The matched projections of idempotents on Hilbert $C^*$-modules

TL;DR

This paper introduces the concept of matched projections for idempotents on Hilbert $C^*$-modules, providing new characterizations, formulas, and estimates that deepen understanding of their structure and relationships.

Contribution

It defines the matched projection of an idempotent and explores its properties, including homotopy and quasi-projection pair relations, offering new tools for operator analysis.

Findings

Matched projection $m(Q)$ is homotopic to the idempotent $Q$.

Formulas for $m(Q)$ are derived and analyzed.

Norm estimations related to $m(Q)$ are established.

Abstract

The aim of this paper is to give new characterizations of some fundamental issues about idempotents. In the general setting of adjointable operators on Hilbert $C^*$-modules, a new term of quasi-projection pair is introduced. For each idempotent $Q$, a projection $m(Q)$, called the matched projection of $Q$, is constructed. It is shown that $Q$ and $m(Q)$ as idempotents are homotopic, and $\big(m(Q),Q\big)$ is a quasi-projection pair. Some formulas for $m(Q)$ are derived. Based on these formulas, representations and norm estimations associated with $m(Q)$ are dealt with.