# On the rigidity of Bergman submodules.

Introduction. The search for and the classification of invariant subspaces of various operators acting on function spaces have proved to be two very rewarding research problems in operator theory and harmonic analysis. Not only that the questions themselves have turned out to be important, but even more so did they applications and the techniques used to solve them. For instance the classical theorem of Beurling [3] on the structure of analytically invariant subspaces of the Hardy space on the torus has been influential in many areas of modern mathematics, ranging from the dilation theory of a contraction, to interpolation problems in function theory or to the probabilistic analysis of time series.In Hilbert spaces of analytic functions depending on several complex variables, such a simple classification as that of Beurling's theorem is no longer to be expected. Even the existence of inner functions can be a difficult question. The case of a Hardy space supported by a polydisk in [C.sup.n] has been, however, successfully studied during the last two decades, see [12], [1]. More recently, R.G. Douglas and his collaborators have developed a general framework for investigating invariant subspaces from the point of view of Hilbert modules over function algebras, [5], [6], [7]. The importance of the algebraic language of modules and their operations lies in its being deeply rooted into the structure theory of algebras of analytic functions. Thus the comparison between Hilbert modules and, say, sheaves of ideals of analytic functions becomes transparent and benefic.

In this way Douglas and Paulsen [5] have isolated from a series a previous works a phenomenon which is specific only to multidimensional Hardy submodules: once two such submodules are analytically isomorphic they are equal, see [1], [5], [6] and [7] for the precise statements. This rigidity behaviour of Hardy submodules contrasts with Beurling's theorem--the latter implying that all invariant subspaces of the Hardy space on the torus are unitarily equivalent. When compared with the purely algebraic case of ideals of a regular, Noetherian local ring, this anomaly is no longer surprising. The transition from a Hardy submodule to an ideal of a local ring, which is necessary in any proof of a rigidity result as before, is made by a localization functor, see [6].

The present note introduces a class of invariant subspaces of the Bergman space of a bounded pseudoconvex domain in [C.sup.n], called privileged Bergman submodules. These subspaces are in general position with respect to the Bergman space (more exactly with respect to a natural localization functor). Their classification, including the rigidity behaviour, follows the pattern discovered by Douglas and Paulsen [5]. The terminology of privileged submodules is borrowed from Douady's thesis [4], where a somewhat inverse operation from sheaves to Hilbert modules has imposed these objects. Actually the aims of the two approaches are identical: to find the moduli space of the subspaces of a given space (i.e. Hardy space respectively an analytic space).

The analysis of privileged Bergman submodules, as carried out in this note, provides a model of good behaviour with respect to classification questions, at least for domains with a sufficiently regular boundary or for polydomains. The distinction between privileged and nonprivileged submodules of a Bergman space lies ultimately in a division problem of analytic functions with [L.sup.2]-bounds. Fortunately the latter question is understood both in sheaf-homological terms [9] and in terms of partial differential equations [13].

The unitary equivalence of Bergman submodules reduces actually to the equality relation in all cases. This striking difference between Bergman and Hardy spaces is explained in the last part of the note.

The author thanks Keren Yan for his constructive remarks made on a first version of the manuscript.

1. Preliminaries. Let [Omega] be a bounded pseudoconvex domain of [C.sup.n], n [is greater than or equal to] 1 and let [Mathematical Expression Omitted] denote the space of analytic functions in [Omega] which are square summable with respect to the Lebesgue measure in [Omega]. This is the Bergman space of the domain [Omega]; it is a closed subspace of [L.sup.2]([Omega]) which contains the entire functions.

The Bergman space is obviously a Hilbert module over the algebra O(U) of analytic functions defined in an open neighborhood U of [Mathematical Expression Omitted]. A Bergman submodule is a closed subspace S of [Mathematical Expression Omitted], which is also a O(U)-submodule of [Mathematical Expression Omitted] for every open set U which includes [Mathematical Expression Omitted]. We will write in short [Mathematical Expression Omitted].

Some obvious examples of Bergman submodules are [Mathematical Expression Omitted], where I is an ideal of [Mathematical Expression Omitted], or [Mathematical Expression Omitted], where V is a closed subset of [Omega]. The relations between these two categories of subspaces is discussed in [11]. The Bergman submodules of finite codimension are also known, at least for a sufficiently regular boundary of [Omega], and they are of the form [Mathematical Expression Omitted], where I is a polynomial ideal with finitely many zeroes inside [Omega], see [2] and [10].

Among all Frechet O([C.sup.n])-modules there is a distinguished class of objects which can be naturally localized. For instance the algebra O([C.sup.n]) itself is localized by the sheaf of holomorphic functions:

U [right arrow] O(U), U [subset] [C.sup.n] open.

More generally, the quotient algebra O([C.sup.n])/I, where I is a finitely generated ideal of O([C.sup.n]), is naturally localized by the coherent analytic sheaf:

U [right arrow] O(U)/I [center dot] O(U), U [subset] [C.sup.n] open.

In fact an arbitrary coherent analytic sheaf F defined on [C.sup.n] localizes its global section space F([C.sup.n]) (which is a Frechet O([C.sup.n])-module).

But there are some more examples of naturally localizable O([C.sup.n])-modules. For instance the space of smooth functions [Epsilon]([C.sup.n]) is localized by the Frechet analytic sheaf:

U [right arrow] [Epsilon](U), U [subset] [C.sup.n] open.

And similarly behaves any classical space of differentiable or measurable functions. More interesting for us is the following sheaf of analytic modules:

F(U) = {f [is an element of] O(U [intersection] [Omega]); [[[f]].sub.2],K [intersection][Omega]] [is less than] [infinity], [Mathematical Expression Omitted],

defined for an open subset U of [C.sup.n], where [Omega] is a fixed bounded pseudoconvex domain of [C.sup.n] . It is obvious that [Mathematical Expression Omitted] and that the support of the sheaf jr is [Mathematical Expression Omitted]. Thus F is a proper localization of the Bergman space [Mathematical Expression Omitted].

So it is very natural to ask whether there is a common property shared by all these localizable Frechet O([C.sup.n])-module. An answer to that question was forseen by E. Bishop in the early years of abstract spectral theory, in the fifties. The formal answer came much later, in the seventies, in the works of analytic geometry of the French school, see [10] and [11] for references and details. The conclusion of these studies is that a slightly relaxed coherence condition characterizes the localizable Frechet O([C.sup.n])-modules.

To be more precise, let S be a Frechet O([C.sup.n])-module. Then the following statements are equivalent (see [10]):

a) There exists an infinite, topologically-free resolution of S of the form:

[Mathematical Expression Omitted],

where [E.sub.0], [E.sub.1], . . . are Frechet spaces and [Mathematical Expression Omitted] is a fixed complete topological tensor product;

b) For every open polydisk [Delta] of [C.sup.n], the Koszul complex: [Mathematical Expression Omitted]

is exact in positive degree and it has Hausdorff homology in degree zero.

In fact condition b) has an intrinsic homological meaning (i.e. the tensor product [Mathematical Expression Omitted] is Hausdorff and its derived functors, defined in an appropriate category, vanish).

A Frechet O([C.sup.n])-module S is called quasicoherent if one of the above condition a) or b) is satisfied. In that case the Frechet analytic sheaf:

[Mathematical Expression Omitted], U [subset] [C.sup.n] open,

localizes the global space S, exactly in the sense suggested by the above examples. As a convention of notation, a script letter S will denote the sheaf

[Mathematical Expression Omitted], U [subset] [C.sup.n] open,

associated to a quasi-coherent O([C.sup.n])-module S. These Frechet analytic sheaves have most of the cohomological properties of analytic coherent sheaves (whence the terminology).

An important fact to be used in the sequel is that the Bergman space [Mathematical Expression Omitted] of a bounded pseudoconvex domain [Omega] is a quasi-coherent O([C.sup.n])-module. For some consequences of this result and references for its proof see [10].

In the next definition we adopt a terminology originating in A. Douady's thesis [4].

Definition 1. A Bergman submodule [Mathematical Expression Omitted] is said to be privileged if the Hilbert O([C.sup.n])-modules S and [Mathematical Expression Omitted] are quasi-coherent and [Mathematical Expression Omitted], where these spaces are regarded as subspaces of O([Omega]).

In the above definition the quasi-coherence of S follows from that of [Mathematical Expression Omitted], via a standard long exact sequence of Tor's argument. In that case the support of the quotient module [Mathematical Expression Omitted] is the support of the sheaf [O.sub.[Omega]]/S[where] [Omega]. A few examples will clarify the meaning of this definition.

Let [Omega] be as above a bounded pseudoconvex domain of [C.sup.n] and let I be a coherent sheaf of ideals of [Mathematical Expression Omitted]. In particular there exists by Hilbert's Syzzgy Theorem a finite free resolution L of I. Then the submodule [Mathematical Expression Omitted] is privileged if it is a closed subspace of the Bergman space [Mathematical Expression Omitted] and the augmented complex [Mathematical Expression Omitted] is exact. This implies that a function [Mathematical Expression Omitted] belongs to the submodule [Mathematical Expression Omitted] if and only if f/[Omega] [is an element of] I ([Omega]), or equivalently, if and only if [f.sub.x] [is an element of] [I.sub.x], x [is an element of] [Omega], at the level of stalks.

Example 1. ([10]) Let [Omega] be a bounded strictly pseudoconvex domain with smooth boundary in [C.sub.n]. Then any Bergman submodule of finite codimension in [Mathematical Expression Omitted] is privileged.

The quasi-coherence of such a submodule S is proved in [10], while the other condition follows from the fact that [Mathematical Expression Omitted] is a module supported entirely by [Omega].

Example 2. ([11]) Let [Omega] be a bounded strictly pseudoconvex domain with smooth boundary in [C.sup.n], and let [Mathematical Expression Omitted] be the reduced ideal attached to a smooth, complex analytic subvariety V of a neighborhood of [Mathematical Expression Omitted].

If V intersects transversally [Delta][Omega], then the Bergman submodule [Mathematical Expression Omitted] is privileged.

In fact in this example V can be a union of smooth, complex submanifolds which intersect transversally [Delta][Omega], see [11].

In order to present the next class of examples we need some more terminology. Let [Omega] = [[Omega].sub.1] x . . . x [[Omega].sub.n] be a polydomain in [C.sup.n]. The boundary of [Omega] admits a filtration with closed subsets:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] denotes the set of those points with at least p coordinate entries belonging to the boundaries of [Delta][[Omega].sub.1], . . ., [Delta][[Omega].sub.n], respectively.

On the other hand, if I is a coherent ideal of [O.sub.U], U being an open neighborhood of [Mathematical Expression Omitted], the support of the sheaf [O.sub.U/I] can be canonically stratified by the locally closed analytic sets:

[S.sub.p][O.sub.U/I]) = {x [is an element of] U; prof [O.sub.U,x]/[I.sub.x] [is less than or equal to] p}, 0 [is less than or equal to] p [is less than or equal to] n,

see [8] for details. We recall that dim [S.sub.p]([O.sub.U/]I) [is less than or equal to] p for all p.

Example 3. ([11]) Let [Omega] = [[Omega].sub.1] x . . . x [[Omega].sub.n] be a bounded polydomain of [C.sup.n], let U be an open neighborhood of [Mathematical Expression Omitted] and let I be a coherent ideal of [O.sub.U]. Assume that [Mathematical Expression Omitted] for every i = 1, . . ., n.

Then the Bergman subspace [Mathematical Expression Omitted] is privileged if and only if [Mathematical Expression Omitted] for 0 [is less than or equal to] p [is less than or equal to] n - 1.

For instance if I is a locally complete intersection ideal with m local generators, then it is well known that [S.sub.p]([O.sub.U]/I) = 0 for p [is not equal to] n - m and consequently Supp([O.sub.U/I]) = [S.sub.n-m]([O.sub.U/I]). Therefore the subspace [Mathematical Expression Omitted] is privileged if and only if [Mathematical Expression Omitted]. In particular the subspace [Mathematical Expression Omitted] corresponding to a hypersurface H given by the equation f = 0 is privileged if and only if [Mathematical Expression Omitted].

It is worth remarking that, in the conditions of Example 3, the Bergman subspace [Mathematical Expression Omitted] is privileged if and only if the subspace [Mathematical Expression Omitted] is closed in [Mathematical Expression Omitted], or if and only if [Mathematical Expression Omitted], see [9].

Also, we have to mention finally that a Nullstellansatz phenomenon holds in Bergman spaces. Quite specifically, let [Omega] be a bounded pseudoconvex domain of [C.sup.n] and let I be a coherent ideal of [Mathematical Expression Omitted]. Then there exists a positive integer N with the property that:

[Mathematical Expression Omitted],

see [13]. However, in general the integer N cannot be assumed to be one, so the above definition of a privileged Bergman submodule is indeed restrictive, see again [13] for a counterexample.

2. Classification of privileged Bergman submodules. In this section we are proving that the privileged Bergman submodules supported by (analytic) sets of codimension 2 are classified up to quasi-similarity by their "algebraic" part. This conclusion agrees with the other similar results of Douglas and Paulsen or Douglas and Yah [5], [6], [7] and with the original philosophy of Douady [4].

THEOREM 1. Let [S.sub.1] and [S.sub.2] be privileged Bergman submodules of the Bergman space associated with a bounded pseudoconvex domain [Omega] of [C.sup.n]. Assume that there exist two continuous [Mathematical Expression Omitted]-linear morphisms [u.sub.1] : [S.sub.1] [right arrow] [S.sub.2], [u.sub.2]: [S.sub.2] [right arrow] [S.sub.1] both having dense range.

If the supports of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] have at least codimension 2 in [Omega], then [S .sub.1] = [S.sub.2].

Proof. Let [S.sub.1], [S.sub.2] denote the quasi-coherent sheaves corresponding to the Bergman subspaces [S.sub.1] and [S.sub.2], respectively. Since [S.sub.1[where][Omega]] and [S.sub.2][where][Omega]] are analytic submodules of [O.sub.[Omega]], both are finitely generated at the level of stalks. Moreover, the quasi-coherence assumption implies that:

[Mathematical Expression Omitted],

where z are the coordinates in [Omega] and w denotes the n-tuple of multiplication by the complex variables in [S.sub.i], i = 1, 2. Thus for a fixed z [is an element of] [Omega], the operator [Mathematical Expression Omitted] has finite codimensional range, whence [S.sub.i[where][Omega]] are even coherent [O.sub.[Omega]]-modules i = 1, 2. In particular Supp([S.sub.i[where][Omega]]), i = 1, 2, are closed analytic subsets of [Omega].

Since [max.sub.i=1,2] dim Supp ([O.sub.[Omega]]/[S.sub.i[where][Omega]]) [is less than or equal to] n - 2, the two coherent sheaves [S.sub.1[where][Omega]] and [S.sub.2[where][Omega]] coincide if and only if [S.sub.1[where][Omega]] [cross product] [O.sub.[Omega]] M [is congruent to] [S.sub.2[where][Omega]] [cross product] [O.sub.[Omega]] M as [O.sub.[Omega]]-modules, where M is a coherent [O.sub.[Omega]]-module with dim M([Omega]) [is less than] [infinity]. This is a result of M. Artin and Grothendieck whose relevance for operator theoretical questions was discovered by Douglas and Paulsen, see [5], Theorem 6.9 and [6]. Once we have proved that [S.sub.1[where][Omega]] = [S.sub.2[where][Omega]] as submodules of [O.sub.[Omega]], by the very definition of privileged Bergman submodules we would infer that:

[Mathematical Expression Omitted].

Let M be a finite dimensional O([Omega])-module, necessarily of the form M = M([Omega]), where M is a coherent [O.sub.[Omega]]-module supported by finitely many points of [Omega]. The two morphisms [u.sub.1] and [u.sub.2] in the statement induce continuous linear maps:

[Mathematical Expression Omitted]

again with dense range because M is a finitely generated [Mathematical Expression Omitted]-module and the tensor product involves only quotient operations. On the other hand [Mathematical Expression Omitted] , i = 1, 2, because the module M is supported by [Omega]. In fact, by the same argumnet one finds that:

[S.sub.i]([Omega]) [[symmetry].sub.[O.sub.([Omega])]] M [congruent] [[symmetry].sub.x [is an element of] Supp M [S.sub.i,x] [[symmetry].sub.[O.sub.[Omega],x]] [M.sub.x] , i = 1, 2

where the right hand spaces are obviously of finite dimension. Thus the maps [u.sub.1] [symmetry] id and [u.sub.2] [symmetry] id above are onto, hence isomorphic.

In conclusion the O([Omega])-modules [S.sub.1]([Omega]) [[symmetry].sub.[O[Omega]] M and [S.sub.2]([Omega]) x O([Omega]) M are isomorphic and consequently [S.sub.1 [where] [Omega]] = [S.sub.2 [where] [Omega]]. This completes the proof of Theorem 1.

Each of the examples discussed in the previous section provides a concrete instance in which Theorem 1 applies. We confine ourselves to explicitly stating only a couple of applications in low-dimensional domains.

a) Let [Omega] be a bounded domain with smooth boundary in C. Let I be a coherent ideal of [Mathematical Expression Omitted], that is a principal ideal generated by a function [Mathematical Expression Omitted]. Then the subspace [Mathematical Expression Omitted] is privileged if and only if f has not zeroes on the boundary of [Omega] (see [10]), and in that case [Mathematical Expression Omitted] is obviously an isomorphism of Hilbert [Mathematical Expression Omitted]-modules. So, for this class of Bergman submodules there is only a single class of equivalence with respect to the similarity.

An arbitrary privileged Bergman submodule S of [Mathematical Expression Omitted] for [Omega] [subset] C arbitrary is necessarily of the form:

[Mathematical Expression Omitted],

where [([[Lambda].sub.k]).sub.k[is an element of]K] is a discrete subset of [Omega] (hence at most countable) and [n.sub.k] are non-negative integers, k [is an element of] K. Indeed as a closed submodule of [Mathematical Expression Omitted], S is subnormal, therefore it is analytically transversal to [Mathematical Expression Omitted], see [10] and the references there. Moreover, the coherent [O.sub.[Omega]]-module [Mathematical Expression Omitted] is supported by a discrete subset [([[Lambda].sub.k]).sub.k [is an element of] K] of [Omega], and for every k [is an element of] K there is a nonnegative integer [n.sub.k] such that [[S.sub.[[Lambda].sub.k] = [(z - [[Lambda].sub.k]).sup.[n.sub.k]+1][O.sub.[[Lambda].sub.k]. Finally, the assumption [Mathematical Expression Omitted] implies the equality (1).

It seems to be a nontrivial problem to decide whether a subspace like S above is generated by a single function, or equivalently, whether it is isomorphic with [Mathematical Expression Omitted] as an [Mathematical Expression Omitted]-module. A separate study of [([[Lambda].sub.k]).sub.k [is an element of] K] as an [Mathematical Expression Omitted]-interpolation sequence would be relevant for this question.

b) In two complex dimensions we will analyze only two distinct cases corresponding to Examples 2) and 3) in the preceding section.

Let [Omega] be a bounded strictly pseudoconvex domain with smooth boundary in [C.sup.2] and let I be a coherent submodule of [Mathematical Expression Omitted]. Then the support of [O.sub.[Omega]/I] consists of a union [V.sub.0] [union] [V.sub.1] of analytic sets of dimension 0, respectively 1. If the Bergman subspace [Mathematical Expression Omitted] is privileged, then [V.sub.0] [subset] [Omega] and each irreducible component of [V.sub.1] intersects [Omega]. Otherwise this space would not be closed in the hilbertian norm, see [11].

If [V.sub.1] = 0, then Theorem 1 applies and the corresponding Bergman subspaces, automatically of finite codimension, are rigid. In the presence of components of dimension one in [V.sub.1], one cannot conclude in general that the respective subspaces are rigid. For instance, if V is a smooth complex submanifold of a neighborhood of [Mathematical Expression Omitted], given globally by a single equation f = 0 and transversal to the boundary of [Omega], then:

[Mathematical Expression Omitted]

is an isomorphism of privileged Bergman submodules which are different.

Turning to the class of examples 3), one can characterize a privileged Bergman submodule [Mathematical Expression Omitted] and [Mathematical Expression Omitted] is a coherent ideal) by [V.sub.0] [intersection] [Delta][Omega] = 0 and [V.sub.1] [intersection] [[Delta].sub.0][Omega] = 0. Here [V.sub.0] [union] [V.sub.1] is again the decomposition in pure dimensional components of [Mathematical Expression Omitted] and [[Delta].sub.0][Omega] stands for the distinguished boundary of [Omega]. Exactly as above the rigidity phenomenon persists for [V.sub.1] = 0, but not in general. In the opposite situation when [V.sub.0] = 0 we conclude this discussion by an example which shows that not all privileged subspaces are isomorphic.

PROPOSITION 1. Let A be a bounded annulus in C and let [Omega] = A x A. There exists a coherent ideal I of [Mathematical Expression Omitted], supported by an analytic subset of pure codimension 1 with the property that [Mathematical Expression Omitted] is a privileged Bergman submodule which is not isomorphic with [Mathematical Expression Omitted].

Proof. Assume that there exists a continuous, [Mathematical Expression Omitted]-linear isomorphism [Mathematical Expression Omitted]. Since [Mathematical Expression Omitted] is a dense subset of [Mathematical Expression Omitted], it follows that u(f) = fu(1) for any [Mathematical Expression Omitted] and whence the ideal I([Omega]) is principal in O([Omega]), generated by u(1).

It remains to prove that there exists an ideal [Mathematical Expression Omitted] as in the statement such that I([Omega]) is not a principal ideal in O([Omega]). Since [H.sup.2]([Omega], Z) [is not equal to] 0, it is known that there exists a divisor D in a neighborhood of [Mathematical Expression Omitted] which is not principal, even when restricted to [Omega], see for instance [8]. By a small perturbation which does not affect the class of D we may assume that D [intersection]([Delta]A x [Delta]A) = 0. Thus the ideal [I.sub.D] of germs of functions vanishing on D generates a localizable Bergman submodule [Mathematical Expression Omitted] but [I.sub.D]([Omega]) is not generated in O([Omega]) by a single function.

For Bergman spaces associated with domains in more than two complex variables, again only the divisors on which vanish the elements of a privileged Bergman submodule can lead to a nonrigidity behaviour. The next result shows that when there are geometric criteria to test the privilegeness of a Bergman submodule, there is still a chance to classify these invariant subspaces.

THEOREM 2. Let [Omega] = [[Omega].sub.1] x ... x [[Omega].sub.n] C [C.sup.n] be a bounded polydomain of [C.sup.n], with [Delta][[Omega].sub.i] smooth, i = 1 ,..., n, and let [Mathematical Expression Omitted] be a coherent ideal.

If [H.sup.2]([Omega], Z) = 0 and [Mathematical Expression Omitted] is a privileged Bergman submodule of [Mathematical Expression Omitted], then there exists a unique pivileged Bergman submodule S which is isomorphic with [Mathematical Expression Omitted] and so that [S.sup.[perpendicular]] is supported by an analytic set of codimension at least 2 in [Omega].

Proof. Since the submodule [Mathematical Expression Omitted] is privileged the analytic sets [Mathematical Expression Omitted] which stratify the support of the sheaf [Mathematical Expression Omitted] satisfy the condition in Example 3: [Mathematical Expression Omitted]. Let X denote the union of the irreducible components of [Mathematical Expression Omitted] of dimension n - 1. Then obviously [Mathematical Expression Omitted] and by taking into account the multiplicities of these irreducible components in I, there exists a divisor D supported by X and such that I [subset] [I.sub.D]. Moreover, according to the main result of [9], the submodule [Mathematical Expression Omitted] is privileged.

On the other hand, the condition [H.sup.2]([Omega], Z) = 0 propagates to an open neighborhood U of [Mathematical Expression Omitted] ([Delta][Omega].sub.i] are smooth) on which the divisor D is defined. Thus D is a principal divisor and consequently there exists an analytic function [Mathematical Expression Omitted] such that D = (f) and respectively [Mathematical Expression Omitted].

By pulling back the subspace [Mathematical Expression Omitted] via the isomorphism [Mathematical Expression Omitted] one finds a closed submodule S of [Mathematical Expression Omitted], isomorphic with [Mathematical Expression Omitted]. Hence S is analytically transversal to [Mathematical Expression Omitted] and its associated sheaf S is an ideal of [Mathematical Expression Omitted] with the property: f(S/[Omega]) = I/[Omega]. In particular a function [Mathematical Expression Omitted] belongs to S if and only if (fg)[where][is an element of]I(Omega] which in turn is equivalent to [f.sub.x][g.sub.x] [is an element of] [I.sub.x] on stalks, x [is an element of] [Omega], that is [g.sub.x] [is an element of] [S.sub.x]. Thus S is a privileged Bergman submodule. In addition, for every point x [is an element of] [Omega] one has the isomorphism:

[O.sub.x]/[S.sub.x] [congruent] f[O.sub.x]/[I.sub.x]

induced by the multiplication by f. This shows that [dim.sub.x] Supp(O/S) [is less than or equal to] n - 2 because the irreducible components in the support of f[O.sub.x]/[I.sub.x] are those of [O.sub.x]/[I.sub.x] minus [X.sub.x].

In conclusion the submodule S has the properties in the statement. It is unique in virtue of Theorem 1 and thus the proof of Theorem 2 is finished.

COROLLARY 1. In the conditions of Theorem 2, let [I.sub.1] and [I.sub.2] be two coherent ideals of [O.sub.[Omega]] which produce privileged Bergman submodules and let [S.sub.1], [S.sub.2] be their corresponding isomorphic subspaces whose complements are supported in codimension 2.

Then there are continuous [Mathematical Expression Omitted]-linear morphisms

[Mathematical Expression Omitted],

both with dense range, if and only if [S.sub.1] = [S.sub.2].

The proof is a simple application of Theorem 1.

We notice that Theorem 2 above has not the most general statement. The reader will easily find its counterparts which cover for instance the class of Examples 2) or some polydomains with irregular boundary.

3. Unitary equivalence of Bergman submodules. Although the result contained in this section is not new, it is not explicitly mentioned in the current literature on invariant subspaces. On the other hand, the theorem below completes the classification picture of the Bergman submodules.

THEOREM 3. Let [Omega] be a bounded pseudoconvex domain of [C.sup.n] and let [S.sub.1], [S.sub.2] be closed, analytically invariant subspaces of the Bergman space [Mathematical Expression Omitted].

If there exists a unitary [Mathematical Expression Omitted]-linear isomorphism between [S.sub.1] and [S.sub.2], then [S.sub.1] = [S.sub.2].

Proof. Let u : [S.sub.1] [approaches] [S.sub.2] be a unitary and analytic isomorphism. We claim that u extends to a unitary [Mathematical Expression Omitted]-linear map U : [L.sup.2]([Omega]) [approaches] [L.sup.2]([Omega]), where z = ([z.sub.1],..., [z.sub.n]) are the coordinates in [C.sup.n].

This fact can be proved by using the theory of subnormal n-tuples of commutative operators, or directly, as we sketch below.

First one remarks that, for any nonzero element [Mathematical Expression Omitted], the subspace [Mathematical Expression Omitted] is dense in [L.sup.2]([Omega]). Indeed otherwise there would be a nontrivial function g [is an element of] [L.sup.2]([Omega]) orthogonal to [Mathematical Expression Omitted] and hence the measure [Mathematical Expression Omitted] would vanish on any real analytic polynomial. But f [is not equal to] 0 almost everywhere, hence g would be zero almost everywhere, a contradiction.

Next we define the extension U of the isomorphism u by:

[Mathematical Expression Omitted],

for p, q [is an element of] C[z] and x [is an element of] [S.sub.1]. The definition is correct because:

[Mathematical Expression Omitted],

for any m [is greater than or equal to] 1, [p.sub.j], [q.sub.j] [is an element of] C[z] and [x.sub.j] [is an element of] [S.sub.1], 1 [is less than or equal to] j [is less than or equal to] m.

Since the polynomials in z and [Mathematical Expression Omitted] are dense in [L.sup.2]([Omega]), it follows that U(h) = hU(1) for any h [is an element of] [L.sup.2]([Omega]), therefore U(1) = [Psi], where [Psi] [is an element of] [L.sup.[infinity]]([Omega]) and [absolute value of] [Psi](z) = 1 a.e. But [Psi][S.sub.1] = [S.sub.2], whence the function [Psi] is analytic, at least off the common zeroes set V of [S.sub.1]. But the latter set V is a proper analytic subset of [Omega] of Lebesgue measure zero and with connected component in [Omega]. Therefore the function [Psi] is constant on [Omega]\V and consequently [S.sub.1] = [S.sub.2].

This completes the proof of Theorem 3.

We conjecture that the same rigidity phenomenon holds for Bergman-Sobolev submodules of the space of analytic functions in [Omega] with all their complex derivatives up to a fixed order in [L.sup.2]([Omega]).

The main distinction between Bergman and Hardy submodules lies in the fact that Theorem 3 ceases to be valid on Hardy spaces. The unitary classification of the Hardy submodules requires some different and more subtle techniques of boundary values of analytic functions, see [1], [5], [7].

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, RIVERSIDE, CA 92521

REFERENCES

[1] O.P. Agrawal, D. N. Clark, and R. G. Douglas, Invariant subspaces in the polydisk, Pacific J. Math. 121 (1986), 1-11.

[2] S. Axler and E Bourdon, Finite codimensional invariant subspaces of Bergman spaces, Trans. Amer. Math. Soc. 305 (1988), 1-13.

[3] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239-255.

[4] A. Douady, Le probleme des modules, Ann. Inst. Fourier 16 (1966), 1-95.

[5] R.G. Douglas and V. Paulsen, Hilbert modules over function algebras, Pitman Res. Notes in Math. 219, Harlow, 1989.

[6] R.G. Douglas, V. I. Paulsen, C. H. Sah, and K. Yan, Algebraic reduction and rigidity for Hilbert modules, Amer. J. Math., to appear.

[7] R.G. Douglas and Yah Keren, On the rigidity of Hardy submodules, Int. Eq. Operator Theory 13 (1990), 350-363.

[8] H. Grauert and R. Remmert, Theory of Stein Spaces, Springer-Verlag, Berlin, 1979.

[9] G. Pourcin, Sous-espaces privilegies d'un polycylindre, Am. Inst. Fourier 25 (1975), 151-193.

[10] M. Putinar, On invariant subspaces of several variable Bergman spaces, Pacific J. Math. 147 (1991), 355-364.

[11] M. Putinar and N. Salinas, Analytic transversality and Nullstellensatz in Bergman space, Contemp, Math. 137 (1992), 367-381.

[12] W. Rudin, Function Theory in Polydisks, Benjamin, New York, 1969.

[13] H. Skoda, Application des techniques [L.sup.2] a la theorie des ideaux d'une algebre de fonctions holomorphes avec poids, Ann. Sci. Ec, Norm, Sup. 5 (1972), 545-579.

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Author: | Putinar, Mihai |
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Publication: | American Journal of Mathematics |

Date: | Dec 1, 1994 |

Words: | 5358 |

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