Non‐compactness of the solution operator to $ \bar \partial $ on the Fock‐space in several dimensions

Abstract

We consider the canonical solution operator to ∂ restricted to (0, 1)-forms with coefficients in the generalized Fock-spacesf is entire andWe will show that the canonical solution operator restricted to (0, 1)-forms with F m -coefficients can be interpreted as a Hankel-operator. Furthermore we will show that the canonical solution operator is not compact for m ≥ 2. PreliminariesIt is well-known that in many cases compactness of the solution operator to ∂ only depends on the behavior of the solution operator on the closed subspace of forms with holomorphic coefficients (see [7], [8], [13], [16] and [6]).In [6], compactness of the solution operator in the case of bounded convex domains is investigated. It is shown in [6] that compactness of the solution operator for ∂ on (0, 1)-forms implies that the boundary of Ω -in this case Ω is a bounded convex domain -does not contain any analytic variety of dimension greater or equal to 1. The proof only requires the existence of a compact solution operator to ∂ on (0, 1)-forms with holomorphic coefficients. So in the case of bounded convex domains compactness of the solution operator restricted to (0, 1)-forms with holomorphic coefficients implies already compactness of the solution operator on general (0, 1)-forms.A similar situation appears in [16] where the Toeplitz C * -algebra T (Ω) is considered and the relation between the structure of T (Ω) and the ∂-Neumann problem is discussed. Definition 1

The Quasi-Canonical Solution Operator to $${\bar \partial }$$ Restricted to the Fock-Space

“…In this general case the solution operator is much more complicated. We will characterize compactness of the solution operator as an application and will therefore be able to generalize the results from [11] and [17]. The question of compactness of the solution operator is of interest for various reasons; see [2] for an excellent survey.…”

“…Here λ denotes the Lebesgue-measure. In [17] it is shown that the canonical solution operator S : . In both cases the solution operator has a quite simple form.…”

“…For (0, 0)-forms-n = 0-it is shown in [11] that the canonical solution operator is compact if m > 2 and for (0, 1)-forms it is shown in [17] that the canonical solution operator is not compact for all m.…”

The canonical solution operator of $ {\bar \partial } $ on weighted spaces with holomorphic coefficients

“…In [10] it turns out that the canonical solution operator S restricted to the Fock space A 2 C, |z| 2 into L 2 C, |z| 2 is not compact, and for m > 2 the restriction of S to A 2 (C, |z| m ) is compact, but not Hilbert-Schmidt. For a generalization to the higher dimensional case we refer the reader to [5].…”

Hankel operators with antiholomorphic symbols on the Fock space

“…In [23] we have seen that the Hankel operator with the symbol z can be interpreted as a globally defined operator even in the case of several variables. There we used facts from Hörmander theory and the identification of the solution operator and the Hankel operator.…”