One Hat Cyber Team
Your IP :
216.73.216.148
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
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
home
/
vo
/
dal
/
View File Name :
partial.tex
\documentclass{article} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{amsthm} \newtheorem{proposition}{Proposition} \begin{document} \begin{proposition} Let $S$ be a partial isometry such that $S^k$, $k\ge1$. $S$ is centered if and only if $S^k$, $k\ge1$, are partial isometries. \end{proposition} \begin{proof} Let $S^k$, $k\ge1$, are partial isometries. We show that the projections $P_k = S^k(S^*)^k$, and $Q_l = (S^*)^lS^l$, $k$, $l\ge1$, form a commuting family. a) We show that $P_kP_l = P_l$, $l\ge k$. Indeed, we have \[ P_kP_l = S^k(S^*)^kS^l(S^*)^l = P_kS^kS^{l-k}(S^*)^l; \] since $S^k$ is a partial isometry, we have $P_kS^k = S^k$, which yields $P_kP_l= P_l$, $l\ge k$. b) The same way, since $Q_k(S^*)^k= (S^*)^k$, we get $Q_kQ_l = Q_l$, $k\ge l$, i.e., the both families of projections are commuting ones. c) Now we show that $P_kQ_l= Q_lP_k$ for all $k$, $l$. Indeed, we have \[ Q_{l+k} = (S^*)^{l+k}S^{l+k} = (S^*)^kQ_lS^k. \] Since $Q_{l+k}$ is a projection, we have \[ (S^*)^kQ_lS^k= (S^*)^k Q_lS^k(S^*)^kQ_lS^k = (S^*)^kQ_lP_kQ_lS^k. \] Multiplying the latter equality by $S^k$ and $(S^*)^k$ from the left and right respectively, we get \[ P_kQ_lP_k = P_kQ_lP_kQ_lP_k, \] which means that $P_kQ_lP_k$ is a projection. But this is possible only if $P_k$ and $Q_l$ commute. Let now $S$ be a centered partial isometry. Since $S$ is a partial isomerty, $P_1$ and $Q_1$ are projections, and $P_1S = S$, $Q_1S^* = S^*$. We apply an induction. Let $Q_{n-1}$ be a projection. Then we have \begin{multline*} Q_n^2 = S^*Q_{n-1}SS^*Q_{n-1}S = S^*Q_{n-1}P_1Q_{n-1}S \\ =S^*Q_{n-1}^2P_1S =S^* Q_{n-1}S = Q_n. \end{multline*} ($P_1$ commutes with $Q_{n-1}$ since $S$ is centered). Similarly, we show that $P_n$, $n\ge2$, are projections. \end{proof} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: