The structures and decompositions of symmetries involving idempotents

TL;DR

This paper investigates the structures and relationships of symmetries involving idempotents on Hilbert spaces, establishing conditions for their existence and explicit forms, and connecting different symmetry sets through specific transformations.

Contribution

It provides new characterizations and explicit structures of symmetries related to idempotents, and reveals their interconnections in Hilbert space operator theory.

Findings

Symmetries $(2P-I)|2P-I|^{-1}$ and $(P+P^{*}-I)|P+P^{*}-I|^{-1}$ are identical.

Existence of symmetries in $ ext{Gamma}_P$ is equivalent to existence in $ ext{Delta}_P$.

Explicit structures of symmetries in $ ext{Gamma}_P$ and $ ext{Delta}_P$ are derived.

Abstract

Let $\mathcal{H}$ be a separable Hilbert space and $P$ be an idempotent on $\mathcal{H}.$ We denote by $$\Gamma_{P}=\{J: J=J^{\ast}=J^{-1} \hbox{ }\hbox{ and }\hbox{ } JPJ=I-P\}$$ and $$\Delta_{P}=\{J: J=J^{\ast}=J^{-1} \hbox{ }\hbox{ and }\hbox{ } JPJ=I-P^*\}.$$ In this paper, we first get that symmetries $(2P-I)|2P-I|^{-1}$ and $(P+P^{*}-I)|P+P^{*}-I|^{-1}$ are the same. Then we show that $\Gamma_{P}\neq\emptyset$ if and only if $\Delta_{P}\neq\emptyset.$ Also, the specific structures of all symmetries $J\in\Gamma_{P}$ and $J\in\Delta_{P} $ are established, respectively. Moreover, we prove that $J\in\Delta_{P}$ if and only if $\sqrt{-1}J(2P-I)|2P-I|^{-1}\in\Gamma_{P}.$