Analytic transversality and Nullstellensatz in Bergman space.

*(English)*Zbl 0784.47010
Complex analysis, Proc. Symp., Madison/WI (USA) 1992, Contemp. Math. 137, 367-381 (1992).

[For the entire collection see Zbl 0755.00016.]

The authors consider the Bergman space \(L^ 2_ a(\Omega)\) over some bounded, pseudoconvex domain \(\Omega\) in \(\mathbb{C}^ n\), that is the space of all holomorphic functions on \(\Omega\) which are square integrable with respect to the usual Lebesgue volume measure on \(\Omega\). A closed subspace of this Bergman space is called analytically invariant if it is invariated by multiplication with each function holomorphic in the neighbourhood of the closure of \(\Omega\). The authors consider some special classes of such spaces and use the notions of transversality and of privileged domain to show that Hilbert’s Nullstellensatz holds for these spaces. The authors solve some problems originating from the theory of Hilbert modules over function algebras and which are connected with this circle of ideas. For instance, given a proper ideal of the algebra of all holomorphic functions in a neighbourhood of the closure of \(\Omega\), they completely characterize the situation when this ideal is dense in the Bergman space, provided that the boundary of \(\Omega\) satisfies some nice conditions. Given a radical ideal \(\mathcal I\), the authors consider the set \(V({\mathcal I})\) of the zeros common to all functions in \(\mathcal I\), and provide the conditions under which this ideal is dense in the analytically invariant subspace of all Bergman functions which vanish on \(V({\mathcal I})\). This is also done under some supplementary assumptions on \(V({\mathcal I})\).

The authors consider the Bergman space \(L^ 2_ a(\Omega)\) over some bounded, pseudoconvex domain \(\Omega\) in \(\mathbb{C}^ n\), that is the space of all holomorphic functions on \(\Omega\) which are square integrable with respect to the usual Lebesgue volume measure on \(\Omega\). A closed subspace of this Bergman space is called analytically invariant if it is invariated by multiplication with each function holomorphic in the neighbourhood of the closure of \(\Omega\). The authors consider some special classes of such spaces and use the notions of transversality and of privileged domain to show that Hilbert’s Nullstellensatz holds for these spaces. The authors solve some problems originating from the theory of Hilbert modules over function algebras and which are connected with this circle of ideas. For instance, given a proper ideal of the algebra of all holomorphic functions in a neighbourhood of the closure of \(\Omega\), they completely characterize the situation when this ideal is dense in the Bergman space, provided that the boundary of \(\Omega\) satisfies some nice conditions. Given a radical ideal \(\mathcal I\), the authors consider the set \(V({\mathcal I})\) of the zeros common to all functions in \(\mathcal I\), and provide the conditions under which this ideal is dense in the analytically invariant subspace of all Bergman functions which vanish on \(V({\mathcal I})\). This is also done under some supplementary assumptions on \(V({\mathcal I})\).

Reviewer: V.Matache (Timişoara)

##### MSC:

47A15 | Invariant subspaces of linear operators |

46E20 | Hilbert spaces of continuous, differentiable or analytic functions |

46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |