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

Abstract

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 $n_1 , ...,n_k$ points in their spectra, respectively. It is shown that this problem is *-wild for some sets of spectra if $(n_1 , ...,n_k)$ 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 $λ ϵ 2 [2, 3]$.