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

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We consider the problem of classification of nonequivalent representations of a scalar operator λ*I* 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 λϵ *2* [2*,* 3]*.*

## Keywords

Hilbert Space Irreducible Representation English Translation Scalar Operator Matrix Algebra
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