Hankel operators with antiholomorphic symbols on the Fock space

Abstract

Abstract. We consider Hankel operators of the form H z k : F m := {f : f is entire and C n |f (z)| 2 e −|z| m < ∞} → L 2 (e −|z| m ). Here k, m, n ∈ N. We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt if m > 2k. PreliminariesThe investigation of Hankel operators on the Bergman space of certain domains Ω has a long history. See, for example, [24] and [25]. Furthermore, there have been some attempts to characterize compactness of Hankel operators on L 2 spaces of entire functions. In [26] the case of essentially bounded symbols is considered. This has the advantage that the corresponding Hankel operator is bounded.There are interesting connections between the theory of partial differential equations and the theory of Hankel operators. In [12] it is shown that the canonical solution operator to ∂ restricted to (0, 1)-forms with coefficients in the spaces of holomorphic functions that are square integrable with respect to the weight function e We want to investigate operators of the formHere P denotes the Bergman projection

Generalized Hankel operators and the generalized solution operator to $ \bar \partial $ on the Fock space and on the Bergman space of the unit disc

“…We remember the analogous situation for the classic Hankel operator H z k (for details see the introduction; the mentioned results for the classic Hankel operator can be found in [19]): These Hankel operators are not bounded. However, using the projection onto A 2, 1 C, |z| 2 instead of onto A 2 C, |z| 2 will not lead to an "improvement" of operator theoretic properties in the cases k > 2.…”

“…Remark 1.4 It should be added that for m ≥ 2k the Hankel operator H z k : A 2 (C, |z| m ) → L 2 (C, |z| m ) is bounded, but for m < 2k it is not bounded. (See [19].) Remark 1.5 Note that it is possible to investigate Hankel operators with symbols in a wide class of antiholomorphic symbols with the methodology of [19].…”

“…(See [19].) Remark 1.5 Note that it is possible to investigate Hankel operators with symbols in a wide class of antiholomorphic symbols with the methodology of [19]. For details see [14].…”

“…Remark 1.6 Let k ∈ N and 0 < ε < 1/2. Then it can be shown with different methods than in [19] that the Hankel operators…”

Hankel operators and the Stieltjes moment problem

“…See also [16] and [21] for the one-dimensional case. It would be of interest to characterize Stieltjes moment sequences having this property.…”