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By the method of similar operators, the spectral properties of a fourth order differential operator are studied under periodic or semiperiodic boundary conditions. The spectrum asymptotics is obtained, together with some estimates for the spectral resolution for the operator in question. Also, the operator semigroup is constructed whose generator is equal to minus the operator under study.2010 Mathematics Subject Classification. Primary 34L20. Key words and phrases. Spectrum of an operator, fourth order differential operator, spectrum asymptotics, equiconvergence of spectral resolutions, method of similar operators.Supported by RFBR (grants 14-01-31196 and 15-31-20241) and by RSF (grant 14-21-00066; Section 4) for investigations done at the Voronezh State University.

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