On unitary invariants of quotient Hilbert modules along smooth complex analytic sets
aa r X i v : . [ m a t h . F A ] D ec ON UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES ALONGSMOOTH COMPLEX ANALYTIC SETS
PRAHLLAD DEB
Abstract.
Let Ω ⊂ C m be an open, connected and bounded set and A (Ω) be a functionalgebra of holomorphic functions on Ω. In this article, we study quotient Hilbert modulesobtained from submodules, consisting of functions in M vanishing to order k along a smoothirreducible complex analytic set Z ⊂
Ω of codimension at least 2, of a quasi-free Hilbertmodule, M . Our motive is to investigate unitary invariants of such quotient modules. Wecompletely determine unitary equivalence of aforementioned quotient modules and relate it togeometric invariants of a Hermitian holomorphic vector bundles. Then, as an application, wecharacterize unitary equivalence classes of weighted Bergman modules over A ( D m ) in terms ofthose of quotient modules arising from the submodules of functions vanishing to order 2 alongthe diagonal in D m . Introduction
The basic problem alluded to the title is as follows:
Given a Hilbert module M and a submodule M over the algebra of holomorphic functions A (Ω) on a bounded domain Ω in C m , satisfying the exact sequence → M i → M π → M q → , where i is the inclusion map, π is the quotient map and M q is the quotient module M ⊖ M , isit possible to determine M q in terms of M and M ? One can make this general question moreprecise by asking if one can assign some computable invariants on M q in terms of M and thesubmodule M . For a quasi-free Hilbert module M [9, Section 2], [10, Page 3] over A (Ω), the quotient module M q obtained from the submodule M , where M is the maximal set of functions in M vanishingalong a smooth hypersurface in Ω, was first studied by R.G. Douglas and G. Misra in [7]. In fact,they considered a quasi-free Hilbert module M of rank 1 and described a geometric invariant ofthe quotient module M q , namely, the fundamental class of the variety Z [16, Page 61] and theydescribed the fundamental class [ Z ] in terms of the curvatures of the line bundles (Remark 2 . M and M . Later in the paper [2], this result was extended to quotient modulescorresponding to the submodules consisting of complex valued functions in A (Ω) vanishing alonga complex algebraic variety of complete intersection of finitely many smooth hypersurfaces. Itwas the paper [12] where quotient modules arising from submodules, M k , of functions in A (Ω)vanishing on a smooth complex hypersurface of order k ≥
2. The module of jets corresponding toa Hilbert module was introduced by means of the jet construction [12, Page 372] and was showedthat the quotient module considered there could be thought of as the module of jets restricted tothe hypersurface. Thus, in the first half of [12] a model for the quotient module obtained fromsubmodules M k was provied while the later half was devoted in finding geometric invariants Mathematics Subject Classification.
Key words and phrases.
Hilbert modules, Quotient module, CowenDouglas operator, jet bundles, curvature.This work is supported by Senior Research Fellowship funded by IISER Kolkata. of the quotient modules in terms of the fundamental class of the hypersurface generalizing theresults of [7]. In [8] a complete set of unitary invariants for quotient modules with k = 2was determined and in a subsequent paper [11] they described complete unitary invariantsfor quotient modules with an arbitrary k . It was shown in [11] that two quotient modules Q := M ⊖ M k and ˜ Q := ˜ M ⊖ ˜ M k are unitarily equivalent if and only if the line bundles E M and E ˜ M arising from M and ˜ M , respectively, are isomorphic in certain sense [11, Definition4.2]. Moreover, for k = 2, a complete set of unitary invariants for quotient modules wereobtained which are the tangential and transverse components of the curvature of the line bundle E M relative to hypersurface Z and the second fundamental form for the inclusion E M ⊂ J (2)1 E M where J (2)1 E M is the second order jet bundle of E M relative to Z [11, Section 3]. Morerecently, results in the paper [11] have been generalized to the quotient modules obtained fromsubmodules of vector valued holomorphic functions on Ω vanishing along a smooth complexhypersurface in Ω by L. Chen and R.G. Douglas in [3]. So to this extent it is natural to considerthe case where quotient modules are obtained from submodules of vector valued holomorphicfunctions on Ω ⊂ C m vanishing of order k ≥ M (Section 2) over A (Ω) consisting of vector valued holomorphic functions on Ωwith the module action obtained by point wise multiplication and go on to describe submodules M of interest in Section 3. We first define the order of vanishing of a vector valued holomorphicfunction along a connected complex submanifold Z of arbitrary codimension which is the keyingredient of the definition of M . Since the definition of the order of vanishing is not canonicalit becomes difficult to calculate the same for a given holomorphic function. However, we providean equivalent condition to define the order of vanishing which is easy to compute. We then,following the technique introduced in the paper [12], describe the jet construction on M relativeto the submanifold Z to identify the quotient module to the restriction of a Hilbert module J ( M ) (we refer the readers to ( 4.3 ) for definition) to Z .The identification, mentioned above, enables us to associate a natural geometric object tothe quotient module, namely, the k -th order jet bundle relative to Z (Section 5) of the vectorbundle associated to the module M . Thus, we pass to the geometric counter part of ourstudy of quotient modules to be able to provide some geometric invariants for them. These jetbundles are canonically associated to the module of jets J ( M ) of the module M . Then usingthe technique of normalised frame [17, Lemma 2.3] of a Hermitian holomorphic vector bundlewe successfully describe a complete set of unitary invariants for quotient modules. Since the jetbundles of our interest are holomorphically trivial (that is, they possess a global holomorphicframe) we always have a bundle isomorphism between any two of them which does not dependon the base manifold. But it is, a priori, not true that such an isomorphism also preserves theHermitian metric of the jet bundle mentioned above. In our case, while the jet bundles areassociated to equivalent quotient modules, we do have isometric bundle isomorphism betweencorresponding jet bundles which does not depend on the base manifold. More precisely, weshow that two such quotient modules are unitarily equivalent if and only if there exists aconstant isometric jet bundle isomorphism between the corresponding jet bundles restrictedto the submanifold Z (Theorem 5 .
10) with the aid of normalized frame (we refer readers toProposition 5 . N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 3
Finally, we describe the quotient module obtained from the submodule of functions inweighted Bergman module H ( α,β,γ ) over A ( D ) vanishing to order 2 along the diagonal setof D . Furthermore, with the help of our main theorem (Theorem 5 .
13) we determine unitaryequivalence classes of weighted Bergman modules over A ( D m ) in terms of quotient modulesarising from the submodules of functions vanishing to order 2 along the diagonal set of D m .Thus, the results presented in this article extend most of the results in the papers [11], [12],[8] and [3] to the case of quotient modules arising from submodules of vector valued holomor-phic functions on a bounded domain in C m which vanishes along a smooth irreducible complexanalytic set of order at least 2.The present article is organized in the following way. In Section 2, we recall some basicdefinitions and introduce few notations which will be used through out this note. Then acomplete description of the submodule M of interest is presented in Section 3. Section 4 isdevoted to study quotient modules obtained from submodules introduced in Section 3. Therewe provide a canonical model for such quotient modules and in the subsequent section, Section5, describe the complete set of unitary invariants of those quotient modules. We then finishthis article by presenting some examples and applications in Section 6.2. Preliminaries
Let Ω be a bounded domain in C m and A (Ω) be the unital Banach algebra obtained asthe norm closure with respect to the supremum norm on Ω of all functions holomorphic on aneighbourhood of Ω. A complex Hilbert space H is said to be a Hilbert module over A (Ω)with module map A (Ω) × H π → H by point wise multiplication such that the module action A (Ω) × H π → H is norm continuous. We say that a Hilbert module H over A (Ω) is contractiveif π is a contraction.Suppose that H and H are two Hilbert modules over A (Ω) with module actions ( f, h i ) M ( i ) f ( h i ), i = 1 ,
2. Then a Hilbert space isomorphism Φ : H → H is said to be moduleisomorphism if Φ( M (1) f ( h )) = M (2) f (Φ( h )) and we denote H ≃ A (Ω) H .In this article, we study quotient modules obtained from certain submodules of quasi-freeHilbert modules. So we recall that a Hilbert space H is said to be a quasi-free Hilbert moduleover A (Ω) of rank r , for 1 ≤ r ≤ ∞ and a bounded domain Ω ⊂ C m , if H is a Hilbert spacecompletion of the algebraic tensor product A (Ω) ⊗ alg C r and following conditions happen to betrue:(i) the evaluation operators e w : H → C r defined by h h ( w ) are uniformly bounded onΩ,(ii) a sequence { h k } ⊂ A (Ω) ⊗ alg C r that is Cauchy in the norm in H converges to 0 in thenorm in H if and only if e w ( h k ) converges to 0 in C r for w ∈ Ω, and(iii) multiplication by functions in A (Ω) define a bounded operator on H .The condition (i) and (ii) together make the completion H a functional Hilbert space over A (Ω) [1, Page 347]. Moreover, condition (i) ensures that the Hilbert space H possesses an r × r matrix valued reproducing kernel thanks to Riesz representation theorem. Finally, condition(iii) along with (i) make H a Hilbert module over A (Ω) in the sense of [13, Definition 1.2].Thus, a quasi-free Hilbert module over A (Ω) of rank r gives rise to a reproducing kernel Hilbertmodule of C r valued holomorphic functions over A (Ω).For Ω ⊂ C , we recall the definition of the Cowen-Douglas class B n (Ω) consisting of operators T on a Hilbert space H for which each w ∈ Ω is an eigenvalue of uniform multiplicity n of T ,the eigenvectors span the Hilbert space H and ran( T − wI H ) is closed for w ∈ Ω. Later the
PRAHLLAD DEB definition was adapted to the case of an m -tuple of commuting operators T acting on a Hilbertspace H , first in the paper [5], and then in the paper [6] from slightly different point of viewwhich emphasized the role of the reproducing kernel. Let us now define the class B n (Ω) forΩ ⊂ C m a bounded domain. Definition 2.1.
The m -tuple T = ( T , . . . , T m ) is in B n (Ω) if(i) ran D T − w is closed for all w ∈ Ω where D T : H → H ⊗ C n is defined by D T h =( T h, ..., T m h ) , h ∈ H ;(ii) span { ker D T − w : w ∈ Ω } is dense in H and(iii) dim ker D T − w = n for all w ∈ Ω.It was then shown that each of these m -tuples T determines a Hermitian holomorphic vectorbundle E of rank n on Ω and that two m -tuples of operators in B n (Ω) are unitarily equivalentif and only if the corresponding vector bundles are locally equivalent. In case of n = 1, this isa question of equivalence of two Hermitian holomorphic line bundles and hence is a questionof equality of the curvatures of those line bundles. However, no such simple characterization isknown if the rank of the bundle is strictly greater than 1. Remark 2.2.
Let us consider a quasi-free Hilbert module H of rank r over the algebra A (Ω).Then, as mentioned earlier, H is a reproducing kernel Hilbert module with reproducing kernel K on Ω. Let M be the m -tuple of multiplication operators ( M , . . . , M m ) acting on H asmultiplication by coordinate functions. It then follows from the reproducing property of K that M i ∗ K ( ., w ) η = w i K ( ., w ) η, for η ∈ C r , w ∈ Ω , ≤ i ≤ m. (2.1)As a Consequence, the dimension of the joint eigenspace of M ∗ at w is at least r . Moreover,from the definition of quasi-free Hilbert modules and holomorphic functional calculus we seethat the joint eigenspace at w must have dimension exactly r for w ∈ Ω. Thus, s ( w ) := { K ( ., w ) σ , . . . , K ( ., w ) σ r } defines a global holomorphic frame for the vector bundle E → Ω ∗ with fibre at w ∈ Ω ∗ := { w : w ∈ Ω } , E w := span s ( w ) = ker D M ∗ − w where { σ j } rj =1 is thestandard ordered basis for C r .We also note that the third condition in the definition of quasi-free Hilbert modules impliesthat ran D M ∗ − w is dense in H . Thus, M satisfies every condition of the definition of B r (Ω ∗ )except the first one. This difference was discussed in the paper [10] where the notion of aquasi-free Hilbert module was introduced. While most of our examples lie in the class B r (Ω),our methods work even with the weaker hypothesis that the modules are quasi-free of rank r over the algebra A (Ω).In this article, we are interested to present some geometric invariant of quotient modules ob-tained from certain class of submodules of a quasi-free Hilbert module over A (Ω). We, therefore,need some geometric tools from complex differential geometry. For the sake of completeness letus recall some basic notions from complex differential geometry following [4] and Chapter 3 of[17].Let E be a Hermitian holomorphic vector bundle of rank n over a complex manifold M ofdimension m with the Chern connection D . Then a simple calculation shows that with respectto a local frame s = { e , . . . , e n } of E the Chern connection D takes the form D ( s ) = ∂H ( s ) · H ( s ) − (2.2)where H ( s ) is the Grammian matrix of the frame s . From now on, by a connection on aHermitian holomorphic vector bundle we will mean the Chern connection. N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 5
The curvature of E is defined as K := D = D ◦ D and K is an element in E ( M ) ⊗ Hom(
E, E ) where E ( M ) is the collection of smooth 2-forms on M . Consequently, in a localcoordinate chart of M one can write K as K ( σ ) = m X i,j =1 K ij dz i ∧ dz j , σ ∈ E ( M, E ) . Since K ∈ E ( M ) ⊗ Hom(
E, E ) we have K ij are also bundle map for i, j = 1 , . . . , m . As beforewe can also express the curvature tensor K with respect to a local frame as follows K ( s ) = ¯ ∂ ( ∂H ( s ) · H ( s ) − ) and equivalently , K ij ( s ) = ¯ ∂ j ( ∂ i H ( s ) · H ( s ) − )(2.3)where ∂ i = ∂∂z i and ¯ ∂ j = ∂∂z j . It is well known that the curvature operator is self adjoint ([4,(2.15.4)]) in the sense that K ij = K ∗ ji .Now following [4, Lemma 2.10], for a given local frame s of E and a C ∞ bundle map Φ : E → ˜ E , we have thatΦ z i ( s ) = ∂ z i Φ( s ) − [ ∂ z i H ( s ) · H ( s ) − , Φ( s )] and Φ z i ( s ) = ∂ z i Φ( s )(2.4)where Φ( s ) , Φ z i ( s ) , Φ z j ( s ) are matrix representations of Φ , Φ z i , Φ z j , respectively, with respectto the local frame s and [ A, B ] denotes the commutator of matrices A and B . In the followinglemma we calculate the covariant derivatives of curvature tensor which will be useful in Section5. The proof of the following lemma, for d = 1, is well known [4, Proposition 2.18]. Althoughthe similar set of arguments used there with more than one variables yields the proof of thefollowing lemma, we present a sketch of the proof for the sake of completeness. Lemma 2.3.
Let E be a Hermitian holomorphic vector bundle over Ω in C m with a fixedholomorphic frame S := { s , . . . , s r } whose Grammian matrix is H . Then (i) For ≤ d ≤ m , α, β ∈ ( N ∪ { (0) } ) d , and i, j = 1 , . . . , d , the r × r matrices ( K ij ( S )) z α ··· z dαd z β ··· z dβd can be expressed in terms of H − and ∂ p · · · ∂ dp d ¯ ∂ q · · · ¯ ∂ q d d H , ≤ p l ≤ α l + 1 , ≤ q l ≤ β l + 1 , l = 1 , . . . , d . (ii) Given ≤ d ≤ m , α, β ∈ ( N ∪ { (0) } ) d , ∂ α · · · ∂ dα d ¯ ∂ β · · · ¯ ∂ β d d H can be written in termsof H − , ∂ p · · · ∂ dp d H , ¯ ∂ q · · · ¯ ∂ q d d H , for ≤ p l ≤ α l , ≤ q l ≤ β l ,and ( K ij ) z r ··· z drd z s ··· z dsd , for ≤ r l ≤ α l − , ≤ s l ≤ β l − , l = 1 , . . . , d , i, j = 1 , . . . , d .Proof. Let E , S and H be as above. Then, for j = 1 , . . . , m , we have ∂ z j H − = − H − · ∂ z j H · H − and ∂ z j H − = − H − · ∂ z j H · H − . (2.5)Now from the definition of curvature we obtain, for i, j = 1 , . . . , d , K ij = ∂ z j ( H − · ∂ z i H ) = − H − · ∂ z j H · H − · ∂ z i H + H − · ∂ z j ∂ z i H which also implies that ∂ z j ∂ z i H = H · K ij + ∂ z j H · H − · ∂ z i H. (2.6)Then repeated application of Leibnitz rule together with the equations in ( 2.5 ) provide thedesired expression in (i). Further, (ii) can also be obtained as before by using Leibnitz rule andformulas ( 2.5 ) and ( 2.6 ) repeatedly. (cid:3) PRAHLLAD DEB
Notations and Conventions.
We finish this introduction with a list of notations andconventions those will be useful through out the paper.(1) In this article, we are intended to study quotient Hilbert modules obtained from sub-modules of quasi-free Hilbert modules over A (Ω) for a bounded domain Ω ∈ C m . Sofrom now on we assume that our Hilbert modules are quasi-free unless and otherwisestated.(2) Let H be a Hilbert module over A (Ω) consisting of holomorphic functions on Ω and H ⊂ H be a subspace which is also a Hilbert module over Ω. Assume that A (Ω)acts on H by point wise multiplication and H q be the quotient module H ⊖ H . Let U ⊂ Ω be an open connected subset. Then from the identity theorem for holomorphicfunctions of several complex variables we have
H ≃ A (Ω) H| res U , H ≃ A (Ω) H | res U , andhence H q ≃ A (Ω) H q | res U where H ≃ A (Ω) H| res U = { h | U : h ∈ H} . Indeed, the restrictionmap R : H → H| res U defined by f f | U is an onto map whose kernel is trivial thanksto the identity theorem and hence the inner product h R ( f ) , R ( g ) i := h f, g i on H| res U turns R to a unitary map. Then one can make H| res U to a Hilbert module by restrictingthe module action of A (Ω) to the open set U and note that R intertwines the moduleactions. Thus, H and H| res U are unitarily equivalent as modules and we also have H ≃ H | res U , H q ≃ H q | res U as modules. We, therefore, may cut down the domain Ω toa suitable open subset U , if necessary, and pretend U to be Ω.(3) Let 1 ≤ d ≤ m , k ∈ N , N = (cid:0) d + k − k − (cid:1) − I N := { , , . . . , N } and A := { α =( α , . . . , α d ) ∈ ( N ∪ { } ) d : 0 ≤ | α | ≤ k − } where | α | = α + · · · + α d . Then considerthe bijection θ : A → I N defined by θ ( α ) := d − X j =1 j ! | α | − d − j X i =1 α i ! j + 1 d ! ( | α | ) d (2.7) where ( z ) t is the Pochhammer symbols defined as, for any complex number z and anatural number t , ( z ) t = z ( z + 1) · · · ( z + t − A by pullingback the usual order on I N via the bijection θ , that is,( α , . . . , α d ) ≤ ( α ′ , . . . , α ′ d ) if and only if θ ( α , . . . , α d ) ≤ θ ( α ′ , . . . , α ′ d ) . Note that the order induced by θ is nothing but the graded colexicographic ordering on A . Here we also point out that A = ∪ k − t =0 A t where A t := { α ∈ A : | α | = t } . Therefore,one can have a natural bijection between A t and I N t where N t is the cardinality of theset A t , namely, θ t := θ | A t : A t → θ ( A t ) . (2.8) These new set of bijections will be useful in the next section.(4) From now on, for α ∈ A and θ ( α , . . . , α d ) = l , we use following notations ∂ l (respectively, ¯ ∂ l ) := ∂ α (respectively, ¯ ∂ α )(2.9) := ∂ | α | ∂z α · · · ∂z dα d respectively, ¯ ∂ | α | ¯ ∂z α · · · ¯ ∂z dα d ! unless and otherwise stated, where ∂ i = ∂∂z i , i = 1 , . . . , d . In this context, note thatsince θ is a bijection there exists unique ( α , . . . , α d ) ∈ A for every l ∈ I N , and we aredenoting ∂ θ − ( l ) as ∂ l . N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 7 The Submodule M Let M be a quasi-free Hilbert module of rank r over A (Ω) and denote the elements of M as h = ( h , . . . , h r ) where h j ∈ A (Ω), 1 ≤ j ≤ r . In this section, we define the submodule M of M . So we begin by recalling some elementary definitions regarding complex analytic varieties. Definition 3.1.
Let Ω ⊂ C m be a bounded domain. Then a subset Z ⊂
Ω is called an analyticset if, for any point p ∈ Ω, there is a connected open neighbourhood U of p in Ω and finitelymany holomorphic functions φ , . . . , φ d on U such that U ∩ Z = { q ∈ U : φ j ( q ) = 0 , ≤ j ≤ d } . Definition 3.2.
An analytic set
Z ⊂
Ω is said to be regular of codimension d at p ∈ Z if thereis an open neighbourhood U p ⊂ Ω and holomorphic functions φ , . . . , φ d on U p such that(a) Z ∩ U p = { q ∈ Ω : φ ( q ) = · · · = φ d ( q ) = 0 } ,(b) the rank of the Jacobian matrix of the mapping q ( φ ( q ) , . . . , φ d ( q )) at p is d .An analytic set is said to be irreducible if it can not be decomposed as union of two analyticsets. It is known in literature that any smooth analytic set is irreducible if and only if it isconnected with respect to the subspace topology [16, page 20].Here we point out that such an analytic set Z is a regular complex submanifold of codimension d in Ω thanks to following well known fact [15, page 161]. Proposition 3.3.
An analytic set Z is regular of codimension d at p ∈ M in a complexmanifold M of dimension m if and only if there is a complex coordinate chart ( U, φ ) of M suchthat B := φ ( U ) is an open subset of C m with φ ( p ) = 0 and φ ( U ∩ Z ) = { λ = ( λ , ..., λ m ) ∈ B : λ = · · · = λ d = 0 } . Remark 3.4.
In this article, we are interested in smooth irreducible analytic sets Z of codi-mension d in some bounded domain Ω in C m . So from the Definition 3 . . p ∈ Z , there is a coordinate chart ( U, φ ) at p of Ω satisfying followingproperties:(a) φ ( p ) = 0 with φ ( U ∩ Z ) = { λ = ( λ , ..., λ m ) ∈ B : λ = · · · = λ d = 0 } ,(b) the rank of the Jacobian matrix of the mapping q ( φ ( q ) , . . . , φ d ( q )) at p is d .We are now about to define the order of vanishing of a holomorphic function along a smoothanalytic set. Our definition is essentially a direct generalization of the definition given in [12] todefine the order of vanishing of a holomorphic function along a smooth complex hypersurface. Definition 3.5.
Let Ω and Z be as above and f : Ω → C be a holomorphic function. Then f is said to have zero of order k at some point p ∈ Z if there exists a coordinate chart ( U, φ ) at p of Ω satisfying the properties (a) and (b) in the Remark 3 . f ] ∈ I k − Z but [ f ] / ∈ I k Z (3.1)where [ f ] is the germ of f at p and I Z is the ideal in O m,p generated by [ φ ] , . . . , [ φ d ]. Remark 3.6.
Note that the above defintion is independent of the choice of coordinate chart at p . Indeed, for two such charts ( U , φ ) and ( U , φ ) with the properties listed in Remark 3 . φ and φ ◦ φ − , respectively, induce isomorphisms Φ : O m,p → O m, defined by Φ ([ g ]) = [ g ◦ φ − ]and Φ : O m, → O m, with Φ([ g ◦ φ − ]) = [ g ◦ φ − ]. As a consequence, it turns out that f satisfies ( 3.1 ) if and only if [ f ◦ φ − ] ∈ I k − but [ f ◦ φ − ] / ∈ I k which is again equivalent tothe fact that [ f ◦ φ − ] ∈ I k − but [ f ◦ φ − ] / ∈ I k where I j is the ideal generated by the germs[ λ j ] , . . . , [ λ jd ], j = 1 ,
2, with local coordinates λ j , . . . , λ jm of C m corresponding to φ j . PRAHLLAD DEB
Definition 3.7.
Let M be a quasi-free Hilbert module of rank r over A (Ω). Then the sub-module M is defined as M := { h ∈ M : h j has zero of order k at every q ∈ Z , ≤ j ≤ r } . Lemma 3.8.
Let Ω be a bounded domain in C m , Z be a complex submanifold in Ω and f :Ω → C be a holomorphic function. Then, for each point p ∈ Z , f vanishes to order k at p along Z if and only if k is the largest integer such that ∂ αλ ( f ◦ φ − ) | φ ( U ∩Z ) := ∂ | α | ∂λ α · · · ∂λ dα d ( f ◦ φ − ) | φ ( U ∩Z ) = 0 for ≤ | α | ≤ k − , where α = ( α , . . . , α d ) and | α | = α + · · · + α d , for some coordinate chart ( U, φ ) as in theRemark . . In general, there are no global defining functions φ , . . . , φ d for a smooth irreducible analyticset Z . But since it has been shown at the end of the Section 2 that the modules and thesubmodules of interest can be localized we can work with a small enough open set U ⊂ Ωintersecting Z . So from now on we consider a fixed neighbourhood U ⊂ Ω of p with U ∩ Z 6 = ∅ and defining functions φ , . . . , φ d satisfying conditions (a) and (b) in Remark 3 .
4. Since theJacobian matrix of the mapping z ( φ ( z ) , . . . , φ d ( z )) has rank d at p , by rearranging thecoordinates in C m , we can assume that D ( p ) := (( ∂ j φ i | p )) di,j =1 is invertible. Then it is easilyseen that D ( z ) is invertible on some neighbourhood of p in U . Abusing the notation, let usdenote this neighbourhood by the same letter U . Now we consider the mapping φ : U → φ ( U )defined as φ ( z ) = ( φ ( z ) , . . . , φ d ( z ) , z d +1 , . . . , z m ) and note that φ is a biholomorphism from U onto φ ( U ) with φ ( p ) = 0 and φ ( U ∩ Z ) = { λ = ( λ , ..., λ m ) ∈ φ ( U ) : λ = · · · = λ d = 0 } . Thus,once we fix a chart as above and pretend that U = Ω, the submodule M may be described as M = ( h ∈ M : ∂ | α | ∂λ α · · · ∂λ dα d ( h j ◦ φ − )( λ ) | φ ( Z ) = 0 for 0 ≤ | α | ≤ k − , ≤ j ≤ r ) . At this stage, we introduce a definition which separates out the coordinate chart describedabove and will be useful through out this article.
Definition 3.9.
Let Ω be a domain in C m and Z ⊂
Ω be a complex submanifold of codi-mension d . Then, for any point p ∈ Z , we call a coordinate chart ( U, φ ) of Z around p an admissible coordinate chart if the biholomorphism φ : U → φ ( U ) takes the form φ ( z ) =( φ ( z ) , . . . , φ d ( z ) , z d +1 , . . . , z m ) with φ ( p ) = 0 and φ ( U ∩ Z ) = { λ = ( λ , ..., λ m ) ∈ φ ( U ) : λ = · · · = λ d = 0 } for some holomorphic functions φ , . . . , φ d on U .Now we should note that even in this local description of the submodule there is a choice ofnormal directions to the submanifold Z involved. The following proposition ensures that in thislocal picture two different sets of normal directions to Z give rise to equivalent submodules.At this point, let us recall some elementary definitions and properties of the ring of polynomialfunctions on a finite dimensional complex vector space which will be useful in the course of theproof of following proposition.For any complex vector space V of dimension d , we denote by C [ V ] the ring of polynomialfunctions on V . Let us recall that f : V → C is an element of C [ V ] means that, for any basis { e , . . . , e d } of V , there exists some polynomial φ ∈ C [ x , . . . , x d ] such that f ( α e + · · · + α d e d ) = φ ( α , . . . , α d ) for all ( α , . . . , α d ) ∈ C d . In other words, f is a polynomial into the elements x = e ∗ , . . . , x d = e ∗ d of the dual basis. It is then clear that C [ V ] ≃ S ( V ∗ ) ≃ C [ x , . . . , x d ](3.2) N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 9 where S ( V ∗ ) is the graded vector space of all symmetric tensors on V ∗ . Note that C [ V ] is analgebra over C .A polynomial function f on V is said to be homogeneous of degree t if f ( αv ) = α t f ( v ) for all α ∈ C and v ∈ V . We denote C [ V ] t the subspace of C [ V ] of homogeneous polynomial functionsof degree t . In particular, C [ V ] = C , C [ V ] = V ∗ and C [ V ] t is canonically identified in thefirst isomorphism in ( 3.2 ) with the t -th symmetric power S t ( V ∗ ), and it can also be identifiedwith the subspace of C [ x , . . . , x d ] generated by the monomials x t · · · x t d d with t + · · · + t d = t via the second isomorphism of ( 3.2 ). Proposition 3.10.
Let Ω be a bounded domain in C m , Z be a complex submanifold in Ω and f : Ω → C be a holomorphic function. Then, for each point p ∈ Z there exists an admissiblecoordinate chart ( U, φ ) of Ω at p such that ∂ αλ ( f ◦ φ − )( λ ) | φ ( p ) = 0 , ≤ | α | ≤ k − if and only if ∂ α f ( z ) | p = 0 , ≤ | α | ≤ k − where α = ( α , . . . , α d ) ∈ ( N ∪ { } ) d , λ = ( λ , . . . , λ m ) denotes the standard coordinates on φ ( U ) ⊂ C m , and ∂ α denotes the differential operator ∂ | α | ∂z α ··· ∂z dαd .Proof. Let us consider an admissible coordinate system (
U, φ ) (Definition 3 .
9) at p ∈ Z ⊂ Ω ofΩ, that is, φ : U → φ ( U ) defined as φ ( z ) = ( φ ( z ) , . . . , φ d ( z ) , z d +1 , . . . , z m ).For q ∈ U , let V q and V φ ( q ) be tangent spaces at q and φ ( q ) to ( C d × { } ) ∩ U and ( C d ×{ } ) ∩ φ ( U ), respectively. We denote the standard ordered basis of V q by B ( q ) := { ∂∂z j | q } dj =1 and that of V φ ( q ) by B ( φ ( q )) := { ∂∂λ j | φ ( q ) } dj =1 . Then it is easily seen that φ induces a lineartransformation from V ∗ q onto V ∗ φ ( q ) , namely, L ( q ) : V ∗ q → V ∗ φ ( q ) defined by L ( q )( dz j ) = d X i =1 ( ∂ j φ i ( q )) dλ i where { dz j } dj =1 and { dλ j } dj =1 are dual bases of B ( q ) and B ( φ ( q )), respectively.Now we consider the ring of polynomial functions C [ V q ] and C [ V φ ( q ) ] on V q and V φ ( q ) , respec-tively, and observe, in view of the first isomorphism in ( 3.2 ), that L ( q ) canonically induceslinear mappings L t ( q ) : S t ( V ∗ q ) → S t ( V ∗ φ ( q ) ) defined by L t ( q )( dz α ⊗ · · · ⊗ dz α d d ) = L ( q )( dz ) α ⊗ · · · ⊗ L ( q )( dz d ) α d where α = ( α , . . . , α d ) ∈ ( N ∪ { } ) d with | α | = α + · · · + α d = t and by dz α j j (respectively, by L ( q )( dz j ) α j ) we mean that the α j -th symmetric power of dz j (respectively, L ( q )( dz j )).Let B t ( q ) := { dz α ⊗· · ·⊗ dz α d d : | α | = t } and B t ( φ ( q )) := { dλ α ⊗· · ·⊗ dλ α d d : | α | = t } be basesfor vector spaces S t ( V ∗ q ) and S t ( V ∗ φ ( q ) ), respectively, and make them ordered bases with respectto the order induced by the bijection θ t ( 2.8 ). We denote the matrix of L t ( q ) represented withrespect to the basis B t ( q ) and B t ( φ ( q )) as D t ( q ), for t ∈ N ∪ { } . Note that since L t ( p ) is avector space isomorphism for each t ∈ N ∪ { } the matrices D t ( p )’s are invertible. In this set up we claim, for z ∈ U with φ ( z ) = λ ∈ φ ( U ), that A k,φ ( z ) · f ◦ φ − ( λ ) ∂ λ f ◦ φ − ( λ )... ∂ Nλ f ◦ φ − ( λ ) = f ( z ) ∂ f ( z )... ∂ N f ( z ) (3.3)where ∂ tλ stands for the differential operator ∂ | α | ∂λ α ··· ∂λ αdd with ( α , . . . , α d ) = θ − ( t ), A k,φ ( z )is the block lower triangular matrix with 1, D ( z ), . . . , D k − ( z ) as the diagonal blocks and N = (cid:0) d + k − k − (cid:1) − k . Here we note that thebase case is the direct consequence of change of variables formula. So let the equation ( 3.3 )hold true for t = l , 1 ≤ l ≤ k − t = l + 1. Now,for α = ( α , . . . , α d ) with | α | = l and θ l ( α , . . . , α d ) = i , the induction hypothesis yields that ∂ α f ( z ) = X | β | = l ( D l ( z )) iθ l ( β ) ∂ βλ f ◦ φ − ( λ ) + other termswhere D l ( z ) is the matrix ((( D l ( z )) θ l ( α ) θ l ( β ) )) | α | = l, | β | = l . Therefore, differentiating both sides ofthe above equation with respect to the z j -th coordinate and using the Leibnitz rule we have,for an arbitrary but fixed point q ∈ U , ∂ j ∂ α f ( z ) | z = q = X | β | = l ∂ j ( D l ( z )) iθ l ( β ) | z = q ∂ βλ f ◦ φ − ( φ ( q ))+ X | β | = l ( D l ( q )) iθ l ( β ) d X s =1 ∂ j φ s ( q ) ∂ λ s ! ∂ βλ f ◦ φ − ( λ ) | λ = φ ( q ) + ∂ j ( other terms)= X | β | = l ( D l ( q )) iθ l ( β ) ∂ βλ d X s =1 ∂ j φ s ( q ) ∂ λ s f ◦ φ − ( λ ) ! (cid:12)(cid:12) λ = φ ( q ) + ( other terms involving ∂ βλ f ◦ φ − ( φ ( q )) with | α | ≤ l )Let us now note that the rings of polynomial functions S ( V ∗ q ) and S ( V ∗ φ ( q ) ) can be canonicallyidentified with the algebras of linear partial differential operators with constant coefficients,namely, Γ q : S ( V ∗ q ) ≃ C { P α a α ∂ α · · · ∂ α d d : a α ∈ C } under the correspondence dz α ⊗ · · · ⊗ dz α d d Γ q ∂ α · · · ∂ α d d and similarly, Γ φ ( q ) : S ( V ∗ φ ( q ) ) ≃ C { P α a α ∂ α λ · · · ∂ α d λ d : a α ∈ C } via themapping dλ α ⊗ · · · ⊗ dλ α d d Γ φ ( q ) ∂ α λ · · · ∂ α d λ d . Then with respect to the above identification wehave N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 11 ∂ α + ε j f ( q ) = X | β | = l ( D l ( q )) iθ l ( β ) ∂ βλ d X s =1 ∂ j φ s ( q ) ∂ λ s f ◦ φ − ( λ ) ! (cid:12)(cid:12) λ = φ ( q ) + ( other terms involving ∂ βλ f ◦ φ − ( φ ( q )) with | α | ≤ l )= (Γ φ ( q ) L ( q )Γ − q ( ∂ )) α · · · (Γ φ ( q ) L ( q )Γ − q ( ∂ j )) α j · · · (Γ φ ( q ) L ( q )Γ − q ( ∂ d )) α d (Γ φ ( q ) L ( q )Γ − q ( ∂ j )) f ◦ φ − ( λ ) (cid:12)(cid:12) λ = φ ( q ) + ( other terms involving ∂ βλ f ◦ φ − ( φ ( q )) with | α | ≤ l )= (Γ φ ( q ) L ( q )Γ − q ( ∂ )) α · · · (Γ φ ( q ) L ( q )Γ − q ( ∂ j )) α j +1 · · · (Γ φ ( q ) L ( q )Γ − q ( ∂ d )) α d f ◦ φ − ( λ ) (cid:12)(cid:12) λ = φ ( q ) + ( other terms involving ∂ βλ f ◦ φ − ( φ ( q )) with | α | ≤ l )= X | β | = l +1 ( D l +1 ( q )) θ ( α + ε j ) θ l ( β ) ∂ βλ f ◦ φ − ( φ ( q ))+ ( other terms involving ∂ βλ f ◦ φ − ( φ ( q )) with | α | ≤ l ) . Since q was chosen to be arbitrary in U we are done with the claim. Thus, A k,φ ( z ) isinvertible if and only if D ( z ), . . . , D k − ( z ) are simultaneously invertible which is the case for z ∈ U . Hence it completes the proof. (cid:3) Thus, from the above proposition and Remark 3 . M as follows: M = { h ∈ M : ∂ α · · · ∂ dα d ( h j ) | Z = 0 , ≤ | α | ≤ k − , ≤ j ≤ r } . Remark 3.11.
Let U be an open subset of Ω such that U ∩ Z is non-empty. Then recall thatthe restriction map R : M → M | res U defined by f f | U is an unitary map with respect tothe prescribed inner product on M | res U ((2) in Section 2). Moreover, it is now clear from thedefinition of M that R ( M ) and R ( M ) are unitarily equivalent where R ( M ) := { f ∈ R ( M ) : f vanishes along U ∩Z to order k } . Consequently, R ( M q ) and R ( M ) q (:= R ( M ) ⊖ R ( M ) ) arealso unitarily equivalent Hilbert modules. So restricting ourselves to an admissible coordinatechart ( U, φ ) around some point p ∈ Z ⊂ Ω, it is enough to study these modules with respect tothe new coordinate system obtained by φ . We now elaborate upon this fact.So let us consider the module, φ ∗ ( M | res U ) which is, by definition, φ ∗ ( M | res U ) := { f | U ◦ φ − : f ∈ M } and note that it is a module over A (Ω) with the module action g · ( f | U ◦ φ − ) := ( gf ) | U ◦ φ − ,for g ∈ A (Ω). Then it is evident that the modules φ ∗ ( M | res U ) and M are isomorphic via theisomorphism Φ : M → φ ∗ ( M | res U ) defined by f f | U ◦ φ − . So, defining an inner product as h f | U ◦ φ − , g | U ◦ φ − i φ ∗ ( M | res U ) := h f, g i M we see that φ ∗ ( M | res U ) is unitarily equivalent to M as Hilbert modules. Since M is a re-producing kernel Hilbert module with a reproducing kernel, say, K so is φ ∗ ( M | res U ) with thekernel function K ′ defined by K ′ ( u, v ) = K ( φ − ( u ) , φ − ( v )), for u, v ∈ φ ( U ). It is also eas-ily seen that the multiplication operators M z , . . . , M z m on M are simultaneously unitarily equivalent to M u , . . . , M u m on φ ∗ ( M | res U ). Indeed, for φ − = ( ψ , . . . , ψ m ), we note that ψ i ( φ ( z , . . . , z m )) = z i , i = 1 , . . . , m and therefore, we have, for i = 1 , . . . , m and f ∈ M ,Φ − M u i Φ( f ) = Φ − ( M u i ( f | U ◦ φ − ))= Φ − (( u i ◦ φ − ) · ( f | U ◦ φ − ))= M z i f | U . Furthermore, Proposition 3.10 together with the Remark 3.11 ensure that the submodules M and φ ∗ R ( M ) are also unitarily equivalent via the same map as mentioned earlier. As aconsequence along with the help of the Remark 3.11 we have the following Proposition. Proposition 3.12.
Let Ω be a bounded domain in C m , Z be a complex connected submanifoldin Ω and M , M be two quasi-free Hilbert modules of rank r over A (Ω) . Let M and M be submodules of M and M , respectively, consisting of holomorphic functions vanishing oforder k − along Z . Assume that ( U, φ ) is an admissible coordinate system around some point p ∈ Z . Then M is untarily equivalent to M as Hilbert modules if and only if φ ∗ ( M | res U ) isunitarily equivalent to φ ∗ ( M | res U ) . In other words, the following diagram commutes. M R −−−−→ M | res U Φ −−−−→ φ ∗ ( M | res U ) y y y M R −−−−→ M | res U Φ −−−−→ φ ∗ ( M | res U )In this article, we are interested in studying equivalence classes of quotient modules obtainedfrom the aforementioned submodules of a quasi-free Hilbert modules. In the next section, wedefine the quotient modules which are the central object of this paper.4. Quotient Module M q We start with a quasi-free Hilbert module M over A (Ω) of rank r ≥ M ⊂ M consisting of C r -valued holomorphic functions on Ω vanishing to order k along anirreducible smooth complex analytic set Z ⊂
Ω of codimension d , d ≥
2. In this setting we areinterested in studying the quotient module M q := M / M = M ⊖ M , in other words, we have following exact sequence0 → M i → M P → M q → i is the inclusion map and P is the quotient map. Now, for f ∈ A (Ω) and h ∈ M , wedefine the module action on the quotient module M q as f P ( h ) = P ( f h ) , (4.2)here we mean ( f h , . . . , f h r ) by f h .In order to study quotient modules we first describe the jet construction relative to thesubmanifold Z following [12]. Suppose that H is a reproducing kernel Hilbert space consistingof holomorphic functions on Ω taking values in C r with a reproducing kernel K . Let N = (cid:0) d + k − k − (cid:1) − { ε l } Nl =0 be the standard ordered basis of C N +1 , { σ i } ri =1 be the standard ordered N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 13 basis of C r and recall that ∂ , ..., ∂ d are the partial derivative operators with respect to z , ..., z d variables, respectively. For h ∈ H , recalling the notations introduced ( 2.9 ) let us define h := r X i =1 N X l =0 ∂ l h i ⊗ ε l ! ⊗ σ i and we consider the space J ( H ) := { h : h ∈ H} ⊂ H ⊗ C ( N +1) r . Consequently, we have themapping J : H → J ( H ) defined by h h . (4.3)Since J is injective we define an inner product on J ( H ) making J to be an unitary transformationas follows h J ( h ) , J ( h ) i J ( H ) := h h , h i H . Since H is a reproducing kernel Hilbert space it is natural to expect that J ( H ) is also areproducing kernel Hilbert space . So we calculate the reproducing kernel of J ( H ). Proposition 4.1.
The reproducing kernel
J K : Ω × Ω → M ( N +1) r ( C ) for the Hilbert space J ( H ) is given by the formula ( J K ) klij ( z, w ) = ∂ k ¯ ∂ l K ij ( z, w ) for ≤ l, k ≤ N, ≤ i, j ≤ r. (4.4) Proof.
Let { e n } n ≥ be an orthonormal basis of H with e n = P ri =1 e in ⊗ σ i where for every ne in is a holomorphic function on Ω. Since J is a unitary operator { J e n } n ≥ is an orthonormalbasis for J ( H ). Thus, we have J K ( z, w ) = ∞ X n =1 J e n ( z )( J e n ( w )) ∗ where J e n ( w ) ∗ : C ( N +1) r → C is defined by ( J e n ( w )) ∗ ( ζ ) := h ζ, J e n ( w ) i C ( N +1) r and J e n ( z ) : C → C ( N +1) r is defined as x x · J e n ( z ). Then note that, for w ∈ Ω, J e n ( w ) ∗ ( ε l ⊗ σ j ) = ¯ ∂ l e jn ( w )and hence we can write J K ( z, w ) ε l ⊗ σ j = ∞ X n =1 J e n ( z ) ¯ ∂ l e jn ( w ) , ≤ l ≤ N, ≤ j ≤ r. (4.5)Therefore, equation ( 4.5 ) together with the calculation below imply the identity in ( 4.4 ). h ∞ X n =1 J e n ( z ) ¯ ∂ l e jn ( w ) , ε k ⊗ σ i i C ( N +1) r = ∞ X n =1 ¯ ∂ l e jn ( w ) ∂ k e in ( z ) = ∂ k ¯ ∂ l K ij ( z, w ) . Since h can uniquely be expressed as h = P ∞ n =1 a n J e n with { a n } ∞ n =1 ⊂ C , using ( 4.5 ), wehave h h , J K ( ., w ) ε l ⊗ σ j i J ( H ) = ∞ X n =1 a n ∂ l e jn ( w ) = ∂ l ( ∞ X n =1 a n e jn ( w )) = ∂ l h j ( w ) . Here we note that the second equality in above two equations hold due to the fact that the seriesin first equality converges uniformly on compact subsets of Ω. This completes the proof. (cid:3)
Thus, we have shown that J ( H ) is a reproducing kernel Hilbert space on Ω. Now we want tomake J ( H ) to be a Hilbert module over A (Ω). So let us define an action of A (Ω) on J ( H ) so that J becomes a module isomorphism. We define, for f ∈ A (Ω) and h ∈ J ( H ), J f : J ( H ) → J ( H )by J f ( h ) := J ( f ) . h where J ( f ) is an ( N + 1) × ( N + 1) complex matrix defined as follows J ( f ) lj := (cid:18) αβ (cid:19) ∂ α − β f := (cid:18) α β (cid:19) · · · (cid:18) α d β d (cid:19) ∂ α − β f (4.6)with α = ( α , . . . , α d ) = θ − ( l ) and β = ( β , . . . , β d ) = θ − ( j ) and h can be thought of an( N + 1) × r matrix with h i := P Nl =0 ∂ l h i ⊗ ε l , 1 ≤ i ≤ r , as column vectors. Note that this is alower triangular matrix and it takes the following matrix form J ( f ) = f . . . ... J ( f ) lj . . . ∂ N f . . . . . . f Thus, with the above definition, J ( H ) becomes a Hilbert module over A (Ω) and J is amodule isomorphism between H and J ( H ) as it is clear from following simple calculation whichis essentially an application of Leibniz rule. For 1 ≤ i ≤ r , we have J ( f · h i ) = N X l =0 ∂ l ( f · h i ) ⊗ ε l = N X l =0 α X β =0 · · · α d X β d =0 (cid:18)(cid:18) α β (cid:19) · · · (cid:18) α d β d (cid:19) ∂ α − β · · · ∂ α d − β d d f · ∂ β · · · ∂ β d d h i (cid:19) ⊗ ε l = J ( f ) · h i which shows that J ( f · h ) = J ( f ) · h .Applying the above construction to the Hilbert module M we have the module of jets J ( M ).As in the case of Hilbert submodule M of the Hilbert module M it is clear that the subspace J ( M ) := { h ∈ J ( M ) : h | Z = 0 } is a submodule of J ( M ). Let J ( M ) q be the quotient module obtained by taking orthogonalcomplement of J ( M ) in J ( M ), that is, J ( M ) q := J ( M ) ⊖ J ( M ) . The following theoremprovides the equivalence of two quotient modules M q and J ( M ) q . Theorem 4.2. M q and J ( M ) q are isomorphic as modules over A (Ω) .Proof. Let us begin with a naturally arising spanning set of M q . Since, for w ∈ Ω and ζ ∈ C r , K ( ., w ) ζ ∈ M it is evident that ¯ ∂ α K ( ., w ) ζ ∈ M for ζ ∈ C r and α ∈ A . Thus, using the repro-ducing property of K , we note that h h, ¯ ∂ α K ( ., w ) ζ i = h ∂ α h ( w ) , ζ i where h ( · ) = P ∞ n =1 a n e n ( · )is any vector in M and { e n ( · ) } ∞ n =1 is an orthonormal basis of M .Then from the above calculation together with the identity obtained by differentiating bothsides of the equation ( 2.1 ) we have, for ζ ∈ C r , w ∈ Z , α ∈ A , and h ∈ M , that h h, ¯ ∂ α K ( ., w ) ζ i = h ∂ α h ( w ) , ζ i = 0which in turn implies that D := { ¯ ∂ α K ( ., w ) σ i : w ∈ Z , ≤ i ≤ r, α ∈ A } is contained in M ⊥ ,that is, (span D ) ⊥ ⊂ M and M ⊂ D ⊥ = (span D ) ⊥ . Consequently, D is a spanning set for M q . N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 15
Now we claim that J ( D ) spans J ( M ) q . In order to establish the claim we follow the samestrategy as before, in other words, we first show that J ( D ) ⊥ = J ( M ) . So let us recall thedefinition of the operator J and try to understand the set J ( D ). Note that, by definition of themap J , we have J ( D ) = { J ( ¯ ∂ α K ( ., w ) σ i ) : w ∈ Z , ≤ i ≤ r, α ∈ A } = { J K ( ., w ) ε j ⊗ σ i : 0 ≤ j ≤ N, ≤ i ≤ r, w ∈ Z} . As before, for h ∈ J ( M ), 1 ≤ i ≤ r , and w ∈ Ω, from the reproducing property of
J K wehave h h , J K ( ., w ) ε j ⊗ σ i i J ( M ) = h h ( w ) , ε j ⊗ σ i i C ( N +1) r = ∂ j h i ( w ) , ≤ j ≤ N, (4.7)which justifies our claim. Hence we conclude that J ( D ) spans the quotient space J ( M ) q , thatis, J ( M q ) = J (span D ) = span J ( D ) = J ( M ) q . Now in course of completion of our proof it remains to check that J is a module isomorphismfrom M q onto J ( M ) q . In other words, we need to verify the following identity J ◦ P ◦ M f = ( J P ) ◦ J f ◦ J, for f ∈ A (Ω), which is equivalent to show that J M ∗ f P = J ∗ f ( J P ) J where J P : J ( M ) → J ( M ) q is the orthogonal projection operator. Since it amounts to showthat J intertwines the module actions on D and both P and J P are identity on D and J ( D ),respectively, it is enough to prove that J M ∗ f = J ∗ f J, for f ∈ A (Ω) , on D. Let α = ( α , · · · , α d ) ∈ A , 1 ≤ i ≤ r and ¯ ∂ α K ( ., w ) σ i ∈ D (we refer the readers ( 2.9 ) forthe notation ¯ ∂ α ). For f ∈ A (Ω), w ∈ Z ⊂ Ω, we have M ∗ f K ( ., w ) σ i = f ( w ) K ( ., w ) σ i . (4.8)Then differentiating both sides of the above equation and using induction on degree of thedifferentiation and adopting the notation introduced in ( 2.9 ) we obtain M f ∗ ¯ ∂ α K ( ., w ) σ i = α X β =0 · · · α d X β d =0 ¯ ∂ α − β · · · ¯ ∂ α d − β d d f ( w ) ¯ ∂ β · · · ¯ ∂ β d d K ( ., w ) σ i . (4.9)Therefore, J ( M f ∗ ¯ ∂ α K ( ., w ) σ i ) = N X l =0 ∂ l α X β =0 · · · α d X β d =0 ¯ ∂ α − β · · · ¯ ∂ α d − β d d f ( w ) ¯ ∂ β · · · ¯ ∂ β d d K ( ., w ) ⊗ ε l ⊗ σ i = α X β =0 · · · α d X β d =0 ¯ ∂ α − β · · · ¯ ∂ α d − β d d f ( w ) J K ( ., w )( ε θ ( β ) ⊗ σ i ) , β = ( β , . . . , β d )= J K ( ., w )( J ( f )( w )) ∗ ( ε θ ( α ) ⊗ σ i ) . Thus, for h ∈ J ( M ), ζ ∈ C ( N +1) r and w ∈ Ω, we have h h , J ∗ f J K ( ., w ) · ζ i J ( M ) = h h ( w ) , J ( f )( w ) ∗ ζ i C ( N +1) r = h h , J K ( ., w ) J ( f )( w ) ∗ ζ i J ( M ) . This completes the roof. (cid:3)
Remark 4.3.
Note that as mentioned in [12] the Theorem 4 . −−−−→ M i −−−−→ M P −−−−→ M q −−−−→ y y y −−−−→ J ( M ) i −−−−→ J ( M ) JP −−−−→ J ( M ) q −−−−→ H with scalar valued repro-ducing kernel K on some set W , the restriction of K on a subset W of W is also a reproducingkernel and restriction of K to W constitutes a reproducing kernel Hilbert space which is iso-morphic to the quotient space H ⊖ H where H := { f ∈ H : f | W = 0 } . Here, adopting theproof from [1] for our case with vector valued kernel, we have the following theorem. Since thisresult is well known (Theorem 3.3, [12]) for the case while the codimension of the submanifold, Z , is one and using the techniques used in that proof in a similar way one can the followingtheorem, we omit the proof. Theorem 4.4.
The normed linear space J ( M ) | res Z is a Hilbert space and the Hilbert spaces J ( M ) q and J ( M ) | res Z are unitarily equivalent. Consequently, the reproducing kernel K for J ( M ) | res Z is the restriction of the kernel J K to the submanifold Z . Moreover, J ( M ) q and J ( M ) | res Z are isomorphic as modules over A (Ω) . Theorem 4.5.
The quotient module M q is equivalent to the module J ( M ) | res Z over A (Ω) .Proof. It is obvious from Theorem 4 . . (cid:3) We now provide a necessary condition for equivalence of two quotient modules in the followingtheorem.
Theorem 4.6.
Let M and ˜ M be Hilbert modules in B (Ω) and M and ˜ M be the submodulesof functions in A (Ω) vanishing along Z to order k . If M and ˜ M are equivalent as Hilbertmodules over A (Ω) then the corresponding quotient modules M q and ˜ M q are also equivalent asHilbert modules over A (Ω) .Proof. Let us begin with a unitary module map T : M → ˜ M . Then, following [6], we have thatthere is a non-vanishing holomorphic function ψ : Ω → C such that T = T ψ where T ψ : M → ˜ M defined by T ψ f = ψf .Now recalling the definition ( 4.3 ) of the unitary operator J we note that T ψ gives rise tothe module map J ψ : J ( M ) → J ( ˜ M ) by the formula J ψ := J ◦ T ψ ◦ J ∗ and J ψ ( h ) = J ◦ T ψ ◦ J ∗ ( h ) = J ( ψh ) = J ( ψ ) h which is actually a unitary module map. Since ψ is non-vanishing the definition ( 4.6 ) of J ( ψ )ensures that J ( M ) gets mapped onto J ( ˜ M ) by J ψ and hence J ( M ) q is equivalent to J ( ˜ M ) q .Thus, we are done thanks to Theorem 4 . (cid:3) Remark 4.7.
Let us now clarify the module action of A (Ω) on the quotient module M q beforeproceeding further. To facilitate this action we, following [12, page 384], consider the algebraof holomorphic functions on Ω taking values in C N +1 with N = (cid:0) d + k − k − (cid:1) − J A (Ω) := {J f : f ∈ A (Ω) } ⊂ A (Ω) ⊗ C ( N +1) × ( N +1) with the multiplication defined by the usual matrix multiplication, namely, ( J f · J g )( z ) := J f ( z ) J g ( z ). Then from ( 4.6 ) it is clear that J ( M ) | res Z is a module over the algebra N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 17 J A (Ω) | res Z obtained by restricting J A (Ω) to the submanifold Z . Note that J defines an alge-bra isomorphism from A (Ω) onto J A (Ω) and intertwines the restriction operators R : A (Ω) →A (Ω) | res Z and R : J A (Ω) → J A (Ω) | res Z . Consequently, J : A (Ω) | res Z → J A (Ω) | res Z is also analgebra isomorphism. So J ( M ) | res Z can be thought of as a Hilbert module over J A (Ω) | res Z .On the other hand, considering the inclusion i : Z →
Ω we see that i induces a map i ∗ : J A (Ω) → J A (Ω) | res Z defined by i ∗ ( J f )( z ) = J f ( i ( z )), for z ∈ Z . Now one can make J ( M ) | res Z to a module over the algebra J A (Ω) by pushing it forward under the map i ∗ , thatis, J f · h | Z := i ∗ ( J f ) h | Z . Thus, recalling the fact that J defines an algebra isomorphism between A (Ω) and J A (Ω), wecan think of J ( M ) q as a module over A (Ω).Since the similar construction can be done for the Hilbert modules M ∈ B r (Ω) with sub-modules M consisting of holomorphic functions A (Ω) vanishing along Z to order k , one canalso ask whether the quotient modules arising from such submodules are in B ( N +1) r ( Z ). In thefollowing theorem, we give an affirmative answer of this for a simple class of Hilbert modulesin B (Ω). Theorem 4.8.
Let Ω ⊂ C m be a bounded domain containing the origin and Z ⊂ Ω be thecoordinate plane defined by Z := { z = ( z , . . . , z m ) ∈ Ω : z = · · · = z d = 0 } . We also assumethat M is a reproducing kernel Hilbert space with the property that the reproducing kernel K has diagonal power series expansion, that is, for z, w ∈ Ω K ( z, w ) = X α ≥ a α ( z − z ) α ( w − w ) α , for some z , w ∈ Z . Then the quotient module M q restricted to a module over A ( Z ) lies in B N +1 ( Z ) provided M ∈ B (Ω) .Proof. In view of Theorem 4 . .
7, it is enough to prove that the module of jets, J ( M ) restricted to Z is in B ( N +1) ( Z ). Let J K | Z be the reproducing kernel of J ( M ) | res Z . Then J K | Z has the following power series expansion at ( z , w ) ∈ Z : J K | Z (˜ z, ˜ w ) = X λ,µ ≥ A λµ (˜ z − z ) λ ( ˜ w − w ) µ where ˜ z, ˜ w ∈ Z , λ, µ ∈ ( N ∪ { } ) m − d , and A λµ ∈ M N +1 ( C ) are defined by the following formula A λµ = ∂ λ d +1 · · · ∂ λ m − d m ¯ ∂ µ d +1 · · · ¯ ∂ µ m − d m J K | Z ( z , w ) . Therefore, using the definition of
J K | Z we get, for 0 ≤ l, k ≤ N , that( A λµ ) lk = ∂ λ d +1 · · · ∂ λ m − d m ¯ ∂ µ d +1 · · · ¯ ∂ µ m − d m ∂ α · · · ∂ α d d ¯ ∂ β · · · ¯ ∂ β d d K ( z , w )(4.10)where ( A λµ ) lk is the lk -th entry of the matrix A λµ and θ − ( l ) = ( α , . . . , α d ), θ − ( k ) =( β , . . . , β d ).Since K has a diagonal power series expansion it is clear, from the equation ( 4.10 ), that A λµ = 0 unless λ = µ . Moreover, in a similar way the same equation also shows that ( A λλ ) lk = 0if l = k and ( A λλ ) ll = a α ,...,α d ,λ ,...,λ m − d . (4.11)Now from the hypothesis we have that the Taylor coefficients, a α satisfy the inequalitystated in part ( b ) of the Theorem 5.4 in [6]. As a consequence, a straight forward calculationusing the equation ( 4.11 ) shows that the matrices A λλ also satisfy the same inequality in [6] but with matrix valued constants. Furthermore, it follows, from the equation ( 4.11 ), that thecoordinate functions of Z act on J ( M ) | res Z by weighted shift operators with weights determinedby matrices A λλ . Therefore, by the same theorem in [6] the Hilbert module, J ( M ) | res Z , as amodule over A ( Z ) is in B N +1 ( Z ). (cid:3) We note that the above theorem provides examples of quotient modules which are in theCowen-Douglas class. Thus, it motivates to consider the following class of Hilbert modules.
Definition 4.9.
Let Ω ⊂ C m be bounded domain and Z ⊂
Ω be a connected complex sub-manifold of codimension d . Then we say that the pair of Hilbert modules ( M , M q ) over thealgebra A (Ω) is in B r,k (Ω , Z ) if(1) M ∈ B r (Ω);(2) there exists a resolution of the module M q as in ( 4.1 ) where the module M appearingin the resolution is quasi-free of rank r over the algebra A (Ω);(3) for f ∈ A (Ω), the restriction of the map J f to the submanifold defines the moduleaction on J ( M ) | res Z which is an isomorphic copy of M q ; and(4) the quotient module M q as a module over the algebra A (Ω) | res Z is in B ( N +1) r ( Z ) where N = (cid:0) d + k − k − (cid:1) −
1. 5.
Jet Bundle
This section is devoted to provide geometric invariants of quotient modules introduced inthe previous section. Suppose, to begin with, we have the Hilbert module M in B r (Ω) withthe submodule M and quotient module M q , as introduced in Section 4, satisfying the exactsequence ( 4.1 ). Then we have, following Remark 2 .
2, that M gives rise to a Hermitianholomorphic vector bundle E with the frame { K ( ., w ) σ , . . . , K ( ., w ) σ r : w ∈ Ω ∗ } on Ω ∗ . Nowto make calculations simpler let us consider the map c : Ω → Ω ∗ defined by w w andpull back the bundle E to a vector bundle over Ω. Then we denote this new bundle with thesame letter E and note that E is a Hermitian holomorphic vector bundle over Ω with theglobal holomorphic frame s := { s ( w ) , . . . , s r ( w ) : w ∈ Ω } with s j ( w ) := K ( ., w ) σ j , 1 ≤ j ≤ r .Correspondingly, we have ∂ l s j ( w ) = ∂ l K ( ., w ) σ j , 1 ≤ j ≤ r , 0 ≤ l ≤ N , where N = (cid:0) d + k − k − (cid:1) − d ≥ E over Ω corresponding to theHilbert module M ∈ B r (Ω) described above and Z ⊂
Ω is a connected complex submani-fold of codimension d . Without loss of generality assume that 0 ∈ Z and let ( U, φ ) be anadmissible coordinate chart (Definition 3 .
9) at 0 of Z . So pretending U as Ω we have that φ (Ω ∩ Z ) = { w ∈ φ (Ω) : w = · · · = w d = 0 } . Since we are interested to investigate unitaryinvariants of the quotient module M q with ( M , M q ) ∈ B r,k (Ω , Z ), following the Proposition3 .
12, it is enough to consider the submanifold φ (Ω ∩ Z ) ⊂ φ (Ω). Therefore, pretending φ (Ω) asΩ, we consider the submanifold Z defined as Z := { z = ( z , · · · , z m ) ∈ Ω : z = · · · = z d = 0 } . We then define the jet bundle J k E of order k of E relative to the submanifold Z on Ωby declaring { s , ∂ s , . . . , ∂ N s } as a frame for J k E on Ω where the differential operators ∂ j , N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 19 ≤ j ≤ N are as introduced in ( 2.9 ), and by ∂ l s we mean the ordered set of sections { ∂ l s , . . . , ∂ l s r } , 0 ≤ l ≤ N . Since we have a global frame on J k E we do not need to worryabout the transition rule.At this point, we should note that our construction depends on the choice of the normaldirection to Z which is, a priori, not unique. Nevertheless one way to show that our constructionis essentially unambiguous is the following proposition. Proposition 5.1.
Let ( U , φ ) and ( U , φ ) be two admissible coordinate charts of Ω aroundsome point p ∈ Z . Then two jet bundles J k E and J k E obtained as above with respect to ( U , φ ) and ( U , φ ) , respectively, are equivalent holomorphic vector bundles over U ∩ U .Proof. In fact, from Proposition 3 .
10 it is clear, for a frame s = { s , . . . , s r } of E on U ∩ U ,that on a small enough neighbourhood U of p in U ∩ U we have,for i = 1 , A k,φ i ( z ) · s i ( λ ) · · · s i r ( λ ) s i ( λ ) · · · s i r ( λ )... ... s iN ( λ ) · · · s iNr ( λ ) = s ( z ) · · · s r ( z ) ∂s ( z ) · · · ∂s r ( z )... ... ∂ N s ( z ) · · · ∂ N s r ( z ) , for z ∈ U and λ i ∈ φ i ( U )where λ i = ( λ i , . . . , λ id ), ( α , · · · , α d ) = θ − ( l ), and s ilj = ∂ | α | ∂λ α i ··· ∂λ αdid ( s j ◦ φ − i ), for 1 ≤ j ≤ r . Since A k,φ i ( z ), for i = 1 , z ∈ U , are invertible (Proposition 3 .
10) we can see that( A k,φ ( z ) ◦ A k,φ ( z ) − ) ⊗ I r is the desired bundle map where I r is the identity matrix of order r . (cid:3) Now in course of completing our construction to make the jet bundle J k E a Hermitianholomorphic vector bundle we need to put a Hermitian metric on J k E compatible with themetric on E . To this extent, if H ( w ) = (( h s i ( w ) , s j ( w ) i E )) ri,j =1 is the metric on E over Ωthen the Hermitian metric on J k E with respect to the frame { s , ∂ s , . . . , ∂ N s } is given by theGrammian J H := ((
J H lt )) Nl,t =0 with r × r blocks J H lt ( w ) := (( h ∂ l s i ( w ) , ∂ t s j ( w ) i )) ri,j =1 for 0 ≤ l, t ≤ N, w ∈ Ω . This completes our construction of the jet bundle.
Remark 5.2.
Note that, for the Hilbert module M over A (Ω) with the corresponding Hermit-ian holomorphic vector bundle E over Ω, the Hermitian holomorphic vector bundle E obtainedfrom J ( M ) is equivalent to the jet bundle J k E | res Z → Z of E relative to Z . To facilitate, let M be a reproducing kernel Hilbert module over A (Ω) which is in B r (Ω). Let K = (( K ij )) ri,j =1 bethe reproducing kernel of M . Then from the preceding construction we have that the metricfor the jet bundle is given by the formula h ∂ l K ( ., w ) σ i , ∂ t K ( ., w ) σ j i = ∂ l ¯ ∂ t K ij ( w, w ) for w ∈ Ω , ≤ l, t ≤ N, ≤ i, j ≤ r. On the other hand, the jet construction presented in Section 4 gives rise to the Hilbert module J ( M ) where J is the unitary module map J : M → J ( M ). Therefore, the vector bundle E isunitarily equivalent to J k E .Note that the action of the algebra A (Ω) on the module J ( M ) defines, for every f ∈ A (Ω),a holomorphic bundle map Ψ f : J k E → J k E whose matrix representation with respect to theframe J ( s ) := { P Nl =0 ∂ l s ⊗ ε l , . . . , P Nl =0 ∂ l s r ⊗ ε l } is the matrix J ( f ) ⊗ I r where J ( f ) is asin ( 4.6 ) and I r is the identity matrix of order r . Thus, Ψ f induces an action of A (Ω) on theholomorphic sections of the jet bundle J k E defined by f · σ ( w ) := Ψ f ( σ )( w ) , (5.1) for f ∈ A (Ω), w ∈ Ω and σ is a holomorphic section of J k E .Therefore, we observe that the question of determining the equivalence classes of modules J ( M ) is same as understanding the equivalence classes of the jet bundles J k E with an additionalassumption that the equivalence bundle map is also a module map on holomorphic sections over A (Ω). Hence it is natural to give the following definition (Definition 4.2, [11]). Definition 5.3.
Two jet bundles are said to be equivalent if there is an isometric holomorphicbundle map which induces a module isomorphism of the class of holomorphic sections.5.1.
Main results from Jet bundle.
In order to find geometric invariants of quotient moduleswe first investigate the simple case, d = k = 2. We show here that the curvature is the completeset of unitary invariants of the quotient module M q for a quasi-free Hilbert module M of rank1. For this case, we give a computational proof to depict the actual picture behind the generalresult which we will prove later in this subsection. Although the line of idea of the proof for k = 2 essentially is same as in [11], in our case calculations become more complicated as herewe have to deal with more than one transversal directions to Z . Thus, our results extend mostof the results of the paper [12], [11] as well as those from a recent paper [3].Without loss of generality, under some suitable change of coordinates, we can assume that0 ∈ Ω and U is a neighbourhood of 0 such that U ∩ Z = { ( z , . . . , z m ) ∈ Ω : z = z = 0 } .Consequently, (0 , , z , . . . , z m ) is the coordinates of Z in U . Now let us begin with a line bundle E over U ∗ with the real analytic metric G which possesses the following power series expansion G ( z ′ , z ′′ ) = ∞ X α,β =0 G αβ ( z ′′ ) z ′ α z ′ β (5.2)where ( z ′ , z ′′ ) ∈ U ∗ , α, β are multi-indices, z ′ α = z α z α , z ′ β = z β z β and z ′′ = ( z , . . . , z m ). Lemma 5.4.
Let Ω ⊂ C m be a bounded domain and Z be a complex connected submanifold of Ω of codimension . Suppose that K and ˜ K are the curvature tensors of line bundles E and ˜ E with respect to the Hermitian metric ρ and ˜ ρ of E and ˜ E , respectively. Then K and ˜ K areequal on Z if and only if there exists holomorphic functions ψ , ψ , ψ on Z such that ((˜ ρ θ − ( i ) θ − ( j ) )) i,j =0 = Ψ · (( ρ θ − ( i ) θ − ( j ) )) i,j =0 · Ψ ∗ (5.3) on Z where θ is as in ( 2.7 ) and Ψ is the × matrix Ψ = ψ ψ ψ ψ ψ . (5.4)Before going into the proof of the lemma let us give an application of it as follows. Theorem 5.5.
Suppose that M and ˜ M are pair of quasi-free Hilbert modules of rank over A (Ω) and that E and ˜ E are the line bundles corresponding to M and ˜ M , respectively. Let M q = M ⊖ M and ˜ M q = ˜ M ⊖ ˜ M be a pair of quotient modules of Hilbert modules M and ˜ M , respectively, over A (Ω) . Assume that ( M , M q ) and ( ˜ M , ˜ M q ) are in B , (Ω , Z ) . Then thequotient modules M q and ˜ M q are isomorphic if and only if the corresponding curvature tensors K and ˜ K of the line bundles E and ˜ E , respectively, are equal on Z .Proof. In fact, Theorem 4 . M q and ˜ M q is same as the equivalenceof J ( M ) | res Z and J ( ˜ M ) | res Z . So let us begin with an isometric module map Ψ : J ( M ) | res Z → N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 21 J ( ˜ M ) | res Z . Since Ψ intertwines the module action Ψ is of the form given in ( 5.4 ). Moreover,being an isometry, Ψ satisfies J K | Z = Ψ · J ˜ K | Z · Ψ ∗ (5.5)which is equivalent to saying that Ψ satisfies the identity ( 5.3 ) on Z as, for z ∈ Z , ρ ( z ) isnothing but K ( z, z ). Then the Lemma 5 . . K and ˜ K on Z implies that Ψ is of the form given in ( 5.4 ) and satisfies ( 5.5 ) which in turn yields that Ψis an isometry from J ( M ) | res Z onto J ( ˜ M ) | res Z and intertwines the module action. Hence thiscompletes the proof. (cid:3) Proof of Lemma . . Let us begin with the assumption that there exists holomorphic functions ψ , ψ , ψ on Z such that ( 5.3 ) holds on Z . Then we wish to show that K and ˜ K are equalrestricted to Z . We have, by the local expression of the curvature ( 2.3 ), that K ij = ∂ i ¯ ∂ j log ρ and ˜ K ij = ∂ i ¯ ∂ j log ˜ ρ , for i, j = 1 , . . . , m . So let us calculate ˜ K ij ( z ) for any point z ∈ Z and i, j = 1 , . . . , m . ˜ K ( z ) = [˜ ρ∂ ¯ ∂ ˜ ρ − ¯ ∂ ˜ ρ∂ ˜ ρ ]˜ ρ − | z = [ ρ (0 , , ρ (1 , , − ρ (0 , , ρ (1 , , ] ρ − , , = K ( z )We also find that ˜ K ( z ) = K ( z ) for z ∈ Z by doing a similar calculation as in the case of K ( z ). Now we calculate ˜ K ( z ).˜ K ( z ) = [˜ ρ∂ ¯ ∂ ˜ ρ − ¯ ∂ ˜ ρ∂ ˜ ρ ]˜ ρ − | z = [ ρ (0 , , ρ (1 , , − ρ (1 , , ρ (0 , , ] ρ − , , = K ( z )Since ˜ K ( z ) = ˜ K we have ˜ K ( z ) = K ( z ) for z ∈ Z . Finally, let us calculate ˜ K i ( z ), for z ∈ Z and 2 < i ≤ m .˜ K i ( z ) = [˜ ρ∂ i ¯ ∂ ˜ ρ − ¯ ∂ ˜ ρ∂ i ˜ ρ ]˜ ρ − | z = [ ρ (0 , , ∂ i ρ (0 , , − ρ (0 , , ∂ i ρ (0 , , ] ρ − , , = K i ( z )Similarly one can show that ˜ K i ( z ) = K i ( z ) for z ∈ Z and 1 ≤ i ≤ m . Thus, we are done withthe converse part using the skew symmetry property of the matrix (( K ij ( z ))) mi,j =1 .Now let us prove the forward direction, namely, assuming that K and ˜ K are equal along Z we want to find ψ , ψ , ψ holomorphic on Z such that ( 5.3 ) holds.Let ˜ ρ = r · ρ , and Γ = log r . Then Γ is real analytic function on Ω. We can, therefore, expandΓ in power series, that is, Γ( z ′ , z ′′ ) = ∞ X α,β =0 Γ αβ ( z ′′ ) z ′ α z ′ β (5.6)where α, β are multi-indices, z ′ α = z α z α , z ′ β = z β z β and z ′′ = ( z , . . . , z m )We have, from our assumption, that K and ˜ K are equal along Z which is equivalent tothe fact that ∂ i ¯ ∂ j Γ = 0, for 1 ≤ i, j ≤ m , along Z . We separate out this into following threedifferent cases. Case I: ∂ i ¯ ∂ j Γ = 0 along Z , for i = 1 , j = 3 , . . . , m .For i = 1 we have ∂ ¯ ∂ j Γ | Z = 0, for j = 3 , . . . , m , which is, from ( 5.6 ), equivalent to¯ ∂ j Γ (1 , , = 0 on Z . In other words, Γ (1 , , is holomorphic on Z . Similarly considering thecase with i = 2, we get Γ (0 , , is also holomorphic on Z . Case II: ∂ i ¯ ∂ j Γ = 0 along Z , for i, j = 1 , ∂ i ¯ ∂ j Γ | Z = 0, for i, j = 1 ,
2. For i = j = 1, wehave ∂ ¯ ∂ Γ | Z = 0, that is, Γ (1 , , = 0 on Z and, for i = j = 2, Γ (0 , , = 0 along Z . Finally,for i = 1 , j = 2, it is easy to verify from the equation ( 5.6 ) that Γ (1 , , = 0 on Z and doingthe same calculation with i and j interchanged we have Γ (0 , , = 0 on the submanifold Z . Case III: ∂ i ¯ ∂ j Γ = 0 along Z , for i, j = 3 , . . . , m .In this last case, we have ∂ i ¯ ∂ j Γ | Z = 0, i, j = 3 , . . . , m which together with power seriesexpansion of Γ yield that ∂ i ¯ ∂ j Γ (0 , , = 0, for i, j = 3 , . . . , m , on Z . Since Z is a complexsubmanifold with coordinates z = (0 , , z , . . . , z m ) ∈ Z the above equations together implythat Γ (0 , , ( z ′′ ) = ψ ( z ′′ ) + ψ ( z ′′ ), for z ′′ ∈ Z and some holomorphic functions ψ , ψ on Z .Now, substituting the above coefficients in the equation ( 5.6 ) and noting that Γ is realvalued, we haveΓ( z ′ , z ′′ ) = ψ + β z + η z + ψ + β z + η z + (terms of degree ≥ ψ i , β i , η i , i = 1 ,
2, are holomorphic functions on Z . Since Γ is a real valued functionΓ = Γ+Γ2 and hence we haveΓ( z ′ , z ′′ ) = ψ + βz + ηz + ψ + βz + ηz + (terms of degree ≥ ψ = ψ + ψ , β = β + β and η = η + η . So from the definition of Γ we can write r = exp Γ= | exp ψ | · | (1 + βz + βz + | β | z z + · · · ) | · ( | ηz + ηz + | η | z z + · · · ) | · · · = | exp ψ | · (1 + βz + ηz + βz + ηz + | β | z z + βηz z + βηz z + | η | z z + · · · )Thus, putting the above expression of r in ˜ ρ = r · ρ and equating the coefficients of ˜ ρ and ρ wesee that ψ , ψ and ψ with ψ = exp ψ, ψ = exp ψβ, ψ = exp ψη yield our desired matrix in ( 5.4 ). (cid:3) It would be nice if one could carry forward the arguments used in the proof of Lemma5 . k , it would be cumbersome to continue the calculationdone in the above Lemma. On the other hand, application of normalized frames makes thecalculations simpler and enables us to get a conceptual proof in the general case as well. Weadopt the idea of using normalized frame from the paper [3] in our case to provide the geometricinvariants for quotient modules using jet bundle construction relative to a smooth complexsubmanifold of codimenssion d . To this extent the following theorem provides the requireddictionary between the analytic theory and geometric theory for quotient modules obtainedfrom submodules consisting of vector valued holomorphic functions on Ω vanishing along asmooth complex submanifold of codimension d . N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 23
Theorem 5.6.
Let Ω be a bounded domain in C m and Z be the complex submanifold in Ω of codimension d . Suppose that ( M , M q ) and ( ˜ M , ˜ M q ) are in B r,k (Ω , Z ) . Then the quotientmodules M q and ˜ M q are equivalent as modules over A (Ω) if and only if the jet bundles J k E | res Z and J k ˜ E | res Z are equivalent where E and ˜ E are the Hermitian holomorphic vector bundles over Ω corresponding to Hilbert modules M and ˜ M , respectively.Proof. Proof follows from Theorem 4 . . (cid:3) Thanks to Theorem 5 . M q by studying the geometry of the jet bundles J l E | res Z , for 0 ≤ l ≤ k . Beforeproceeding further, let us recall a fact from complex analysis. Lemma 5.7.
Let Ω ⊂ C m be a domain and f ( z, w ) be a function on Ω × Ω which is holomorphicin z and anti-holomorphic in w . If f ( z, z ) = 0 for all z ∈ Ω , then f ( z, w ) = 0 identically on Ω . Since this lemma is well known [14, Proposition 1] we omit the proof. We use the lemmaseveral times in the proof of following theorems.Note also that a Hermitian holomorphic vector bundle can not have holomorphic orthonor-mal frame in general. Instead one can have (Lemma 2.4 of [4]) a holomorphic frame on aneighbourhood of a point which is orthonormal at that point. Then using the technique of theproof of Lemma 2.4 in [4] in a similar way, we have the following existence of normalized frameof a Hermitian holomorphic vector bundle over Ω along a submanifold of codimension at least d in Ω. In the following proposition we use the notation z = ( z ′ , z ′′ ) where z ′ = ( z , · · · , z d )and z ′′ = ( z d +1 , . . . , z m ). Proposition 5.8.
Let E be a Hermitian holomorphic vector bundle of rank r over a boundeddomain Ω ⊂ C m . Assume that ∈ Ω and Z ⊂ Ω is the submanifold defined by z = · · · = z d = 0 .Then there is a holomorphic frame s ( z ′ , z ′′ ) = { s ( z ′ , z ′′ ) , . . . , s r ( z ′ , z ′′ ) } on a neighbourhood ofthe origin in Ω such that (( h ∂ l s i (0 , z ′′ ) , s j (0 , i )) ri,j =1 is the zero matrix for any integer l and (( h s i (0 , z ′′ ) , s j (0 , i )) ri,j =1 is the identity matrix on Z . We say a frame is normalized at origin if it satisfies the properties in the above proposition.
Theorem 5.9.
Let Ω be a bounded domain in C m and Z be the complex submanifold in Ω ofcodimension d defined by z = · · · = z d = 0 . Assume that pair of Hilbert modules ( M , M q ) and ( ˜ M , ˜ M q ) are in B ,k (Ω , Z ) . Then M q and ˜ M q are unitarily equivalent as modules over A (Ω) ifand only if ∂ l ¯ ∂ j k ˜ s k = ∂ l ¯ ∂ j k s k on Z for all ≤ l, j ≤ N where { s ( z ) } and { ˜ s ( z ) } are framesof the line bundles E and ˜ E on Ω associated to the Hilbert modules M and ˜ M , respectively,normalized at origin.Proof. We begin with the observation, following Theorem 5 .
6, that the quotient modules M q and˜ M q are equivalent as modules over A (Ω) if and only if the jet bundles J k E | res Z and J k ˜ E | res Z areequivalent where E and ˜ E are the Hermitian holomorphic vector bundles over Ω correspondingto Hilbert modules M and ˜ M , respectively. So it is enough to prove that there exists a jetbundle isomorphism Φ : J k E | res Z → J k ˜ E | res Z if and only if ∂ l ¯ ∂ j k ˜ s k = ∂ l ¯ ∂ j k s k on Z for all0 ≤ l, j ≤ N .We start with the necessity. Let Φ : J k E | res Z → J k ˜ E | res Z be a jet bundle isomorphism.Consequently, by Definition 5 .
3, Φ intertwines the module actions on the space of holomorphicsections and preserves the Hermitian metrics. The isomorphism Φ can be represented by an( N + 1) × ( N + 1) complex matrix (( φ ij )) Ni,j =0 with respect to the frames { s (0 , z ′′ ) , ∂s (0 , z ′′ ) , . . . ,∂ N s (0 , z ′′ ) } and { ˜ s (0 , z ′′ ) , ∂ ˜ s (0 , z ′′ ) , . . . , ∂ N ˜ s (0 , z ′′ ) } where φ ij are holomorphic functions on Z . Then in terms of matrices the fact that Φ is an isomorphism of two jet bundles J k E | res Z and J k ˜ E | res Z translates to the following two matrix equations on Z :(5.8) (( h ∂ l s, ∂ j s i )) Nl,j =0 = (( φ ij )) Ni,j =0 (( h ∂ l ˜ s, ∂ j ˜ s i )) Nl,j =0 ((( φ ij )) Ni,j =0 ) ∗ (5.9) (( φ ij )) Ni,j =0 (( J ( f ) lk )) Nl,k =0 = (( J ( f ) lk )) Nl,k =0 (( φ ij )) Ni,j =0 . Then the proof of the forward direction easily follows from the following claims.
Claim 1. (a) For 0 ≤ k ≤ N and z = (0 , z ′′ ) ∈ Z , φ kk (0 , z ′′ ) = φ (0 , z ′′ ).(b) Let 1 ≤ i, j ≤ N , α = ( α , . . . , α d ) = θ − ( i ) , β = ( β , . . . , β d ) = θ − ( j ). Then for z ∈ Z we have(5.10) φ ij (0 , z ′′ ) = (cid:26)(cid:0) αα − β (cid:1) φ θ ( α − β )0 (0 , z ′′ ) if α t ≥ β t ∀ t = 1 , · · · , d, . Before going into the proof of Claim 1, let us make some observations about the matrix J ( f ) = (( J ( f ) lk )) Nl,k =0 which will be used in the proof. So to begin with, let γ = ( γ , . . . , γ d )with θ ( γ ) = l , 0 ≤ l ≤ N , and γ j ∈ N ∪ { } , 1 ≤ j ≤ d . We also denote the subdiagonals of thematrix J ( f ) as S , . . . , S N , that is, S j is the set S j := {J ( f ) j , . . . , J ( f ) j + kk +1 , . . . , J ( f ) NN − j +1 } = {J ( f ) lt : j ≤ l ≤ N, ≤ t ≤ N − j + 1 , l − t = j − } . Thus, S consists of all diagonal entries and S N is the singleton set {J ( f ) N } . Then a simplecalculation using the definition ( 4.6 ) of J ( f ) yields the following property.( P1 ) For f ( z ′ , z ′′ ) = z γ · · · z γ d d with θ ( γ , . . . , γ d ) = l , the matrix J ( f ) | Z has only non-zeroentries along the subdiagonals, S j , for l ≤ j ≤ N .Now we note, for l ≤ i ≤ N , that common entries of i -th row of J ( f ) with subdiagonals S l , . . . , S N are J ( f ) i , . . . , J ( f ) ii − l +1 , respectively. Therefore, for l ≤ i ≤ N with θ − ( i ) = α =( α , . . . , α d ) and f ( z ′ , z ′′ ) = z γ · · · z γ d d , the above property, ( P1 ), shows that non-zero entriesof i -th row of the matrix J ( f ) | Z must live in the set {J ( f ) ij (0 , z ′′ ) : 0 ≤ j ≤ i − l + 1 } . On theother hand, for 0 ≤ j ≤ i − l + 1 with θ − ( j ) = β = ( β , . . . , β d ), J ( z γ · · · z γ d d ) ij | Z = 0 ⇐⇒ ∂ α − β ( z γ · · · z γ d d ) | Z = 0 ⇐⇒ β = α − γ. Thus, we have the following property of J ( z γ · · · z γ d d ).( P2 ) For α, γ , i, l as above and f ( z ′ , z ′′ ) = z γ · · · z γ d d , J ( f ) iθ ( α − γ ) (0 , z ′′ ) is the only non-zeroentry of i -th row of J ( f ) | Z . In particular, we observe that J ( f ) ij (0 , z ′′ ) = 0, for any j with 0 ≤ j ≤ N whenever α t < γ t for some t ∈ { , . . . , d } .Since Φ is a jet bundle isomorphism between J k E | res Z and J k ˜ E | res Z , by the definition (Def-inition 5 .
12) Φ commutes with the module action of A (Ω) on the sections of the above jetbundles, namely, from ( 5.1 ) we have the equation ( 5.9 ) holds on Z for all f ∈ A (Ω). Let g ( z ′ , z ′′ ) = z η · · · z dη d , for given 0 ≤ k ≤ N with θ − ( k ) = ( η , . . . , η d ). Then from ( P2 ) wehave that J ( g ) k (0 , z ′′ ) is the only non-zero entry of k -th row of J ( g ) | Z . Therefore, equating k f = g we obtain, for z ∈ Z , that φ kk (0 , z ′′ ) J ( g ) k (0 , z ′′ ) = J ( g ) k (0 , z ′′ ) φ (0 , z ′′ )which proves (a) of the claim above as J ( g ) k = η !.To prove the part (b) let 0 ≤ i, j ≤ N , θ − ( i ) = ( α , . . . , α d ), θ − ( j ) = ( β , . . . , β d ) and g ( z ′ , z ′′ ) = z β · · · z β d d . We also assume that i < j . Then, using ( P2 ) with f = g and l = j , N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 25 we see that J ( g ) j (0 , z ′′ ) is the only non-zero entry of the matrix J ( g )(0 , z ′′ ) for (0 , z ′′ ) ∈ Z .Therefore, only i φ ij . On the otherhand, the i i < j , thanks to the property( P1 ). Thus, comparing the i φ ij (0 , z ′′ ) = 0 on Z , for 1 ≤ i < j ≤ N .Now we assume that i > j . Since only non-zero entry of j -th row of the matrix J ( g ) | Z is J ( g ) j (0 , z ′′ ) it is clear that only i φ ij . The i β ! φ ij (0 , z ′′ ) as J ( g ) j (0 , z ′′ ) = β !, for (0 , z ′′ ) ∈ Z . So,as before in order to calculate φ ij , we need to compare i P2 ) we have that only non-zero entry of i -th row of J ( g ) | Z is J ( g ) iθ ( α − β ) (0 , z ′′ ) which,by definition ( 4.6 ) of J ( g ) with g ( z ′ , z ′′ ) = z β · · · z β d d , is (cid:18) αα − β (cid:19) ∂ α − ( α − β ) g ( z ′ , z ′′ ) | Z = (cid:18) αα − β (cid:19) ∂ β g ( z ′ , z ′′ ) | Z = (cid:18) αα − β (cid:19) β ! , provided α t ≥ β t , for all t = 1 , . . . , d . Furthermore, if α t < β t for some t ∈ { , . . . , d } , it followsfrom ( P2 ) that every entry of i -th row is zero. Therefore, equating i φ ij (0 , z ′′ ) = (cid:26)(cid:0) αα − β (cid:1) φ θ ( α − β )0 (0 , z ′′ ) for α t ≥ β t ∀ t = 1 , . . . , d, φ ij (0 , z ′′ ))) Ni,j =0 is a lower triangular matrix. Con-sequently, we have that Φ induces bundle morphisms Φ | J l E | res Z : J l E | res Z → J l ˜ E | res Z , for0 ≤ l ≤ k . Claim 2. φ is a constant function and φ ii = φ , for i = 0 , . . . , N , on Z .Note that it is enough to show that φ is a constant function on Z thanks to Claim 1. Infact, from the equation ( 5.8 ) we have h s (0 , z ′′ ) , s (0 , z ′′ ) i = φ (0 , z ′′ ) h ˜ s (0 , z ′′ ) , ˜ s (0 , z ′′ ) i φ (0 , z ′′ ) . Consequently, Lemma 5 . . φ (0 , z ′′ ) φ (0 ,
0) = 1 . Hence we are done with Claim 2.
Claim 3. (( φ ij (0 , z ′′ ))) Ni,j =0 is a constant diagonal matrix with diagonal entries φ ii = φ ,for 0 ≤ i ≤ N , that is, (( φ ij (0 , z ′′ ))) Ni,j =0 = φ · I where I is the ( N + 1) × ( N + 1) identitymatrix.In view of Claim 1 and Claim 2, it is enough to show that φ l = 0, for 0 < l ≤ N , on Z . Socalculating l . h ∂ l s (0 , z ′′ ) , s (0 , w ′′ ) i = l X j =0 φ lj (0 , z ′′ ) h ∂ j ˜ s (0 , z ′′ ) , ˜ s (0 , w ′′ ) i φ and consequently, after putting w ′′ = 0 and applying the Proposition 5 . { s } and { ˜ s } at origin we get φ l (0 , z ′′ ) = 0 on Z .Thus, Claim 1, Claim 2, Claim 3 and the equation ( 5.8 ) together yield that ∂ l ¯ ∂ j (cid:13)(cid:13) s (0 , z ′′ ) (cid:13)(cid:13) = φ ∂ l ¯ ∂ j (cid:13)(cid:13) ˜ s (0 , z ′′ ) (cid:13)(cid:13) φ = ∂ l ¯ ∂ j (cid:13)(cid:13) ˜ s (0 , z ′′ ) (cid:13)(cid:13) (5.12) on Z , for 0 ≤ l, j ≤ N .The converse statement is easy to see. Indeed, if the equation ( 5.12 ) happens to be truethen the desired jet bundle isomorphism Φ is given by the constant matrix I with respect to theframes { s, ∂s, . . . , ∂ N s } and { ˜ s, ∂ ˜ s, . . . , ∂ N ˜ s } where I is the identity matrix of order N + 1. (cid:3) Theorem 5.10.
Let Ω be a bounded domain in C m and Z be the complex submanifold in Ω ofcodimension d defined by z = · · · = z d = 0 . Assume that ( M , M q ) and ( ˜ M , ˜ M q ) are pair ofHilbert modules in B r,k (Ω , Z ) . Then M q and ˜ M q are unitarily equivalent as modules over A (Ω) if and only if there exists a constant unitary matrix D such that ∂ l ¯ ∂ j H = D ( ∂ l ¯ ∂ j ˜ H ) D ∗ on Z ,for all ≤ l, j ≤ N where H ( z ) and ˜ H ( z ) are the Grammian matrices for the holomorphicframes s and ˜ s of the Hermitian holomorphic vector bundles E and ˜ E on Ω associated to theHilbert modules M and ˜ M , respectively, normalized at origin.Proof. We note, as in Theorem 5 .
9, that it is enough to show that the jet bundles J k E | res Z and J k ˜ E | res Z are equivalent according to the Definition 5 . D such that ∂ l ¯ ∂ j H = D ( ∂ l ¯ ∂ j ˜ H ) D ∗ on Z , for all 0 ≤ l, j ≤ N , where H ( z ) and˜ H ( z ) are as above.So, to begin with, let Φ : J k E | res Z → J k ˜ E | res Z be a jet bundle isomorphism. Then theisomorphism Φ can be represented by an ( N + 1) × ( N + 1) block matrix ((Φ lt )) Nl,t =0 withrespect to the frames { s , ∂ s (0 , z ′′ ) , . . . , ∂ N s (0 , z ′′ ) } and { ˜ s , ∂ ˜ s (0 , z ′′ ) , . . . , ∂ N ˜ s (0 , z ′′ ) } where Φ lt are holomorphic r × r matrix valued functions on Z . Then in terms of matrices the fact thatΦ is an isometry of two jet bundles J k E | res Z and J k ˜ E | res Z translates to the following matrixequation on Z :(5.13) (( ∂ l ¯ ∂ t H )) Nl,t =0 = ((Φ lt )) Nl,t =0 (( ∂ l ¯ ∂ t ˜ H )) Nl,t =0 (((Φ lt )) Nl,t =0 ) ∗ . Let E i | res Z and ˜ E i | res Z be line bundles determined by the frames { s i } and { ˜ s i } , respectively,on Z , for 1 ≤ i ≤ r . Then we can have the decomposition E | res Z = ⊕ ri =1 E i | res Z and J k E | res Z = ⊕ ri =1 J k E i | res Z with { s i , ∂s i , . . . , ∂ N s i } as a frame on Z . Further, let P i : J k E | res Z → J k E i | res Z and ˜ P i : J k ˜ E | res Z → J k ˜ E i | res Z be projection morphisms where the frame { ˜ s i , ∂ ˜ s i , . . . , ∂ N ˜ s i } de-fines the jet bundle J k ˜ E i | res Z . Then note that the matrix of Φ with respect to the frames J ( s ) = { P Nl =0 ∂ l s ⊗ ε l , . . . , P Nl =0 ∂ l s r ⊗ ε l } and J (˜ s ) = { P Nl =0 ∂ l ˜ s ⊗ ε l , . . . , P Nl =0 ∂ l ˜ s r ⊗ ε l } is (([ P ij ])) ri,j =1 where [ P ij ] represents the matrix of ˜ P i Φ P ∗ j with respect to the frames { s j , ∂s j , . . . , ∂ N s j } and { ˜ s i , ∂ ˜ s i , . . . , ∂ N ˜ s i } . Since Φ is a jet bundle isomorphism (Definition5 .
3) it intertwines the module action on the class of holomorphic sections of J k E | res Z and J k ˜ E | res Z . As a consequence, we have(([ P ij ])) ri,j =1 ( J ( f ) ⊗ I r ) = ( J ( f ) ⊗ I r )(([ P ij ])) ri,j =1 , for f ∈ A (Ω), which is equivalent to the fact that the bundle morphisms ˜ P i Φ P ∗ j intertwine themodule action ( 5.1 ) on holomorphic sections of J k E j | res Z and J k ˜ E i | res Z . Thus, ˜ P i Φ P ∗ j definesa jet bundle morphism from J k E j | res Z onto J k ˜ E i | res Z .We, therefore, can apply Claim 1 in Theorem 5 .
10 to ˜ P i Φ P ∗ j to conclude, for 0 ≤ l, t ≤ N ,that [ P ij ] lt = (cid:18) αα − β (cid:19) [ P ij ] θ ( α − β )0 (0 , z ′′ ) , ( α − β ) ∈ ( N ∪ { } ) d = (cid:18) αα − β (cid:19) (Φ θ ( α − β )0 (0 , z ′′ )) ij , ( α − β ) ∈ ( N ∪ { } ) d , N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 27 otherwise, [ P ij ] lt is the zero matrix. Then it follows that the matrix of Φ(0 , z ′′ ) with respectto the frames { s , ∂ s , . . . , ∂ N s } and { ˜ s , ∂ ˜ s , . . . , ∂ N ˜ s } is a lower triangular block matrix withΦ ll (0 , z ′′ ) = Φ (0 , z ′′ ) for 0 ≤ l ≤ N and, for 0 ≤ l, t ≤ N , α = ( α , . . . , α d ) = θ − ( l ) , β =( β , . . . , β d ) = θ − ( t ) and 1 ≤ i, j ≤ r ,(Φ lt (0 , z ′′ )) ij = (cid:18) αα − β (cid:19) (Φ θ ( α − β )0 (0 , z ′′ )) ij if ( α − β ) ∈ ( N ∪ { } ) d , (5.14)and is zero, otherwise, on Z . Now a similar proof as in Claim 2 in Theorem 5 . H, ˜ H and Φ on Z yields that Φ is a constant unitary matrix.Thus, the proof will be done once we prove that Φ l = 0, for 0 < l ≤ N , on Z . So calculatingthe l . h ∂ l s i (0 , z ′′ ) , s j (0 , w ′′ ) i )) ri,j =1 = l X t =0 Φ lt (0 , z ′′ )(( h ∂ t ˜ s i (0 , z ′′ ) , ˜ s j (0 , w ′′ ) i )) ri,j =1 ! Φ ∗ and consequently, after putting w ′′ = 0 and applying the Proposition 5 . s and˜ s at origin we get Φ l (0 , z ′′ ) = 0 on Z . Thereby from ( 5.13 ) we have ∂ l ¯ ∂ j H (0 , z ′′ ) = D ( ∂ l ¯ ∂ j ˜ H (0 , z ′′ )) D ∗ (5.15)on Z for all 0 ≤ l, j ≤ N where D = Φ .For the converse direction, note that the equation ( 5.15 ) canonically gives rise to the jetbundle isomorphism Φ by prescribing the matrix of Φ as D ⊗ I with respect to the frames J ( s )and J (˜ s ) where I is the identity matrix of order N + 1. (cid:3) Corollary 5.11.
Let T = ( T , . . . , T m ) and ˜ T = ( ˜ T , . . . , ˜ T m ) be two operator tuples in B (Ω) .Then T and ˜ T are unitarily equivalent if and only if there are jet bundle isomorphisms Φ k : J k E | res Z → J k ˜ E | res Z , for every k ∈ N ∪ { } where Z is any singleton set { p } , for p ∈ Ω .Proof. The necessity part is trivial and so we only show that T and ˜ T are unitarily equivalentassuming that there are jet bundle isomorphisms Φ k : J k E | res Z → J k ˜ E | res Z , for every k ∈ N ∪ { } .Let E and ˜ E be vector bundles over Ω corresponding to operator tuples T and ˜ T , respectively,and Z = { } . Thus, the codimension of Z is m . We now wish to apply the previous theoremfor each non-negative integer k . So let us start with frames s and ˜ s for E and ˜ E , respectively,normalized at origin. Then by Theorem 5 . k ∈ N ∪{ } , ∂ l ¯ ∂ j k ˜ s k = ∂ l ¯ ∂ j k s k at 0 for all 0 ≤ l, j ≤ N ( k ) where N ( k ) = (cid:0) d + k − k − (cid:1) −
1. In other words, translating the notationsused in the above equation we get ∂ α · · · ∂ α m m ¯ ∂ β · · · ¯ ∂ β m m k s (0) k = ∂ α · · · ∂ α m m ¯ ∂ β · · · ¯ ∂ β m m k ˜ s (0) k (5.16)for all α, β ∈ ( N ∪ { } ) m .Now since s and ˜ s both are holomorphic on their domains of definition k s k and k ˜ s k arereal analytic there. Consequently, using the power series expansion of k s k and k ˜ s k togetherwith the equation ( 5.16 ) we obtain that k s ( z ) k = k ˜ s ( z ) k on some open neighbourhood, say Ω , of the origin in Ω. Thus, the bundle map Φ : E → ˜ E determined by the formula Φ( s ( z )) = ˜ s ( z ) defines an isometric bundle isomorphism between E and ˜ E over Ω . Then our result is a direct consequence of the Rigidity theorem in [4]. (cid:3) Remark 5.12.
Note that the above theorem shows that the unitary equivalence of local op-erators (1.5 in [4]) N ( k ) ω and ˜ N ( k ) ω corresponding to T and ˜ T , respectively, for all k ≥ ω ∈ Ω implies the unitary equivalence of T and ˜ T . In other words, any tuplesof operators T ∈ B (Ω) enjoy the ”Taylor series expansion” property. Moreover, following thetechnique used in Theorem 18 in [3], it is seen that the same property is also enjoyed by any T ∈ B r (Ω), r ≥ Theorem 5.13.
Let Ω ⊂ C m be a bounded domain and Z ⊂ Ω be the complex manifold ofcodimension d defined by z = · · · = z d = 0 . Suppose that pair of Hilbert modules ( M , M q ) and ( ˜ M , ˜ M q ) are in B r,k (Ω , Z ) . Then M q and ˜ M q are isomorphic as modules over A (Ω) if andonly if following conditions hold: (i) There exists holomorphic isometric bundle map
Φ : E | res Z → ˜ E | res Z where E and ˜ E areHermitian holomorphic vector bundles over Ω corresponding to the Hilbert modules M and ˜ M over A (Ω) . (ii) The transverse curvature of E and ˜ E as well as their covariant derivatives of order atmost k − , along the transverse directions to Z , are intertwined by Φ on Z . (iii) The bundle map Φ intertwines the bundle maps J li ( H ) := ¯ ∂ i ( H − ∂ l H ) and J li ( ˜ H ) :=¯ ∂ i ( ˜ H − ∂ l ˜ H ) , d + 1 ≤ i ≤ m , holds on Z where s = { s , . . . , s r } and ˜ s = { ˜ s , . . . , ˜ s r } areframes of E and ˜ E , respectively, for ≤ l ≤ N , and H and ˜ H are Grammians of s and ˜ s , respectively. Remark 5.14.
At this point although it seems that the condition (iii) in the above theoremdepends on the choice of a frame it is not the case. For instance, if t is another frame normalizedat origin we have t = s A for some holomorphic function A : Z → GL r ( C ). Now since both s and t are normalized at origin the same proof as in Claim 2 in Theorem (5.9) with matrix valuedholomorphic functions shows that A is a constant unitary matrix. Thus, we have H = AHA ∗ and hence it follows that J li ( G ) = A J li ( H ) A − , d + 1 ≤ i ≤ m, where G is the Gramm matrix of the frame t . Proof.
Let Ω ⊂ C m and Z ⊂ Ω be as given. Suppose that M q and ˜ M q are equivalent as modulesover A (Ω). Then by Theorem 5 .
10 there exists a constant unitary matrix D such that ∂ l ¯ ∂ j H (0 , z ′′ ) = D ( ∂ l ¯ ∂ j ˜ H (0 , z ′′ )) D ∗ , for (0 , z ′′ ) ∈ Z and 0 ≤ l, j ≤ N, (5.17)where H ( z ) and ˜ H ( z ) are the Grammian matrices for holomorphic frames s = { s , . . . , s r } and˜ s = { ˜ s , . . . , ˜ s r } of the Hermitian holomorphic vector bundles E and ˜ E on Ω associated toHilbert modules M and ˜ M , respectively, normalized at origin. In particular, for l = j = 0,( 5.17 ) becomes H (0 , z ′′ ) = D ˜ H (0 , z ′′ ) D ∗ , for (0 , z ′′ ) ∈ Z . Let Φ : E | res Z → ˜ E | res Z be the bundle morphism whose matrix representation with respectto the frames s and ˜ s is D . Then Φ is the desired isometric bundle map in (i). Further, theequation ( 5.17 ) together with (i) of Lemma 2 . D is a constant unitarymatrix on Z , (iii) is an easy consequence of ( 5.17 ) with j = 0. N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 29
Now let us prove the converse direction. To do so we show that the condition (i), (ii), (iii)in the statement together imply the condition of the Theorem 5 .
10, that is, we need to showthat there exists a constant unitary matrix D on Z such that the equation ( 5.18 ) holds onthe submanifold Z for 0 ≤ l, t ≤ N and frames s and ˜ s of E and ˜ E , respectively, normalized atorigin.We first extend the holomorphic isometric bundle map Φ : E | res Z → ˜ E | res Z , obtained fromcondition (i), to a family of linear isometries ˆΦ z : J k E | z → J k ˜ E | z for every z ∈ Z . Then weshow that this extension is actually a jet bundle isomorphism providing our desired matrix. Solet us begin with frames s and ˜ s for E and ˜ E , respectively, normalized at z ∈ Z , for an arbitrary z ∈ Z . Condition (i) then yields an isometric holomorphic bundle map Φ : E | res Z → ˜ E | res Z andconsequently, we have a holomorphic r × r matrix valued function φ on Z such that H (0 , z ′′ ) = φ (0 , z ′′ ) ˜ H (0 , z ′′ ) φ (0 , z ′′ ) ∗ (5.18)where φ represents Φ with respect to frames s and ˜ s . Since both s and ˜ s are normalized at z , the above equation ( 5.18 ) shows that φ (0 , z ′′ ) is a unitary matrix. Furthermore, fromcondition (ii) of our hypothesis along with second statement of Lemma 2 . ≤ α + · · · + α d ≤ k − , ≤ β + · · · + β d ≤ k − ∂ α · · · ∂ α d d ¯ ∂ β · · · ¯ ∂ β d d H (0 , z ′′ ) = φ (0 , z ′′ ) ∂ α · · · ∂ α d d ¯ ∂ β · · · ¯ ∂ β d d ˜ H (0 , z ′′ ) φ (0 , z ′′ ) ∗ (5.19)as ∂ α · · · ∂ α d d H (0 , z ′′ ) (respectively, ∂ α · · · ∂ α d d ˜ H (0 , z ′′ )) and ¯ ∂ β · · · ¯ ∂ β d d H (0 , z ′′ ) (respectively,¯ ∂ β · · · ¯ ∂ β d d ˜ H (0 , z ′′ )) are zero matrices for any α i , β i ≥ i = 1 , . . . , d . Thus, the equationsabove (5 . , .
19) lead to the following natural isometric extension, ˆΦ z : J k E | z → J k ˜ E | z defined by ˆΦ z ( ∂ l s j (0 , z ′′ )) = r X i =1 φ ji (0 , z ′′ ) ∂ l ˜ s i (0 , z ′′ ) , ≤ l ≤ N, ≤ j ≤ r. (5.20)Then we note, for 0 ≤ l ≤ N , 1 ≤ j ≤ r , z = (0 , z ′′ ) ∈ Z and f ∈ A (Ω), thatˆΦ z J ( f )( z )( ∂ l s j (0 , z ′′ )) = ˆΦ z ( l X p =0 ∂ l − p f ( z ) ∂ p s j (0 , z ′′ )) = J ( f )( z )( ˆΦ z ( ∂ l s j (0 , z ′′ )) . Thus, the above extension ( 5.20 ) intertwines the module action ( 5.1 ) on the sections of J k E and J k ˜ E over Z . From now on, in the rest of the proof, we denote this extension by ˆΦ. We alsonote that ˆΦ satisfies the equation ( 5.13 ).Let us now work with frames s and ˜ s of E and ˜ E , respectively, normalized at origin, and D (0 , z ′′ ) := (( ˆΦ ij (0 , z ′′ ))) Ni,j =0 be the matrix of ˆΦ with respect to the frames { s , ∂ s (0 , z ′′ ) ,. . . , ∂ N s (0 , z ′′ ) } and { ˜ s , ∂ ˜ s (0 , z ′′ ) , . . . , ∂ N ˜ s (0 , z ′′ ) } . Then we wish to show that D is a constantmatrix on Z .We point out that the (1 , D (0 , z ′′ ), namely, ˆΦ (0 , z ′′ ) is the matrix represen-tation of Φ : E | res Z → ˜ E | res Z with respect to the above frames, s and ˜ s , and hence ˆΦ (0 , z ′′ )is holomorphic on Z . So recalling the proof of Claim 2 in Theorem 5 . (0 , z ′′ ) is a constant unitary matrix, say, ˆΦ on Z .We also note, from the construction of ˆΦ above, that ˆΦ( J l E | res Z ) ⊂ J l ˜ E | res Z , for 0 ≤ l ≤ k .Consequently, D (0 , z ′′ ) is a lower triangular matrix. Moreover, since ˆΦ commutes with themodule action on the sections of J k E | res Z and J k ˜ E | res Z , the same proof as in Theorem 5 . l (0 , z ′′ ) = 0 on Z for 0 < l ≤ N . In order to show this, we first need to establish that ˆΦ l (0 , z ′′ ) is a holomorphic function on Z so that we can invoke the argument used in theTheorem 5 .
10 to show ˆΦ l (0 , z ′′ ) = 0 on Z using the Lemma 5 . . (0 , z ′′ ) from the equation ( 5.13 ) by equating 10-th block and we haveˆΦ (0 , z ′′ ) = ( ∂ H (0 , z ′′ ) ˆΦ − ˆΦ ∂ ˜ H (0 , z ′′ )) ˜ H − (0 , z ′′ ) . For d + 1 ≤ i ≤ m , differentiating both sides of the above equation with respect to z i we obtain¯ ∂ i ˆΦ (0 , z ′′ ) = ¯ ∂ i [ ∂ H (0 , z ′′ ) ˆΦ ˜ H − (0 , z ′′ )] − ˆΦ ¯ ∂ i [ ∂ ˜ H (0 , z ′′ ) ˜ H − (0 , z ′′ )]= ¯ ∂ i [ ∂ H (0 , z ′′ ) H − (0 , z ′′ )] ˆΦ − ˆΦ ¯ ∂ i [ ∂ ˜ H (0 , z ′′ ) ˜ H − (0 , z ′′ )]where the second inequality holds as ˆΦ is a constant unitary matrix satisfying the equation( 5.13 ). Then using condition (iii) with l = 1 we have ¯ ∂ i ˆΦ (0 , z ′′ ) = 0 on Z , for d + 1 ≤ i ≤ m ,which completes the proof of the base case. Now let ¯ ∂ i ˆΦ j (0 , z ′′ ) = 0 on Z , for 0 < j ≤ l , d + 1 ≤ i ≤ m and for some 1 ≤ l ≤ N . Since ˆΦ j is holomorphic on Z , for 0 ≤ j ≤ l , recallingthe equation ( 5.14 ), and using Lemma 5 . . . l + 1)-th row of D contains only two non-zero blocks, namely, ˆΦ l +10 and ˆΦ l +1 l +1 . Therefore, from the equation( 5.13 ) by equating l + 10-th block we haveˆΦ l +10 (0 , z ′′ ) = ( ∂ l +1 H (0 , z ′′ ) ˆΦ l +1 l +1 (0 , z ′′ ) − ˆΦ l +1 l +1 (0 , z ′′ ) ∂ l +1 ˜ H (0 , z ′′ )) ˜ H − (0 , z ′′ ) . Now as before, for d + 1 ≤ i ≤ m , applying the differential operator ¯ ∂ i both sides of the equationabove and recalling that ˆΦ l +1 l +1 (0 , z ′′ ) = ˆΦ , for (0 , z ′′ ) ∈ Z , we get¯ ∂ i ˆΦ l +10 (0 , z ′′ ) = ¯ ∂ i [ ∂ l +1 H (0 , z ′′ ) ˆΦ ˜ H − (0 , z ′′ )] − ˆΦ ¯ ∂ i [ ∂ l +1 ˜ H (0 , z ′′ ) ˜ H − (0 , z ′′ )]= ¯ ∂ i [ ∂ l +1 H (0 , z ′′ ) H − (0 , z ′′ )] ˆΦ − ˆΦ ¯ ∂ i [ ∂ l +1 ˜ H (0 , z ′′ ) ˜ H − (0 , z ′′ )]Then by condition (iii) we conclude that ¯ ∂ i ˆΦ l +10 (0 , z ′′ ) = 0 on Z for d + 1 ≤ i ≤ m . So bymathematical induction ˆΦ l (0 , z ′′ ) is holomorphic on Z for 0 ≤ l ≤ N . Now, as in the proof ofTheorem 5 .
10, Proposition 5 . . (cid:3) Remark 5.15.
From the proof of Theorem 5 .
13, it is clear that, for r = 1 and d = k = 2,condition (i), (ii) and (iii) of Theorem 5 .
13 together yield that the curvatures of the bundles E | res Z and ˜ E | res Z are equal. Further, the matrix in ( 5.4 ) turns out to be the diagonal matrix ψ I with respect to a normalized frame at origin where I is the identity matrix of order 3.Moreover, following the proof of Claim 2 in Theorem 5 . ψ is a constant functionon Z with | ψ | = 1. Thus, Theorem 5 .
13 is exact generalization of Lemma 5 . i ) , ( ii ) and ( iii ) listed in the theorem above correspondto the condition that the metric of E and ˜ E are equivalent to order k , in the sense of the paper[11], on Z while the codimension of Z is 1. Consequently, following [11, Remark 6.1], we seethat the condition ( iii ) in the above theorem corresponds to equality of the second fundamentalforms for the inclusion E | res Z ⊂ J E | res Z and ˜ E | res Z ⊂ J ˜ E | res Z , for k = 2.6. Examples and Application
For λ ≥
0, let H ( λ ) be the Hilbert space of holomorphic functions on D with the reproducingkernel K ( λ ) ( z, w ) = (1 − zw ) − λ for z, w ∈ D . It is then evident that the set { e ( λ ) n ( z ) := c − n z n : N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 31 n ≥ } forms a complete orthonormal set in H ( λ ) where c n are the n -th coefficient of the powerseries expansion of (1 − | z | ) − λ , in other words, c n = (cid:18) − λn (cid:19) = λ ( λ + 1)( λ + 2) · · · ( λ + n − n ! . Let us recall that for λ ≥
0, the natural action of polynomial ring C [ z ] on each Hilbert space H ( λ ) makes it into a Hilbert module over C [ z ]. We also point out that, for λ > H ( λ ) becomesa Hilbert module over the disc algebra A ( D ).It is well known that product of two reproducing kernels is also a reproducing kernel [1,8]. So, for α = ( α , . . . , α m ) with α i ≥ i = 1 , . . . , m , let us consider the Hilbert space H ( α ) := H ( α ) ⊗ · · · ⊗ H ( α m ) with the natural choice of complete orthonormal set { e ( α ) i ( z ) ⊗· · · ⊗ e ( α m ) i m ( z ) : i j ≥ , j = 1 , . . . , m } . Then under the identification of the functions z i · · · z i m m on D m := D × · · · × D , H ( α ) naturally possesses an obvious reproducing kernel K ( α ) ( z, w ) := (1 − z w ) − α · · · (1 − z m w m ) − α m on D m . Furthermore, the natural action of C [ z ] on H ( α ) makes it a Hilbert module over C [ z ],for α i ≥ i = 1 , . . . , m , where by C [ z ] we mean C [ z , . . . , z m ].Let us now consider the subspace H ( α )0 consisting of holomorphic functions in H ( α ) whichvanish to order 2 along the diagonal ∆ := { ( z , . . . , z m ) ∈ D m : z = · · · = z m } , that is, followingthe definition given in Section 3, H ( α )0 = { f ∈ H ( α ) : f = ∂ f = · · · = ∂ m f = 0 on ∆ } . We are now interested in describing the quotient space H ( α,β,γ ) q := H ( α,β,γ ) ⊖ H ( α,β,γ )0 in caseof m = 3, where α, β, γ ≥ H ( α,β,γ ) q with the helpof which we find the desired expression of the reproducing kernel K ( α,β,γ ) q of H ( α,β,γ ) q obtainedfrom the Theorem 4 .
5. We then present an application of the Theorem 5 .
13 showing that theunitary equivalence classes of weighted Hardy modules are precisely determined by those of thequotient modules obtained from the submodules of functions vanishing to order at least 2 alongthe diagonal set ∆ := { ( z , z , z ) ∈ D : z = z = z } .6.1. Examples.
We start with the submodule H ( α,β,γ )0 which is the closure of the ideal I := < ( z − z ) , ( z − z )( z − z ) , ( z − z ) > in the Hilbert space H ( α,β,γ ) . It then fol-lows that B := { z i z j z k ( z − z ) , z i z j z k ( z − z )( z − z ) , z i z j z k ( z − z ) : i, j, k ∈ N ∪ { }} isa spanning set for the submodule H ( α,β,γ )0 . So to calculate an orthonormal basis for H ( α,β,γ ) q itis enough to find an orthonormal basis for the orthogonal complement of B . It is easily verifiedthat { g ( p )1 , g ( p )2 , g ( p )3 : p ∈ N ∪ { }} forms a basis for B ⊥ where g ( p )1 := p X l =0 p − l X k =0 z p − l − k z k (cid:13)(cid:13)(cid:13) z p − l − k (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) z k (cid:13)(cid:13) z l (cid:13)(cid:13) z l (cid:13)(cid:13) , g ( p )2 := p X l =0 p − l X k =0 lz p − l − k z k (cid:13)(cid:13)(cid:13) z p − l − k (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) z k (cid:13)(cid:13) z l (cid:13)(cid:13) z l (cid:13)(cid:13) g ( p )3 := p X l =0 p − l X k =0 lz p − l − k z k (cid:13)(cid:13)(cid:13) z p − l − k (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) z k (cid:13)(cid:13) z l (cid:13)(cid:13) z l (cid:13)(cid:13) . Then note that the corresponding orthogonal basis for B ⊥ is { f ( p )1 , f ( p )2 , f ( p )3 : p ∈ N ∪ { }} with f ( p )1 := g ( p )1 , f ( p )2 := D g ( p )1 , g ( p )2 E g ( p )1 − (cid:13)(cid:13)(cid:13) g ( p )1 (cid:13)(cid:13)(cid:13) g ( p )2 , f ( p )3 := D ˜ f ( p )3 , f ( p )2 E f ( p )2 − (cid:13)(cid:13)(cid:13) f ( p )2 (cid:13)(cid:13)(cid:13) ˜ f ( p )3 where ˜ f ( p )3 is orthogonal to g ( p )1 and is given by the following formula˜ f ( p )3 = D g ( p )1 , g ( p )3 E g ( p )1 − (cid:13)(cid:13)(cid:13) g ( p )1 (cid:13)(cid:13)(cid:13) g ( p )3 . Thus, our required orthonormal set of vectors in the quotient module H ( α,β,γ ) q is B := e ( p )1 = f ( p )1 (cid:13)(cid:13)(cid:13) f ( p )1 (cid:13)(cid:13)(cid:13) , e ( p )2 = f ( p )2 (cid:13)(cid:13)(cid:13) f ( p )2 (cid:13)(cid:13)(cid:13) , e ( p )3 = f ( p )3 (cid:13)(cid:13)(cid:13) f ( p )3 (cid:13)(cid:13)(cid:13) ∞ p =0 . Following the Theorem 4 . h h | ∆ := N X l =0 ∂ l h ⊗ ε l | ∆ (6.1)for h ∈ H ( α,β,γ ) where N = k ( k − . In our calculation, for k = 2, it is enough to determinethe action of this map on the orthonormal basis B . In this context the following calculationsprovide us the necessary ingredients to compute the action of this unitary map.We first note that(1 − | z | ) − ( α + β + γ ) = (1 − | z | ) − α (1 − | z | ) − β (1 − | z | ) − γ | ∆ = ∞ X p =0 p X l =0 p − l X k =0 (cid:13)(cid:13)(cid:13) z p − l − k (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) z k (cid:13)(cid:13) (cid:13)(cid:13) z l (cid:13)(cid:13) · | z | p which implies that (cid:13)(cid:13)(cid:13) f ( p )1 (cid:13)(cid:13)(cid:13) is the p -th coefficient of the power series expansion of (1 −| z | ) − ( α + β + γ ) .Thus, we have (cid:13)(cid:13)(cid:13) f ( p )1 (cid:13)(cid:13)(cid:13) = (cid:0) − ( α + β + γ ) p (cid:1) . Further we point out that β (1 − | z | ) − ( α + β + γ +1) = (1 − | z | ) − α (1 − | z | ) − γ dd | z | (1 − | z | ) − β | ∆ = ∞ X p =0 p X l =0 p − l X k =0 (cid:13)(cid:13)(cid:13) z p − l − k (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) z k (cid:13)(cid:13) l (cid:13)(cid:13) z l (cid:13)(cid:13) · | z | p − which, as before, together with a similar calculation for γ (1 − | z | ) − ( α + β + γ +1) ensure that D g ( p )1 , g ( p )2 E = β (cid:18) − ( α + β + γ + 1)( p − (cid:19) and D g ( p )1 , g ( p )3 E = γ (cid:18) − ( α + β + γ + 1)( p − (cid:19) . Now to calculate the inner product of g ( p )2 and g ( p )3 we consider the following power series N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 33 βγ (1 − | z | ) − ( α + β + γ +2) = (1 − | z | ) − α dd | z | (1 − | z | ) − β dd | z | (1 − | z | ) − γ | ∆ = ∞ X p =0 p X l =0 p − l X k =0 k ( p − l − k ) (cid:13)(cid:13)(cid:13) z p − l − k (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) z k (cid:13)(cid:13) (cid:13)(cid:13) z l (cid:13)(cid:13) · | z | p − which shows that D g ( p )2 , g ( p )3 E = βγ (cid:0) − ( α + β + γ +2)( p − (cid:1) . Furthermore, we have β (1 + β | z | )(1 − | z | ) − ( α + β + γ +2) = (1 − | z | ) − α (1 − | z | ) − γ (cid:18) dd | z | ( | z | dd | z | (1 − | z | ) − β ) (cid:19) | ∆ = ∞ X p =0 p X l =0 p − l X k =0 (cid:13)(cid:13)(cid:13) z p − l − k (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) z k (cid:13)(cid:13) l (cid:13)(cid:13) z l (cid:13)(cid:13) · | z | p − and hence with the help of a similar calculation we find that (cid:13)(cid:13)(cid:13) g ( p )2 (cid:13)(cid:13)(cid:13) = β (cid:18)(cid:18) − ( α + β + γ + 2)( p − (cid:19) + β (cid:18) − ( α + β + γ + 2)( p − (cid:19)(cid:19) and (cid:13)(cid:13)(cid:13) g ( p )3 (cid:13)(cid:13)(cid:13) = γ (cid:18)(cid:18) − ( α + β + γ + 2)( p − (cid:19) + γ (cid:18) − ( α + β + γ + 2)( p − (cid:19)(cid:19) . Thus, the above calculations lead us to compute the norms of the vectors { f ( p )1 , f ( p )2 , f ( p )3 : p ∈ N ∪ { }} . We have already showed that (cid:13)(cid:13)(cid:13) f ( p )1 (cid:13)(cid:13)(cid:13) is (cid:0) − ( α + β + γ ) p (cid:1) and recalling the definitionof f ( p )2 and f ( p )3 it is easily seen that (cid:13)(cid:13)(cid:13) f ( p )2 (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) g ( p )1 (cid:13)(cid:13)(cid:13) (cid:18)(cid:13)(cid:13)(cid:13) g ( p )1 (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) g ( p )2 (cid:13)(cid:13)(cid:13) − | D g ( p )1 , g ( p )2 E | (cid:19) = β ( α + γ )( α + β + γ ) (cid:18) − ( α + β + γ ) p (cid:19) (cid:18) − ( α + β + γ + 2)( p − (cid:19) and a similarly one can find, from the definition of f ( p )3 , that (cid:13)(cid:13)(cid:13) f ( p )3 (cid:13)(cid:13)(cid:13) = αβ γ ( α + γ )( α + β + γ ) (cid:18) − ( α + β + γ ) p (cid:19) (cid:18) − ( α + β + γ + 2)( p − (cid:19) . We also point out that similar computations as above give rise to following identities: ∂ g ( p )1 = α (cid:18) − ( α + β + γ + 1)( p − (cid:19) , ∂ g ( p )1 = β (cid:18) − ( α + β + γ + 1)( p − (cid:19) ,∂ g ( p )2 = αβ (cid:18) − ( α + β + γ + 2)( p − (cid:19) , ∂ g ( p )2 = β (cid:18)(cid:18) − ( α + β + γ + 2)( p − (cid:19) + β (cid:18) − ( α + β + γ + 2)( p − (cid:19)(cid:19) ,∂ g ( p )3 = αγ (cid:18) − ( α + β + γ + 2)( p − (cid:19) and ∂ g ( p )3 = βγ (cid:18) − ( α + β + γ + 2)( p − (cid:19) . So we are now in position to calculate the orthonormal basis for the quotient module H ( α,β,γ ) q and their derivatives along z and z direction restricted to the diagonal set ∆ which we exactlyrequire to compute the reproducing kernel of the quotient module. Let us begin, from ( 6.1 ),by pointing out that e ( p )1 (cid:0) − ( α + β + γ ) p (cid:1) z p α q pα + β + γ (cid:0) − ( α + β + γ +1)( p − (cid:1) z p − β q pα + β + γ (cid:0) − ( α + β + γ +1)( p − (cid:1) z p − , e ( p )2 αβ √ β ( α + γ ) 1 √ α + β + γ (cid:0) − ( α + β + γ +2)( p − (cid:1) z p − βγ √ β ( α + γ ) 1 √ α + β + γ (cid:0) − ( α + β + γ +2)( p − (cid:1) z p − and e ( p )3 q αγα + γ (cid:0) − ( α + β + γ +2)( p − (cid:1) z p − − q αγα + γ (cid:0) − ( α + β + γ +2)( p − (cid:1) z p − . This allows us to compute the reproducing kernel of the quotient module H ( α,β,γ ) q as follows K q ( z , w ) = ∞ X p =0 e ( p )1 ( z ) · e ( p )1 ( w ) ∗ + e ( p )2 ( z ) · e ( p )2 ( w ) ∗ + e ( p )3 ( z ) · e ( p )3 ( w ) ∗ , z , w ∈ D which is a 3 × K q ( z , z ) ij )) i,j =1 on D as expected. To compute thekernel K q ( z , z ) for z ∈ ∆ we note, for z = ( z , z , z ) in ∆, that K q ( z , z ) = K ( α,β,γ ) ( z , z ) , K q ( z , z ) = ∂ K ( α,β,γ ) ( z , z ) , K q ( z , z ) = ∂ K ( α,β,γ ) ( z , z ) ,K q ( z , z ) = αβα + β + γ dd | z | (cid:16) | z | (1 − | z | ) − ( α + β + γ +1) (cid:17) + (cid:18) αβγ ( α + γ )( α + β + γ ) − αγα + γ (cid:19) (1 − | z | ) − ( α + β + γ +2) = αβ | z | (1 − | z | ) − ( α + β + γ +2) = ¯ ∂ ∂ K ( α,β,γ ) ( z , z ) ,K q ( z , z ) = α α + β + γ dd | z | (cid:16) | z | (1 − | z | ) − ( α + β + γ +1) (cid:17) + (cid:18) α β ( α + γ )( α + β + γ ) + αγα + γ (cid:19) (1 − | z | ) − ( α + β + γ +2) = [ α + α | z | ](1 − | z | ) − ( α + β + γ +2) = ∂ ¯ ∂ K ( α,β,γ ) ( z , z ) , and similar calculations also yield that K q ( z , z ) = ¯ ∂ K ( α,β,γ ) ( z , z ), K q ( z , z ) = ¯ ∂ K ( α,β,γ ) ( z , z ), K q ( z , z ) = ¯ ∂ ∂ K ( α,β,γ ) ( z , z ), and K q ( z , z ) = ∂ ¯ ∂ K ( α,β,γ ) ( z , z ). Thus, we have K q ( z , z ) | ∆ = J K ( α,β,γ ) ( z , z ) | ∆ which verifies the Theorem 4 . N UNITARY INVARIANTS OF QUOTIENT HILBERT MODULES 35
Application.
Let us consider the family of Hilbert modules
Mod ( D m ) := {H ( α ) : α =( α , . . . , α m ) ≥ } over the polydisc D m in C m . In this subsection we prove that for any pairof tuples α = ( α , . . . , α m ) and α ′ = ( α ′ , . . . , α ′ m ), the unitary equivalence of two quotientmodules H ( α ) q and H ( α ′ ) q , obtained from the submodules of functions vanishing of order 2 alongthe diagonal set ∆, implies the equality of the Hilbert modules H ( α ) and H ( α ′ ) . In other words,the restriction of the curvature of the jet bundle J E ( α ) to the diagonal ∆ is a complete unitaryinvariant for the class Mod ( D m ) where the jet bundle J E ( α ) is defined by the global frame { K ( α ) ( ., w ) , ∂ K ( λ ) ( ., w ) , . . . , ∂ m K ( λ ) ( ., w ) } where ∂ j are the differential operators with respectto the variable z j , for j = 1 , . . . , m Theorem 6.1.
For α = ( α , . . . , α n ) and α ′ = ( α ′ , . . . , α ′ n ) with α i , α ′ i ≥ , for all i = 1 , . . . , n ,the quotient modules H ( α ) q and H ( α ′ ) q are unitarily equivalent if and only if α i = α ′ i , for all i = 1 , . . . , n .Proof. The proof of sufficiency is trivial. So we only prove the necessity. Let us begin bypointing out that the diagonal set ∆ in D m can be described as the zero set of the ideal I := < z − z , . . . , z i − z i +1 , . . . , z m − − z m > . Then it is easy to verify that φ : U → C m definedby φ ( z , . . . , z m ) = ( z − z , . . . , z i − z i +1 , . . . , z m − − z m , z m )yields an admissible coordinate system (Definition 3 .
9) around the origin. We choose U smallenough so that φ ( U ) ⊂ D m . A simple calculation then shows that φ − : φ ( U ) → U takes theform φ − ( u , . . . , u m ) = ( m X j =1 u j , . . . , m X j = i u j , . . . , u m − + u m , u m ) . For rest of the proof we pretend U to be D m thanks to the Remark 3 .
11. Now recalling theProposition 3 .
12 it is enough to prove that α i = α ′ i , i = 1 , . . . , m , provided φ ∗ H ( α ) q is uitarilyequivalent to φ ∗ H ( α ′ ) q where φ ∗ H ( α ) q and φ ∗ H ( α ′ ) q are the quotient modules obtained from thesubmodules φ ∗ H ( α )0 and φ ∗ H ( α ′ )0 of the Hilbert modules φ ∗ H ( α ) and φ ∗ H ( α ′ ) , respectively.We now note that φ ∗ H ( α ) and φ ∗ H ( α ′ ) are reproducing kernel Hilbert modules with repro-ducing kernels K ( u ) = m Y i =1 − | m X j =1 u j | − α i and K ′ ( u ) = m Y i =1 − | m X j =1 u j | − α ′ i , respectively, where u = ( u , . . . , u m ) ∈ φ ( U ). We also pint out that the submodules φ ∗ H ( α )0 and φ ∗ H ( α ′ )0 consists of functions in φ ∗ H ( α ) and φ ∗ H ( α ′ ) , respectively, vanishing along thesubmanifold Z := { (0 , . . . , , u m ) : u m ∈ D } ∩ φ ( U ) of order 2.Since φ ∗ H ( α ) q and φ ∗ H ( α ′ ) q are unitarily equivalent recalling the Theorem 5 .
13 we concludethat K| Z = K ′ | Z (6.2)where K and K ′ are the curvature matrix for the vector bundles E and E ′ over φ ( U ) obtainedfrom the Hilbert modules φ ∗ H ( α ) and φ ∗ H ( α ′ ) , respectively. Now we have, by definition, K ( u ) =(( K ij ( u ))) mi,j =1 where K ij ( u ) = ∂ ∂u i ∂u j log K ( u , u ) , for u = ( u , . . . , u m ) ∈ φ ( U ). Thus, for 1 ≤ i ≤ m and u ∈ φ ( U ), K ii ( u ) = ∂ ∂u i ∂u i log K ( u , u ) = i X l =1 α l − | m X j = l u j | − . A similar computation also yields, for i = 1 , . . . , m and u ∈ φ ( U ), that K ′ ii ( u ) = i X l =1 α ′ l − | m X j = l u j | − . We now note that, for u ∈ Z , K ii ( u ) = P il =1 α l (1 − | u m | ) and K ′ ii ( u ) = P il =1 α ′ l (1 − | u m | ) . Thus, by using the equality in equation ( 6.2 ) it is not hard to see that α i = α ′ i , i = 1 , . . . , m . (cid:3) Acknowledgement.
The author thanks to Dr. Shibananda Biswas and Prof. GadadharMisra for their support and valuable suggestions in preparation of this paper.
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Department of Mathematics and Statistics, Indian Institute of Science Education and ResearchKolkata, Mohanpur - 741246, West Bengal, India
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