Xia's analytic model of a subnormal operator and its applications
The present paper gives a nice exposition on subnormal operators, presenting in an attractive form the analytic model of \textit{D.--X.\,Xia} [Integral Equations Oper.\ Theory 10, 258--289 (1987; Zbl 0645.47020)] and its applications. New proofs of some results are given and many examples to help clarify the model are provided. Each section is concluded with a paragraph of notes and open problems, which gives some additional background to the topics covered in the section and some related open problems. One of the basic results in operator theory is the spectral theorem for normal operators. Characterizations of such a type have been given for the theory of Hilbert space contractions. In the second section of the paper, the authors review the effort made towards this end for subnormal operators. Xia's analytic model is presented and some of the complete unitary invariants for pure subnormal operators are described. The third main section of the paper explores these unitary invariants when the self-commutator is of finite rank. A~study of the one-dimensional and the two-dimensional case is made, and a classification of subnormal operators of finite type is presented.
scientific article; zbMATH DE number 5541248
scientific article