# Subnormal operators of finite type II. Structure theorems.

Revista Matemática Iberoamericana (1998)

- Volume: 14, Issue: 3, page 623-681
- ISSN: 0213-2230

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topYakubovich, Dmitry V.. "Subnormal operators of finite type II. Structure theorems.." Revista Matemática Iberoamericana 14.3 (1998): 623-681. <http://eudml.org/doc/39559>.

@article{Yakubovich1998,

abstract = {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.},

author = {Yakubovich, Dmitry V.},

journal = {Revista Matemática Iberoamericana},

keywords = {Operadores lineales; Curvas algebraicas; Operador de rango finito; Espacios de Hilbert; Estructuras proyectivas; quadrature Riemann surface; normal part; subnormal; normal extension; finite rank self-commutator; finite type},

language = {eng},

number = {3},

pages = {623-681},

title = {Subnormal operators of finite type II. Structure theorems.},

url = {http://eudml.org/doc/39559},

volume = {14},

year = {1998},

}

TY - JOUR

AU - Yakubovich, Dmitry V.

TI - Subnormal operators of finite type II. Structure theorems.

JO - Revista Matemática Iberoamericana

PY - 1998

VL - 14

IS - 3

SP - 623

EP - 681

AB - 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.

LA - eng

KW - Operadores lineales; Curvas algebraicas; Operador de rango finito; Espacios de Hilbert; Estructuras proyectivas; quadrature Riemann surface; normal part; subnormal; normal extension; finite rank self-commutator; finite type

UR - http://eudml.org/doc/39559

ER -

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