On $q$-commuting co-extensions and $q$-commutant lifting

TL;DR

This paper extends existing results on $q$-commuting dilations and $q$-commutant lifting for operator pairs to a broader class of scalars $q$ with $|q| \,\leq\, 1/\|T\|$, beyond the previously studied case $|q|=1$.

Contribution

The authors generalize $q$-commutant lifting results from the case $|q|=1$ to all scalars with $|q|\leq 1/\|T\|$, broadening the applicability of these operator theory results.

Findings

Extended $q$-commutant lifting to a larger class of scalars $q$ with $|q|\leq 1/\|T\|$.

Improved existing dilation results for $q$-commuting pairs.

Provided new bounds and conditions for $q$-commuting dilations.

Abstract

Consider a nonzero contraction $T$ and a bounded operator $X$ satisfying $TX=qXT$ for a complex number $q$. There are some interesting results in the literature on $q$-commuting dilation and $q$-commutant lifting of such pair $(T,X)$ when $|q|=1$. Here we improve a few of them to the class of scalars $q$ satisfying $|q|\leq \dfrac{1}{\|T\|}$.