Commutant Lifting for Commuting Row Contractions

TL;DR

This paper proves that for commuting row contractions, operators commuting with the original contraction can be dilated to operators commuting with the Arveson dilation, preserving the norm.

Contribution

It establishes a commutant lifting theorem for commuting row contractions and their Arveson dilations, extending classical results to multivariable operator theory.

Findings

Operators commuting with the original contraction dilate to commuting operators with the dilation.

The dilation preserves the operator norm.

The result applies to multivariable operator theory contexts.

Abstract

If $T= \big[ T_1 ... T_n\big]$ is a row contraction with commuting entries, and the Arveson dilation is $\tilde T= \big[ \tilde T_1 ... \tilde T_n\big]$, then any operator $X$ commuting with each $T_i$ dilates to an operator $Z$ of the same norm which commutes with each $\tilde T_i$.