Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

We consider a selfadjoint and smooth enough operator-valued function L(λ) on the segment [a, b]. Let L(a)≪0, L(b)≫0, and there exist two positive numbers ε and δ such that the inequality |(L(λ)f, f)|<ε (ε∈[a, b] {norm of matrix}f{norm of matrix}=1) implies the inequality (L'(λ)f, f)>δ. Then the function L(λ) admits a factorization L(λ)=M(λ)(λI-Z) where M(λ) is a continuous and invertible on [a, b] operator-valued function, and operator Z is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge condition L′(λ)≫0 (λ ∈ [a,b]). For analytic function L(λ) the result of this paper was obtained in [13].