Hardy Submodules Over The Bidisc And Their Operators

Operator theory has been greatly enriched after the introduction of Hilbert spaces of analytic functions. On one hand, many classic operator theoretical problems can be reformulated and solved via the analytic function spaces. For example, the (vector-valued) Hardy space over the unit disk is the basic model for the Nagy-Foias's dilation theory; on the other hand, people try to introduce the algebra, geometry, topology and other tools into operator theory and interpret it from a new viewpoint.It is well known that the structure of the submodule of the Hardy space over the unit disk can be determined completely by an inner function. However, it is really hard to determine the structure of submodule of the Hardy over the polydisc. This thesis mainly focuses on Hardy submodules over the bidisc and related operator theory.The first chapter will introduce the basic conceptions and main tools and results that will be used in the thesis.The second chapter studies the Beurling submodule of H2(Dn). It proves that if the cross commutators[Rzi*,Rzj]=0 holds for every i≠j, then M is the Buerling submodule, that is, there is an inner functionθin H2(Dn) such that M=θH2(Dn). In particular, in case n=2, we prove that the evaluation operator L(0) is a compact operator on the Beurling type quotient module N= H2(D2)(?)θH2(D2) if and only if 0 is a finite Blaschke product which depends only on the second variable w.In chapter 3, we focus on a special submodule of H2(D2), i.e., inner sequence based submodule, which has a close connection with the classical single variable operator theory. It is proved that the dimension of the defect space is always finite and moreover, we give an estimate for this dimension. Compactness and normality of the compression operator Sz are characterized completely. It gives a necessary and sufficient condition of the compactness of the evaluation operator L(0) and R(0) on the inner sequence based quotient module. We also compute the Hilbert-Schmidt norm.In the Hardy space over the bidisc, inner function no longer plays a central role in the study of submodules. The central role appears to be played by the so-called core operator and there are deep connections between core operator and the (essential) spectrum and index of the Jordan block (Sz,Sw) in two variables. Chapter 4 will do some research on the compression pairs (Sz,Sw). It is showed that if (Sz,Sw) is Fredholm, then the dimension of the defect space (M (?) zM) n (M (?) wM) is finite. We also give an index formula for (Sz, Sw) by the dimension of the eigenspace of the core operator.