Spectral Properties of Some Linear Matrix Differential Operators in L ^p-spaces on ${\mathbb{R}}$

Abstract.

A detailed study is made of matrix-valued, ordinary linear differential operators T in $L^{p}({\mathbb{R}},{{\mathbb{C}}^{N}})$ for 1 < p < ∞, which arise as the perturbation of a constant coefficient differential operator of order n ≥ 1 by a lower order differential operator $S = {\sigma_{j=0}^{n--1}} F_{j}(x)(--i\frac{d}{dx})^{j}$ which has a factorisation S = AB for suitable operators A and B. Via techniques from L ^p-harmonic analysis, perturbation theory and local spectral theory, it is shown that T satisfies certain local resolvent estimates, which imply the existence of local functional calculi and decomposability properties of T.