$$\mathbb {Z}$$ 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.