On a certain class of commuting systems of linear operators (Q366098)

Let \(\Omega\subset{\mathbb R}^{2}_{+}\) be a compact set, \(a=\sup\{ x: (x,y)\in\Omega\}\), \(b=\sup\{ y: (x,y)\in\Omega\}\). Consider the system of linear bounded operators on the space \(L^{2}(\Omega)\) defined as follows: \[ (\widetilde A_{1}f)(x,y)=\chi_{\Omega}(x,y)i \int_{x}^{a}f(t,y)\,dt, \quad (\widetilde A_{2}f)(x,y)=\chi_{\Omega}(x,y)i \int_{y}^{b}f(x,s)\,ds, \] where it is assumed that the function \(f\) is extended by zero outside \(\Omega\), and \(\chi_{\Omega}\) is the characteristic function of the domain \(\Omega\). \textit{M. S. Livshits} [Mat. Sb., N. Ser. 19(61), 239--262 (1946; Zbl 0061.25903)] posed the problem to describe the class of systems of linear operators \(\{ A_{1}, A_{2}\}\) on a Hilbert space \(H\) which are unitarily equivalent to the system \(\{\widetilde A_{1},\widetilde A_{2}\}\). In the paper under review, this problem is solved for the case of the domain \(\Omega\) bounded by the lines \(x=a\), \(y=b\) and by a decreasing curve \(y=p(x)\), where \(p\in C^{(1)}(a,b)\), \(p(0)=b\), \(p(a)=0\).