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

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.

Generalized Hankel operators on the Fock space

“…The following results are devoted to this task and will yield a generalization of Proposition 3.2 in [7].…”

“…As mentioned in the introduction, we want to generalize the results for boundedness and compactness in [11] and [7] for the operators H 0 z k and H 1 z k , respectively, to the operators H l z k for arbitrary l ∈ N. This turns out to be much more complicated than in the special cases l = 0, 1. The first step is to compute the projection P l (z k z n+k ).…”

“…This result generalizes a presently existing ones for the cases l = 0 and l = 1 for the Fock space (cf. [11] and [7]). From an operatortheoretic point of view the methodology applied here generalizes the one of [7].…”

“…This connection motivates the investigation of the operators H l z k . The following two results can be found in [7]. Theorem 1.4 On the Fock space F 2 (|z| 2 ) the generalized Hankel operator…”

“…It completes the investigation of operator-theoretic properties of Hankel operators H 0 f with generalized conjugate-holomorphic L 2 (|z| m )-symbols on the generalized Fock spaces F 2 (|z| m ). [7]. We only mention at this point that there is a tight connection between the canonical solution operator to complex differential operators and the Hankel operators H l z k with monomial symbols.…”