Convolution on locally compact spaces

Abstract

In this paper a convolution similar to that defined by Ionescu TulceaSimon [3] is developed by means of an extension of the underlying convolution of functions to convolution of measures. The hypotheses under which the extension obtains are relatively weak and it is to be noted that associativity is not assumed (an elementary non-associative example is given in Section 2). Under various hypotheses additional to those assumed in Section 2 one may obtain (using other results in this paper) generalized translation operators in the sense of Levitan [4] and several Banach algebras. An interesting example where solutions to partial differential equations are used to generate the underlying convolution is given by Povzner [6]. The general thrust of this work is towards the spectral representation of certain families of operators. In particular a left-convolution (see [2]) is obtained in Section 4. This may be used in the construction of locally compact spaces on which the integrations are performed (see [3] p. 1765 and [5] p. 6). The remaining sections of this paper are as follows: Section 2. AnLl-type convolution, Section 3. Convolution of measures, Section 4. Translation of functions and measures, and Section 5. The compactness condition (iii) of Section 2. The main results are Theorems 1 and 2, and for applications, the corollaries of Theorem 2. The author gratefully acknowledges his indebtedness to C. Ionescu Tulcea for many fruitful discussions of material in this paper.