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

In 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 space A 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)