Finite-rank intermediate Hankel operators on the Bergman space

Integral Equations and Operator Theory, 1994

In this paper we consider a class of weighted integral operators on L 2 (0, ∞) and show that they are unitarily equivalent to little Hankel operators between weighted Bergman spaces of the right half plane. We use two parameters α, β ∈ (−1, ∞) and involve two weights to define Bergman spaces of the domain and range of the little Hankel operators. We obtained conditions for the Hankel integral operator to be Hilbert-Schmidt, nuclear, finite rank and compact, expressed in terms of the kernel of the integral operator. For certain class of weights, these operators are shown to be unitarily equivalent to little Hankel operators between weighted Bergman spaces of the disk, and the symbol correspondence is given. In view of the strong link between Hankel operators and best approximation, some asymptotic results on the singular values of Hankel integral operators are also provided.

Glasgow Mathematical Journal, 1997

We prove the compactness of certain Hankel operators on weighted Bergman spaces of harmonic functions on the unit ball in Rn.

Journal of Functional Analysis, 2009

Mathematical Reports, 2023

In this paper, we consider little Hankel operators Γφ defined on the Bergman space L 2 a (D) with symbol φ ∈ H ∞ (D) that are contractions. Necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part of these little Hankel operators. We also present an explicit description of this unitary part. This extends the results of Butz for Hankel operators defined on the Hardy space.

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.

2020

In this paper we have shown that if S ∈ L(L 2 a (dA α )) and Θ ST (x, y)(K (α) (x, y)) 2 for all x, y ∈ D and for all T ∈ L(L 2 a (dA α )), then S = T

Indian Journal of Pure and Applied Mathematics, 2010

In this paper we derive certain algebraic properties of Toeplitz and Hankel operators defined on the vector-valued Bergman spaces L 2,C n a (D) where D is the open unit disk in C and n ≥ 1. We show that the set of all Toeplitz operators T Φ , Φ ∈ L ∞ Mn (D) is strongly dense in the set of all bounded linear operators L(L 2,C n a (D)) and characterize all finite rank little Hankel operators.

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.

Journal of Geometric Analysis, 2011

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on C n . The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander's L 2 estimates for the ∂ operator are key ingredients in the proof.

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.