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

TL;DR

This paper extends classical lifting theorems to the setting of $Q$-commuting and $Q$-intertwining operators, providing multiple proofs, explicit dilations, and connections to graph theory.

Contribution

It introduces new $Q$-commuting and $Q$-intertwining lifting theorems with explicit constructions and links to graph theory, expanding classical operator theory.

Findings

Multiple proofs of $Q$-lifting theorems provided

Explicit Ando-type dilations constructed for $Q$-commuting contractions

Connections established between operator theory and graph theory

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.