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 (2i(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 ℝ² bounded by the lines x = a and y = b and a decreasing smooth curve L connecting the points (0, b) and (a, 0).