On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum

We consider the problem of classification of nonequivalent representations of a scalar operator λI in the form of a sum of k self-adjoint operators with at most n1, . . . ,nk points in their spectra, respectively. It is shown that this problem is *-wild for some sets of spectra if (n1, . . . ,nk) coincides with one of the following k -tuples: (2, . . . , 2) for k ≥ 5, (2, 2, 2, 3), (2, 11, 11), (5, 5, 5), or (4, 6, 6). It is demonstrated that, for the operators with points 0 and 1 in the spectra and k ≥ 5, the classification problems are *-wild for every rational λϵ 2 [2, 3].