Hankel Operators on Fock Spaces

Journal of Functional Analysis, 2009

Illinois Journal of Mathematics, 1988

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.

2021

Abstract. In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces A2ωφ , where ωφ = e −φ and φ is a subharmonic function. We consider compact Hankel operators Hφ, with anti-analytic symbols φ, and give estimates of the trace of h(|Hφ|) for any convex function h. This allows us to give asymptotic estimates of the singular values (sn(Hφ))n in terms of decreasing rearrangement of |φ′|/ √ ∆φ. For the radial weights, we first prove that the critical decay of (sn(Hφ))n is achieved by (sn(Hz))n. Namely, we establish that if sn(Hφ) = o(sn(Hz)), then Hφ = 0. Then, we show that if ∆φ(z) ≍ 1 (1−|z|2)2+β with β ≥ 0, then sn(Hφ) = O(sn(Hz)) if and only if φ belongs to the Hardy space H, where p = 2(1+β) 2+β . Finally, we compute the asymptotics of sn(Hφ) whenever φ ′ ∈ H.

Integral Equations and Operator Theory, 1995

We study Hankel operators on the harmonic Bergman spaceb 2(B), whereB is the open unit ball inR n,n≥2. We show that iff is in $C(\bar B)$ then the Hankel operator with symbolf is compact. For the proof we have to extend the definition of Hankel operators to the spacesb p(B), 1<p<∞, and use an interpolation theorem. We also use the explicit formula for the orthogonal projection ofL 2(B, dV) ontob 2(B). This result implies that the commutator and semi-commutator of Toeplitz operators with symbols in $C(\bar B)$ are compact.

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.

Duke Mathematical Journal, 1992

We describe the boundedness of a linear operator from B p (ρ) = {f : D → C analytic : D ρ(1 − |z|) (1 − |z|) |f (z)| p dA(z) 1/p < ∞} , for 0 < p ≤ 1 under some conditions on the weight function ρ, into a general Banach space X by means of the growth conditions at the boundary of certain fractional derivatives of a single X-valued analytic function. This, in particular, allows us to characterize the dual of B p (ρ) for 0 < p < 1 and to give a formulation of generalized Carleson measures in terms of the inclusion B 1 (ρ) ⊂ L 1 (D, µ). We then apply the result to the study of multipliers, Hankel operators and composition operators acting on B p (ρ) spaces.

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.

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

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.