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]*.*

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Correspondence to V. I. Rabanovych.

## Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 5, pp. 701–716, May, 2015.

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Rabanovych, V.I. On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum.
*Ukr Math J* **67, **795–813 (2015). https://doi.org/10.1007/s11253-015-1115-z

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### Keywords

- Hilbert Space
- Irreducible Representation
- English Translation
- Scalar Operator
- Matrix Algebra