Lifting $Q$-commuting and $Q$-intertwining operators

Abstract

There are several proofs of the classical commutant lifting and intertwining lifting theorems in the literature. In this article, we present analogous proofs to a few $Q$-commuting lifting and $Q$-intertwining lifting theorems. We provide several proofs and show explicit constructions of Ando-type dilations for a pair of $Q$-commuting contractions $(T_1,T_2)$ when $Q$ is a bounded operator. We establish a few connections of these results with graph theory.

Contains 52 pages. This is just a first draft and will be revised soon. This is a part of the PhD thesis of the second named author submitted to IIT Bombay on May, 2022