Convolution Operators in a Fading Memory Space: The Critical Case

Abstract

We study the linear system of convolution equations \[ \mathcal{L}x(t) \equiv x'(t) + \mu *x(t) = f(t),\quad t \in ( - \infty ,\infty ), \]where x, f are n-dimensional column vectors and $\mu $ is an $n \times n$ matrix-valued measure which is finite with respect to a suitable weight function. We describe the null space and the range of the operator $\mathcal{L}$ in a fading memory space. Our results include the previously untreated critical case when there may be a finite number of eigenvalues of the Lap lace lace transform $\hat L(z) = zI + \hat \mu (z)$ of $\mathcal{L}$ on the boundary of the strip of convergence of $\hat \mu $. Our description is given in terms of the Jordan chains at the eigenvalues of the locally analytic matrix-valued function $\hat L(z)$. We prove a new Smith factorization theorem for locally analytic matrix functions. At the eigenvalues on the boundary of the strip of convergence, sufficient conditions for the existence of such a factorization are given in terms of the Banach algebra concept of the order of smoothness of a locally analytic matrix function and the structure of the Smith factorization. The authors have previously developed such Banach algebra methods to analyze scalar locally analytic functions.