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

• Published in: Ukrainian mathematical journal Vol. 66; no. 1; pp. 122 - 144
• Main Authors: Hatamleh, R., Zolotarev, V. A.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.06.2014; Springer
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Statistics; Simple Spectrum; Characteristic Function; Commute Operator; Complete Orthonormal System; Hilbert Space
• ISSN: 0041-5995; 1573-9376
• DOI: 10.1007/s11253-014-0916-9

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) .