Operator theory in the Hardy space over the bidisk (II)

This paper is a continuation of a project of developing a systematic operator theory inH²(D²). A large part of it is devoted to a study ofevaluation operator which is a very useful tool in the theory. A number of elementary properties of the evaluation operator are exhibited, and these properties are used to derive results in other topics such as interpretation of characteristic opertor function inH²(D²), spectral equivalence, compactness, compressions of shift operators, etc., Even though some results reflect the two variable nature ofH²(D²), the goal of this paper is to manifest a close tie between the operator theory inH²(D²) and classical single operator theory. The unilateral shift of a finite multiplicity and the Bergman shift will be used as examples to illustrate some of the results.Research in this paper is partially supported by a grant from the national science foundation DMS 9970932.