Hilbert Multi-disc Mode

This thesis mainly deals with the Hilbert module over the polydisc.In the past decades, the concept of Hilbert module was introduced by R.Douglas and V.Paulsen. The aim is to introduce the algebraic machine to operator theory. In most cases, a Hilbert module H is an action of the polynomial ring C[z₁, · · · , z_d ]on the Hilbert space H induced by a d-tuple of commuting operators (T₁,T₂, · · · ,T_d), that isp· x = p(T₁ · · · , T_d)_x, p ∈ C[z₁· · · ,z_d], x∈H.In the classical operator theory, the Predholm theory is one of the most beautiful theory, and the Predholm index is one of the most important invariants. A natural way to define the Fredholmness of a d-tuple, by using the Koszul complex, is due to J.Taylor. So, how to characterize the Fredholmness of a d-tuple? A part of our research is characterizing the Fredholmness of an isometric pair by the so-called defect operator. The defect operator of a contractive operator comes from the dilation theory of Sz.Nagy and C.Foias, and it is generalized by K.Guo to the multi-variable operator theory by using the reproducing kernel theory [Guo]. It is proved that, the defect operator of a submodule of H²(D^d) carries key information of the submodule. Inspired by this, we define the defect operator of an isometric pair T = (T₁,T₂) byâ–³_T = 1 - T₁T₁^* - T₂T₂^* + T₁T₂T₁^*T₂^*.We find that the Fredholmness of an isometric pair can be deduced from the compactness of the associated defect operator.So, we want to characterize the compactness of the defect operator for some special isometric pair, such as Toeplitz pair with the inner symbols on both the Hardy spaces H²(D) and H²(D²), in the case of H²(D), we find that the compactness of the inner Toeplitz pair is closely related with the classical function theory:For η1,η2 be two inner functions in H²(D), T = (T_η1,T_η2) is an isometric pair on H²(D). Then â–³_T is compact if and only if H^∞[(?)1] ∩ H^∞[(?)2) (?) H^∞ + C. Thisis exact the Axler-Chang-Sarason-Volberg condition for the semicommutator of two Toeplitz operator to be compact.However, in the case of H²(D²), the compactness of the defect operator of an inner Toeplitz pair appears only in the trivial case:For η1.η2 be two inner functions in H²(D²), T = (T_η1,T_η2) is an isometric pair on H²(D²). Then â–³_t is compact if and only if η1 = B₁(z), η2 = B₂(ω) or η1 = B₁(w), η2 = B₂(z), where B_i are finite Blaschke products.The other part of our research focuses on the essential normality of quotient module over the Bidisk. Essentially normal Hilbert modules are very important in the category of Hilbert modules because of BDF-Theory. It is easy to see that, Hardy module H²(D²) and its nonzero submodules are all defined by a pair of isometric operators of infinite multiplicity, so they are not essentially normal. However, R.Douglas and G.Misra showed that some of the quotient modules are essentially normal, and some of them are not. Then R.Douglas asked, when is a quotient module of H²(D^d) essentially normal? We mainly concentrate on the homogenous quotient module of H²(D²). In the second part of this thesis, we completely characterize the essential normality of the homogenous quotient module over the bidisk.