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

Abstract

AbstractIn this paper we consider Hankel operators $ \tilde H _{{\bar z}^k}$ = (Id – P 1)$ \bar z^k $ from A 2(ℂ, |z |2) to A 2,1(ℂ, |z |2)⊥. Here A 2(ℂ, |z |2) denotes the Fock spaceA 2(ℂ, |z |2) = {f: f is entire and ‖f ‖2 = ∫ℂ |f (z)|2 exp (–|z |2) dλ (z) < ∞}.Furthermore A 2,1(ℂ, |z |2) denotes the closure of the linear span of the monomials {$ \bar z ^l $z n : n, l ∈ ℕ, l ≤ 1} and the corresponding orthogonal projection is denoted by P 1. Note that we call these operators generalized Hankel operators because the projection P 1 is not the usual Bergman projection. In the introduction we give a motivation for replacing the Bergman projection by P 1. The paper analyzes boundedness and compactness of the mentioned operators.On the Fock space we show that $ \tilde H _{{\bar z}^2}$ is bounded, but not compact, and for k ≥ 3 that $ \tilde H _{{\bar z}^k}$ is not bounded. Afterwards we also consider the same situation on the Bergman space of the unit disc. Here a completely different situation appears: we have compactness for all k ≥ 1.Finally we will also consider an analogous situation in the case of several complex variables. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)