Conditions of Ф-Factorability in the Space L_p. Criterion of Ф-Factorability in L₂ of Bounded Measurable Matrix Functions

Abstract

In this comparatively small chapter we shall be concerned with classes of matrix functions from L_∞ defined on a contour of class R and Ф-factorable in the spaces L_p (Γ), where we pursue the aim to select such classes which are as wide as possible. For the construction of classes of matrix functions from the classes studies previously in Chapters 5, 6 and 7 which admit a Ф-factorization we shall use the ε-local principle (Theorem 3.22) and the properties of a matrix function F,ϵ L_∞ to preserve Ф-factorability on multiplication by a matrix function from the classes ${\text{L}}_\infty ^ \pm + {\text{C}}$ (Theorem 5.5). As a matter of fact, this general approach forced us from the very start to give up any attempts to obtain, simultaneously with conditions of Ф-factorability, any conclusions about the values of partial indices. The conditions of Ф-factorability in L_p derived below are sufficient. However, for p = 2 in case of a Lyapunov contour Γ, we are also able to substantiate their necessity. A criterion of Ф-factorability of a matrix function G from L_∞ in the spaces L_p with p ≠ 2 fails to be known hitherto.