In this paper we investigate Hankel operators H f :where A 2 m are general Fock spaces. We will show that H f is not continuous if the corresponding symbol is not a polynomial f = N k=0 b k z k . For polynomial symbols we will give necessary and sufficient conditions for continuity and compactness in terms of N and m. For monomials z k we will give a complete characterization of the Schatten-von Neumann p-class membership for p > 0. Namely in case 2k < m the Hankel operators H z k are in the Schattenvon Neumann p-class iff p > 2m/(m−2k); and in case 2k m they are not in the Schatten-von Neumann p-class.

Key words (Generalized) canonical solution operator of ∂, Hankel operator, (weighted) Bergman spaces, Fock space MSC (2000) Primary: 32A36, Secondary: 47B35, 32A15In this paper we consider Hankel operators eFurthermore A 2,1`C , |z| 2´d enotes the closure of the linear span of the monomials˘z l z n : n, l ∈ N, l ≤ 1ā nd the corresponding orthogonal projection is denoted by P1. Note that we call these operators generalized Hankel operators because the projection P1 is not the usual Bergman projection. In the introduction we give a motivation for replacing the Bergman projection by P1. The paper analyzes boundedness and compactness of the mentioned operators.On the Fock space we show that e H z 2 is bounded, but not compact, and for k ≥ 3 that e H 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.

We derive conditions for compactness of Hankel operatorsbounded, holomorphic symbols f for a large class of convex and bounded domains Ω with Ω D k .

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