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.