Properties of two unitary operator functions involving idempotents
Tóm tắt
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.
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