Eigenvalues Of Core Operator On Homogeneous Submodules

The Hardy space H2(D2)can be viewed as a module over the polynomial ring C[z,w]with module action defined by multiplication of functions.In classical Hardy space H2(D)(which is a module over C[z]),Beurling’s famous theorem on shift operators states that to every invariant subspace S of H2(D)corresponds an inner function θ(z)such that S = θ(z)H2(D).However Beurling’s theorem has no direct generalization to H2(D2)and the function theoretic approach for characterizing submodules in H2(D2)is nearly impossible.The structure of these submodules is quite complicated and requires a deeper study.An alternative operator theoretic approach has been proven to be successful in past two decades.The key ingredient for this approach is the so-called "core operator" which is a bounded self-adjoint integral operator defined on submodules of H2(D2)and it gives rise to some interesting numerical invariants for the submodules.These invariants are difficult to compute or estimate in general.In this thesis we compute these invariants for homogenous submodules through Toeplitz determinants.In chapter 1,Some basic concepts and backgrounds on Hardy space over the bidisk are mentioned and a brief review of research developments on characteriz-ing the submodules of H2(D2)is given.In chapter 2,Examples of important submoudules of H2(D2)are given.These examples make the complexity of the structure of submodules of H2(D2)more clearer.In chapter 3,Operator pairs(Rz,Ru),(Sz,Sw)and fringe operator are in-troduced.Core operator and numerical invariants for submodules are briefly explained.A number of known results are given which are essential for a clear idea on research and we will refer to some of them in Chapter 4.Chapter 4 includes the main part of this thesis.In this chapter several new results are proven:In section 4.1,employing Toeplitz determinants,a new formula for orthonormal bases of the defect spaces Me(?)zM and M(?)wM for homogenous submodules M =[p]is given.Using these formulas the numerical invariants ∑0 and ∑1 are computed for M =[p].In section 4.2,all eigenvalues of core operator for homogenous submodules M =[p]are computed.In section 4.3,two estimations for the second largest eigenvalue of core operator for M =[p]are given,one of which is based on particular Toeplitz determinants and the other is solely dependent on the coefficients of polynomial p.The Main goal of section 4.4 is to do more studies on related Toeplitz determi-nants.A relation of these determinants with the Mahler measure of the complex polynomial p(z,1)is shown where M =[p].