Abstract: | In this paper we consider Hankel operators = (Id – P 1) from A 2(?, |z |2) to A 2,1(?, |z |2)⊥. Here A 2(?, |z |2) denotes the Fock space A 2(?, |z |2) = {f: f is entire and ‖f ‖2 = ∫? |f (z)|2 exp (–|z |2) dλ (z) < ∞}. Furthermore A 2,1(?, |z |2) denotes the closure of the linear span of the monomials { z n : n, l ∈ ?, 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 is bounded, but not compact, and for k ≥ 3 that 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) |