Principal part of resolvent and factorization of an increasing nonanalytic operator-function

Let a selfadjoint operator-valued function L(λ) be given on the interval [a,b] such that L(a)≪0, L(b)≫0, L′(λ)≫0 (a≤λ≤b), and L″(λ) has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued function L(λ) can be reduced to the spectral theory of one operator Z, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L(λ)=M(λ)(λI-Z), where the operator-valued function M(λ) is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition of L^-1(λ) into the sume of its principal and regular parts.