The block matrix representations for the quasi-projection pairs on Hilbert $C^*$-modules

TL;DR

This paper develops block matrix representations for harmonious quasi-projection pairs on Hilbert $C^*$-modules, providing new insights into their structure and applications in operator theory.

Contribution

It introduces novel block matrix representations for harmonious quasi-projection pairs and related projections on Hilbert $C^*$-modules, expanding understanding of their structure.

Findings

Derived new block matrix forms for quasi-projection pairs

Provided representations for range and null space projections

Explored applications of these representations

Abstract

A quasi-projection pair consists of two operators $P$ and $Q$ acting on a Hilbert $C^*$-module $H$, where $P$ is a projection and $Q$ is an idempotent satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ denotes the adjoint operator of $Q$, and $I$ is the identity operator on $H$. Such a pair is said to be harmonious if both $P(I-Q)$ and $(I-P)Q$ admit polar decompositions. The primary goal of this paper is to present the block matrix representations for a harmonious quasi-projection pair $(P,Q)$ on a Hilbert $C^*$-module, and additionally to derive new block matrix representations for the matched projection, the range projection, and the null space projection of $Q$. Several applications of these newly obtained block matrix representations are also explored.