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 λϵ 2 [2, 3].
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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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DOI: https://doi.org/10.1007/s11253-015-1115-z