Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)


Mishra, Gadadhar (TSMU-Bangalore; ISI)

Abstract (Summary of the Work)

One of the basic problem in the study of a Hilbert module H over the ring of polynomials C[z] := C[z1, . . . , zm] is to find unitary invariants (cf. [15,7]) for H. It is not always possible to find invariants that are complete and yet easy to compute. There are very few instances where a set of complete invariants have been identified. Examples are Hilbert modules over continuous functions (spectral theory of normal operator), contractive modules over the disc algebra (model theory for contractive operator) and Hilbert modules in the class Bn for a bounded domain C m (adjoint of multiplication operators on reproducing kernel Hilbert spaces). In this thesis, we study Hilbert modules consisting of holomorphic functions on some bounded domain possessing a reproducing kernel. Our methods apply, in particular, to submodules of Hilbert modules in B1.Another important aspect of operator theory starts from the work of Beurling [4]. Beurlings theorem describing the invariant subspaces of the multiplication (by the coordinate function) operator on the Hardy space of the unit disc is essential to the Sz.-Nagy Foias model theory and several other developments in modern operator theory. In the language of Hilbert modules, Beurlings theorem says that all submodules of the Hardy module of the unit disc are equivalent (in particular, equivalent to the Hardy module). This observation, due to Cowen and Douglas [9], is peculiar to the case of one-variable operator theory. The submodule of functions vanishing at the origin of the Hardy module H2 0 (D 2 ) of the bi-disc is not equivalent to the Hardy module H2 (D 2 ). To see this, it is enough to note that the joint kernel of the adjoint of the multiplication by the two co-ordinate functions on the Hardy module of the bi-disc is 1 - dimensional (it is spanned by the constant function 1) while the joint kernel of these operators restricted to the submodule is 2 - dimensional (it is spanned by the two functions z1 and z2).There has been a systematic study of this phenomenon in the recent past [1, 16] resulting in a number of Rigidity theorems for submodules of a Hilbert module M over the polynomial ring C[z] of the form [I] obtained by taking the norm closure of a polynomial ideal I in the Hilbert module. For a large class of polynomial ideals, these theorems often take the form: two submodules [I] and [J ] in some Hilbert module M are equivalent if and only if the two ideals I and J are equal. More generallyTheorem 0.1. Let I, Ie be any two polynomial ideals and M, Mf be two Hilbert modules of the form [I] and [Ie] respectively. Assume that M, Mf are in B1(Ω) and that the dimension of the zero set of these modules is at most m − 2. Also, assume that every algebraic component of zerosets intersects Ω. If M and Mf are equivalent, then I = Ie.We give a short proof of this theorem using the sheaf theoretic model developed in this thesis and construct tractable invariants for Hilbert modules over C[z].Let M be a Hilbert module of holomorphic functions on a bounded open connected subset Ω of C m possessing a reproducing kernel K. Assume that I ⊆ C[z] is the singly generated ideal hpi. Then the reproducing kernel K[I] of [I] vanishes on the zero set V (I) and the map w 7→ K[I] (·, w) defines a holomorphic Hermitian line bundle on the open set Ω∗ I = {w ∈ C m : ¯w ∈ Ω \\ V (I)} which naturally extends to all of Ω∗ . As is well known, the curvature of this line bundle completely determines the equivalence class of the Hilbert module [I]. However, if I ⊆ C[z] is not a principal ideal, then the corresponding line bundle defined on Ω∗ I no longer extends to all of Ω∗ . For example, H2 0 (D 2 ) is in the Cowen-Douglas class B1(D 2 \\ {(0, 0)}) but it does not belong to B1(D 2 ). Indeed, it was conjectured in [14] that the dimension of the joint kernel of the Hilbert module [I] at w is 1 for points w not in V (I), otherwise it is the codimension of V (I). Assuming that(a) I is a principal ideal or(b) w is a smooth point of V (I),Duan and Guo verify the validity of this conjecture in [17]. Furthermore, they show that if m = 2 and I is prime then the conjecture is valid.To systematically study examples of submodules like H2 0 (D 2 ), or more generally a submodule [I] of a Hilbert module M in the Cowen-Douglas class B1(Ω), we make the following definition (cf. [6]).Definition 0.2. Fix a bounded domain Ω ⊆ C m. A Hilbert module M ⊆ O(Ω) over the polynomial ring C[z] is said to be in the class B1(Ω) if(rk) it possess a reproducing kernel K (we don’t rule out the possibility: K(w, w) = 0 for w in some closed subset X of Ω) and(fin) the dimension of M/M is finite for all. For a Hilbert modules M in B1 we have proved the following Lemma. Lemma 0.3. Suppose M is a Hilbert modules in B1 which is of the form [I] for some polynomial ideal I. Then M is in B1if the ideal I is singly generated while if the cardinality of the minimal set of generators is not 1, then M is in B1.


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