The Theory of Infinitely Divisible Matrices and Kernels

Abstract

Introduction and notation. Kernels which generate nonnegative definite or semi-definite quadratic forms play an important role in many branches of mathematics, but general kernels whose fractional powers all have this same property have only recently been studied. A special family of such kernels has long been of importance in probability theory, and it is this family which inspires the name infinitely divisible kernels for the general class. It was work of C. Loewner concerning semigroups of conformal mappings which first drew attention to these kernels, and interest in them could only increase when he showed recently that the Green's function for Laplace's equation is, under certain conditions, an infinitely divisible kernel. In this paper we shall develop a general theory of infinitely divisible kernels and indicate briefly a few of its applications. It is convenient to begin by examining the discrete analogues of the infinitely divisible kernels, i.e., infinitely divisible matrices. In §1 we characterize this class of matrices and show that the occurrence of zeros among their entries is subject to a certain condition which is most easily stated in the language of graph theory. We then use the matrical theory to obtain corresponding results for continuous infinitely divisible kernels, and we develop the technical tools for dealing with these kernels which are needed later in the applications. We shall use Rm and Cm to denote w-dimensional real and complex space, respectively, with R = R\ C=C\ and R+ = {x e R \ x>0}. If S^Rm is a set with