Characterizations of the semi-harmonious and harmonious quasi-projection pairs on Hilbert $C^{\ast }$-modules

Abstract

For each adjointable idempotent $Q$ on a Hilbert $C^{\ast }$-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 $Q^{\ast }=(2P-I)Q(2P-I)$, in which $Q^{\ast }$ 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 $C^{\ast }$-modules, and meanwhile to provide examples illustrating the non-triviality of the associated characterizations.

The current version is accepted for publication in Quaestiones Mathematicae. The proofs of Lemmas 3.1--3.3, originally given in Reference [12] (arXiv:2404.12826v2), have been moved to this manuscript. Consequently, a revision of arXiv:2404.12826v2 will be addressed later