On Two-Dimensional Model Representations of One Class of Commuting Operators - Hatamleh, R.; Zolotarev, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2014)
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).