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 2 Ω 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).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 1, pp. 108–127, January, 2014.
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Hatamleh, R., Zolotarev, V.A. On Two-Dimensional Model Representations of One Class of Commuting Operators. Ukr Math J 66, 122–144 (2014). https://doi.org/10.1007/s11253-014-0916-9
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DOI: https://doi.org/10.1007/s11253-014-0916-9