Subnormal operators of finite type II: Structure theorems
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Autores: Dmitry Yakubovich
- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 14, Nº 3, 1998, págs. 623-681
- Idioma: inglés
- DOI: 10.4171/rmi/247
- Títulos paralelos:
- Operadores sub-normales de tipo finito II: Teoremas de estructura.
- Enlaces
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Resumen
This paper concerns pure subnormal operators with finite rank self-commutator, which we call subnormal operators of finite type. We analyze Xia's theory of these operators [21]-[23] and give its alternative exposition. Our exposition is based on the explicit use of a certain algebraic curve in C2, which we call the discriminant curve of a subnormal operator, and the approach of dual analytic similarity models of [26]. We give a complete structure result for subnormal operators of finite type, which corrects and strengthens the formulation that Xia gave in [23]. Xia claimed that each subnormal operator of finite type is unitarily equivalent to the operator of multiplication by z on a weighted vector H2-space over a quadrature Riemann surface (with a finite rank perturbation of the norm). We explain how this formulation can be corrected and show that, conversely, every quadrature Riemann surface gives rise to a family of subnormal operators. We prove that this family is parametrized by the so-called characters. As a departing point of our study, we formulate a kind of scattering scheme for normal operators, which includes Xia's model as a particular case.
