$\mathbb {Z}$-operators related to a finite measure space

Abstract

In this paper we introduce and study a class of linear operators related to a finite measure space for which its $L^2$-Hilbert space is separable. These linear operators are generalizations of pseudo-differential operators on $\mathbb {Z}$. We call these operators $\mathbb {Z}$-operators or generalized pseudo-differential operators on $\mathbb {Z}$. We give some $L^p$-boundedness and compactness results, $1\le p < \infty$, and we also study Hilbert–Schmidt property concerning this class of linear operators. In the end we give some examples of finite measure spaces for which we can apply our results.