{C}^n}(mathbb{D})$, where $mathbb{D}$ is the open unit disk in $mathbb{C}$ and $ngeq 1.$ We show that the set of all Toeplitz operators $T_{Phi}, Phiin L_{M_n}^{infty}(mathbb{D})$ is strongly dense in the set of all bounded linear operators ${mathcal L}(L_a^{2, mathbb{C}^n}(mathbb{D}))$ and characterize all finite rank little Hankel operators.

Journal of Operator Theory, 2015

If µ is a finite measure on the unit disc and k ≥ 0 is an integer, we study a generalization derived from Englis's work, T (k) µ , of the traditional Toeplitz operators on the Bergman space A 2 , which are the case k = 0. Among other things, we prove that when µ ≥ 0, these operators are bounded if and only if µ is a Carleson measure, and we obtain some estimates for their norms.

2016

Abstract. In this paper we give necessary and sufficient conditions for the hyponormality of the Toeplitz operator Tϕ on the weighted Bergman space A 2 α(D) for ϕ(z) = ā−3 z̄ 3 + ā−1z̄+ a1z+ a3z 3 and ϕ(z) = ā−3z̄ 3 + ā−2z̄ 2 + a2z 2 + a3z 3. 1.

Journal of Functional Analysis, 1999

We consider the question for which square integrable analytic functions f and g on the unit disk the densely defined products T f T gÄ are bounded on the Bergman space. We prove results analogous to those obtained by the second author for such Toeplitz products on the Hardy space. We furthermore obtain similar results for Hankel products H f H g * , where f and g are square integrable on the unit disk, and for the mixed Haplitz products H f T gÄ and T g H f * , where f and g are square integrable on the unit disk and g is analytic.

Transactions of the American Mathematical Society, 1992

In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in C whose symbols are bounded measurable functions. We give necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Toeplitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of H°°(Ci) ; for these symbols we describe when the Toeplitz or Hankel operators are compact.

We consider the question for which square integrable analytic functions f and g on the unit disk the densely defined products T f T g are bounded on the Bergman space. We prove results analogous to those we obtained in the setting of the unweighted Bergman space [17]. We will furthermore completely describe when the Toeplitz product T f T g is invertible or Fredholm and prove results generalizing those we obtained for the un-weighted Bergman space in [18].

Afrika Matematika, 2019

An operator T on a Hilbert space is hyponormal if T*T-TT* is positive. In this work we consider hyponormality of Toeplitz operators on the Bergman space with a logarithmic weight. Under a smoothness assumption we give a necessary condition when the symbol is of the form f + g with f , g analytic on the unit disk. We also find a sufficient condition when f is a monomial and g a polynomial.

Journal of Inequalities and Applications, 2021

For 1 ≤ p < ∞, let A p ω be the weighted Bergman space associated with an exponential type weight ω satisfying D K z (ξ) ω(ξ) 1/2 dA(ξ) ≤ Cω(z)-1/2 , z ∈ D, where K z is the reproducing kernel of A 2 ω. This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂-equation (Theorem 2.5) for more general weight ω *. As an application, we prove the boundedness of the Bergman projection on L p ω , identify the dual space of A p ω , and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from A p ω into A q ω , 1 ≤ p, q < ∞, such as Toeplitz and (big) Hankel operators.

AIMS mathematics, 2024

We first describe commuting Toeplitz operators with harmonic symbols on weighted harmonic Bergman spaces. Then, a sufficient condition for hyponormality on weighted Bergman spaces of the punctured unit disk, when the analytic part of the symbol is a monomial, is shown.

Journal of the Australian Mathematical Society, 1998

We consider the Bergman spaces consisting of harmonic functions on the unit ball in Rn that are squareintegrable with respect to radial weights. We will describe compactness for certain classes of Toeplitz operators on these harmonic Bergman spaces.