Journal of Operator Theory

Volume 94, Issue 1, Summer 2025 pp. 219-251.

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

Authors: Xiaoyi Tian (1), Qingxiang Xu (2), Chunhong Fu (3)
Author institution: (1) Department of Mathematics, Shanghai Normal University, Shanghai 200234, P.R. China
(2) Department of Mathematics, Shanghai Normal University, Shanghai 200234, P.R. China
(3) Health School Attached to Shanghai University of Me\-dicine $\&$ Health Sciences, Shanghai 200237, P.R. China

Summary: 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 notion 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 $ (m(Q),Q )$ 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.

DOI: http://dx.doi.org/10.7900/jot.2023oct21.2450
Keywords: Hilbert $C^*$-module, projection, idempotent, representation, norm estimation

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