PhD Thesis Defence

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 multi-variate setting. First studied by Brehmer in 1961, regular dilation has since been generalized to many other contexts in recent years.

In this talk, I will give a brief overview of some recent development of regular dilation on various classes of semigroups. We start by examining the case of regular dilation on lattice ordered semigroups and answer an open question on whether contractive Nica-covariant representations are regular. We then consider a more general case on graph products of N, where regular dilation unifies two seemingly unrelated results in dilation theory. This allows us to further extend regular dilation to more general classes of semigroups, where we study the relation between regular dilation and Nica-covariant representations.