标题： Hilbert $C^*$-模上的半和谐和和谐准投影对的特征 显示英文标题

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

摘要： 对于Hilbert$C^*$-模$H$上的每个可伴随幂等元$Q$，最近引入了一个特定的投影$m(Q)$，称为$Q$的匹配投影，这是由于从投影到$Q$的所有距离中的最小值的表征。 受$m(Q)$与$Q$之间关系的启发，最近还引入了一个称为准投影对的术语$(P,Q)$，其中$P$是一个在$H$上的投影，满足$Q^*=(2P-I)Q(2P-I)$，其中$Q^*$是幂等算子$Q$的伴随算子，而$I$是在$H$上的恒等算子。 本文旨在对Hilbert$C^*$-模上的半和谐和和谐准投影对进行系统的特征分析，同时提供说明相关特征非平凡性的例子。 显示英文摘要

摘要： 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.