On pure contractive semigroups

TL;DR

This paper characterizes the commutant of pure contractive semigroups on Hilbert spaces and establishes dilation results for doubly commuting pure contractive semigroups, providing a comprehensive model for such tuples.

Contribution

It introduces a complete model for doubly commuting isometric semigroups and extends dilation theory to pure contractive semigroups.

Findings

Identified the commutant of pure contractive semigroups.

Proved dilation of doubly commuting pure contractive semigroups to isometric ones.

Developed a complete model for doubly commuting isometric semigroups.

Abstract

We find the commutant of a pure contractive semigroup on a Hilbert space. We demonstrate that any tuple of doubly commuting pure contractive semigroups can be dilated to a tuple of doubly commuting pure isometric semigroups. En route, we obtain a complete model for the tuples of doubly commuting isometric semigroups.