Factorization of selfadjoint operator polynomials

Factorization theorems ate obtained for selfadjoint operator poly-nomials L(λ):= εⁿ _j=0 λ^j A_j where A₀, A₁,..., A_n are selfadjoint bounded linear operators on a Hilbert space H{small}. The essential hypotheses concern the real spectrum of L(λ) and, in particular, ensure the existence of spectral sub-spaces associated with the real line for the (companion) linearization. Under suitable additional conditions, the main results assert the existence of polynomial factors (a) of degrees [1/2n] and [1/2(n + 1)] when the leading coefficient An is strictly positive and (b) of degree 1/2n (when n is even) when An is in-vertible and the spectrum of L (λ) is real. Consequences for the factorization of regular operator polynomials (when L(α) is invertible for some real α) are also discussed.