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 denote by $${\Gamma }_P=\{J:J=J^{\ast }=J^{-1}andJPJ=I-P\}$$ and $${\Delta }_P=\{J:J=J^{\ast }=J^{-1}andJPJ=I-P^{\ast }\}.$$ In this paper, we first get that symmetries $(2P-I)|2P-I{|}^{-1}$ and $(P+P^{\ast }-I)|P+P^{\ast }-I{|}^{-1}$ are the same. Then we show that ${\Gamma }_P\ne \varnothing$ if and only if ${\Delta }_P\ne \varnothing .$ 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 $\sqrt{-1}J(2P-I)|2P-I{|}^{-1}\in {\Gamma }_P.$