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In this paper we consider the spectrum of the self-adjoint differentialoperator L generated by the differential expression of order n with the m by mperiodic matrix coefficients, where n and m are respectively odd and evenintegers and n>1. We prove that the number of gaps in the spectrum of L isfinite and find explicit estimation in term of coefficients for the number ofthe gaps. Moreover, we find a condition on the norms of the coefficients forwhich the spectrum is real axis. Besides we investigate the bands of thespectrum and prove that most of the real axis is overlapped by m bands.

On the self-adjoint differential operator with the periodic matrix coefficients