NC functions over the nc Grassmannian

TL;DR

This paper extends Voiculescu's non-commutative function framework from the Riemann sphere to Grassmannians and flag manifolds, introducing generalized nc resolvents and exploring spectral analysis of unbounded operators.

Contribution

It formulates nc functions over Grassmannians within existing frameworks and introduces a generalized nc resolvent with spectral analysis implications.

Findings

Formulated nc functions over Grassmannians and flag manifolds.

Introduced a generalized non-commutative resolvent.

Explored spectral properties of unbounded operators in this setting.

Abstract

We explain how to formulate Voiculescu's non-commutative Riemann sphere framework for fully matricial functions \cite{v10} within the theory of nc functions developed by Vinnikov and Kaliuzhnyi-Verbovetskyi \cite{kvv14}. We then extend this framework from the Riemann sphere to Grassmannians (and flag manifolds). Moreover, as an example of nc functions in this setting, we introduce a generalization of Voiculescu's non-commutative resolvent on the Riemann sphere, study a corresponding generalization of the resolvent equation, and discuss aspects of the spectral analysis of unbounded operators in Voiculescu's framework \cite{v10}.