Families of unbounded self-adjoint operators connected by non-Lie relations (Q920379)

A structure theorem is given for a commuting family of unbounded selfadjoint operators \(A_ x\), \(x\in X\) satisfying the relations \(A_ xB=BF_ x(A)\), \(x\in X\). Here \(R^ x\ni \lambda (\cdot)\to F(\lambda (\cdot))(\cdot)\) is measurable and \(F_ x(A)=\int F(\lambda (\cdot))(x)dE_ A(\lambda (\cdot)),\) \(E_ A(\cdot)\) being the joint spectral decomposition of the family \(\{A_ x\), \(x\in X\}\). The choice \(F(\lambda (\cdot))(x)=\lambda (x)\) yields the spectral theorem for a family of commuting selfadjoint operators; other choices of F are also discussed.