Spectra of periodic operator polynomials with finitely many diagonals

Spectra of polynomials whose coefficients are periodic operators with finitely many nonzero diagonals are studied. Such polynomials appear in the study of infinite chains of damped harmonic oscillators. It is proved that, assuming the spectrum is bounded, it consists of finitely many analytic arcs. Criteria for existence of point spectrum are given. Several important particular cases are indicated.