The structures and decompositions of symmetries involving idempotents

Abstract

Let $\mathcal {H}$ be a separable Hilbert space and P be an idempotent on $\mathcal {H}.$ We set

$$\begin{aligned} \Gamma _{P}=\left\{ J: J=J^{*}=J^{-1} \quad \hbox { and }\quad JPJ=I-P\right\} \end{aligned}$$

and

$$\begin{aligned} \Delta _{P}=\left\{ J: J=J^{*}=J^{-1} \quad \hbox { and }\quad JPJ=I-P^*\right\} . \end{aligned}$$

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 $\Gamma _{P}\ne \emptyset$ if and only if $\Delta _{P}\ne \emptyset .$ 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 $iJ(2P-I)|2P-I|^{-1}\in \Gamma _{P}$.