SIAM J. MATH. ANAL. Vol. 18, No. 2, March 1987 (C) 1987 Society for Industrial and Applied Mathematics 007 CONVOLUTION OPERATORS IN A FADING MEMORY SPACE: THE CRITICAL CASE* G. S. JORDAN,, OLOF J. STAFFANSt AND ROBERT L. WHEELER Abstract. We study the linear system of convolution equations x(t) x’(t) +/x * x(t) =f(t), (-co, ), where x, f are n-dimensional column vectors and/z is an n x 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 in a fading memory space. Our results include the previously untreated critical case when there may be a finite number of eigenvalues of the Laplace transform/,(z) zI + IS, (z) of on the boundary of the strip of convergence of/2. Our description is given in terms of the Jordan chains at the eigenvalues of the locally analytic matrix-valued function /,(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. Key words, convolution equation, locally analytic matrix function, Smith factorization, null space, range, fading memory AMS(MOS) subject classifications. 45F05, 45Mxx, 34K20 I. Introduction. We study the linear system of convolution equations (1.1) ’x(t) ------ x’(t) + IX * x(t) =f(t), R --- (-, c), where x, f are n-dimensional column vectors and IX is an n n matrix-valued measure which is finite with respect to a suitable weight function. As usual Ix. x denotes the convolution tx * x( t) I/ dix(s)x( s). We describe the null space and the range of the operator in a fading memory space in the critical case when there may be a finite number of eigenvalues of the Laplace transform f(z)= zI+12(z) of (that is, zeros of the determinant of (z)) on the boundary of the strip of convergence of/2. Our descriptions^are in terms of Jordan chains of vectors at the eigenvalues of the Laplace transform L of , and they depend on a Smith factorization theorem which we prove for locally analytic matrix functions. Conditions sufficient to guarantee the existence of local and global Smith factoriza- tions of locally analytic matrix functions are given in 3. These results are central to the subsequent analysis and are the most difficult theorems of the paper. The global Smith factorization theorem, Theorem 3.2, is the matrix analogue of the Ll-quotient theorem for scalar locally analytic functions [7, Thm. 3.4]. Since we are primarily concerned with permitting eigenvalues on the boundary of the strip of convergence of * Received by the editors February 16, 1984, and in revised form November 1, 1985. " Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300. Institute of Mathematics, Helsinki University of Technology, SF-02150 Espoo, Finland. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-4097. The work of this author was partially supported by the National Science Foundation under grants MCS 83-00559 and DMS 8500947. 366