Subnormal operators of finite type II. Structure theorems.

Yakubovich, 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 -