Some aspects of semi-harmonious quasi-projection pairs

A term called the quasi-projection pair $(P,Q)$ was introduced recently by the authors, where $P$ is a projection and $Q$ is an idempotent on a Hilbert $C^{\ast }$-module $H$ 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 $H$. Some fundamental issues on quasi-projection pairs, such as the block matrix representations for quasi-projection pairs and the $C^{\ast }$-morphisms associated with quasi-projection pairs, are worthwhile to be investigated. This paper aims to make some preparations. One object called the semi-harmonious quasi-projection pair is introduced in the general setting of the adjointable operators on Hilbert $C^{\ast }$-modules. Some related operator theories on the common similarity of operators and a norm equation associated with the Friedrichs angle are dealt with.