One Hat Cyber Team

Your IP : 216.73.216.115

Server IP : 194.44.31.54

Server : Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64

Server Software : Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k

PHP Version : 5.6.40

View File Name : semi1.tex

\documentclass{article}

\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amsthm}
\newtheorem{proposition}{Proposition}

\begin{document}
\section{Commutative models}

Let $T_t$, $t\ge0$ be a centered semigroup. First we assume that the
operators $T_t$ are invertible, i.e., have zero kernel and their image
is the whole $H$. Write the polar decomposition, $T_t
= U_tC_t$, then $C_t$ have (possibly unbounded) inverse, and $U_t$ are
unitaries. 

Write $A_t = C_t$, $A_{-t} = U_t C_t^{-1} U_t^*$, $t\ge0$. Then $A_t$,
$t\in \mathbb R$ is a commuting family of positive self-adjoint
operators.

\begin{proposition}
The following relations hold for all $t\in \mathbb R$, $s\ge0$
\begin{align*}
A_t U_s & = U_s A_{t+s} A_s^{-1},
\\
A_tU_s^* & = U_s^* A_{t-s} A_{-s}^{-1}.
\end{align*}
\end{proposition}

\begin{proof}
We have the  relations
\[
U_s^*C_tU_sC_s = C_sU_s^*C_tU_s,\quad C_tU_sC_sU_s^* = U_sC_sU_s^*C_t,
\]
and taking into account that $(U_tC_tU_t^*)^{-1} = U_tC_t^{-1}U_t^*$,
we get the needed relations.
\end{proof}

\end{document}


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: