# On Two-Dimensional Model Representations of One Class of Commuting Operators

### Abstract

In the work by V. A. Zolotarev,*Dokl. Akad. Nauk Arm. SSR*,

**63**, No. 3, 136–140 (1976), a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators {

*A*

_{1},

*A*

_{2}} ([

*A*

_{1},

*A*

_{2}] = 0, [

*A*

_{1}

^{∗},

*A*

_{2}] = 0) such that rank (

*A*

_{1})

_{ I }(

*A*

_{2})

_{ I }= 1 (2

*i*(

*A*

_{ k })

_{ I }=

*A*

_{ k }−

*A*

_{ k }

^{∗},

*k*= 1, 2) and the spectrum of each operator

*A*

_{ k },

*k*= 1, 2

*,*is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in

*L*

_{ Ω }

^{2}where the domain

*Ω*= [0,

*a*] × [0,

*b*] is a compact set in ℝ

^{2}bounded by the lines

*x*=

*a*and

*y*=

*b*and a decreasing smooth curve

*L*connecting the points (0

*, b*) and (

*a,*0)

*.*

Published

25.01.2014

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 66, no. 1, Jan. 2014, pp. 108–127, http://umj.imath.kiev.ua/index.php/umj/article/view/2115.

Issue

Section

Research articles