Characterizations of the semi-harmonious and harmonious quasi-projection pairs on Hilbert $C^*$-modules

TL;DR

This paper systematically characterizes semi-harmonious and harmonious quasi-projection pairs on Hilbert $C^*$-modules, expanding understanding of their structure and providing illustrative examples.

Contribution

It offers new systematic characterizations of these quasi-projection pairs and demonstrates their non-trivial properties with examples.

Findings

Characterizations of semi-harmonious pairs established

Characterizations of harmonious pairs established

Examples illustrating non-triviality provided

Abstract

For each adjointable idempotent $Q$ on a Hilbert $C^*$-module $H$, a specific projection $m(Q)$ called the matched projection of $Q$ was introduced recently due to the characterization of the minimum value among all the distances from projections to $Q$. Inspired by the relationship between $m(Q)$ and $Q$, another term called the quasi-projection pair $(P,Q)$ was also introduced recently, where $P$ is a projection on $H$ satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ is the adjoint operator of the idempotent $Q$ and $I$ is the identity operator on $H$. This paper aims to make systematical characterizations of the semi-harmonious and harmonious quasi-projection pairs on Hilbert $C^*$-modules, and meanwhile to provide examples illustrating the non-triviality of the associated characterizations.