The Powers Sum of spatial CPD-semigroups and CP-semigroups

TL;DR

This paper introduces the concept of spatial CPD-semigroups, constructs their Powers sum, and demonstrates that the product system of the sum equals the product of the factors' systems, generalizing previous results.

Contribution

It defines spatial CPD-semigroups, constructs their Powers sum for general cases, and unifies existing results under a broader framework.

Findings

The product system of the Powers sum equals the product of the factors' systems.

The constructions coincide on the domain of intersection for certain CP-semigroups.

The work generalizes all known results about Powers sums of CP-semigroups.

Abstract

We define spatial CPD-semigroup and construct their Powers sum. We construct the Powers sum for general spatial CP-semigroups. In both cases, we show that the product system of that Powers sum is the product of the spatial product systems of its factors. We show that on the domain of intersection, pointwise bounded CPD-semigroups on the one side and Schur CP-semigroups on the other, the constructions coincide. This summarizes all known results about Powers sums and generalizes them considerably.