Regular Dilation on Semigroups

Abstract

Dilation theory originated from Sz.Nagy's celebrated dilation theorem which states that every contractive operator has an isometric dilation. Regular dilation is one of many fruitful directions that aims to generalize Sz.Nagy's dilation theorem to the multi-variate setting. First studied by Brehmer in 1961, regular dilation has since been generalized to many other contexts in recent years.

This thesis is a compilation of my recent study of regular dilation on various semigroups. We start from studying regular dilation on lattice ordered semigroups and shows that contractive Nica-covariant representations are regular. Then, we consider the connection between regular dilation on graph products of N, which uni es Brehmer's dilation theorem and the well-known Frazho-Bunce-Popescu's dilation theorem. Finally, we consider regular dilation on right LCM semigroups and study its connection to Nica-covariant dilation.