Properties of two unitary operator functions involving idempotents

Abstract

Let P be an idempotent operator on a Hilbert space $$\mathcal {H}.$$ We denote two unitary operator functions $$U_{\lambda }$$ and $$V_{\lambda }$$ by $$\begin{aligned} U_{\lambda }:=(\lambda P+I)|\lambda P+I|^{-1} \hbox { } \hbox { and }\hbox { } V_{\lambda }:=(\lambda P^{*}+I)|\lambda P^{*}+I|^{-1}, \ \ \hbox { for }\lambda \in \mathbb {C}\backslash \{-1\}. \end{aligned}$$ In this paper, we first give the specific structures of $$U_{\lambda }$$ and $$V_{\lambda },$$ respectively. Then the sufficient and necessary conditions under which $$U_{\lambda }$$ and $$V_{\lambda }$$ are symmetries are presented. Moreover, the specific structures and spectra of the unitary operator $$U=\lim \limits _{\lambda \rightarrow -1^+}U_{\lambda }$$ are characterized.