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\documentclass{article} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{amsthm} \author{Vasyl Ostrovskyi} \title{Centered one-parameter semigroups} \newtheorem{definition}{Definition} \newtheorem{proposition}{Proposiion} \begin{document} \maketitle \begin{abstract} We introduce the class of centered one-parameter semigroups which are continuous analogue of a single centered operator, and study their properties. In particular, we prove the Wold decomposition for such semigroups and give complete description of one-parameter centered semigroups of partial isometries. \end{abstract} \section*{Introduction} Let $T$ be a bounded operator and write $T_k = T^k$. Recall \cite{mormu} that $T$ is centered if the operators $(T_kT_k^*, T_l^*T_l)$, $k$, $l\ge1$, form a commuting family. In this paper we introduce a continuous analogue of a centered operator---one-parameter centered semigroup. \begin{definition} Let $T_t$, $t\ge0$, be a strongly continuous semigroup of operators in a Hilbert space $H$. We say that the semigroup $T_t$ is centered, if the operators $(T_tT^*_t, T^*_s T_s)$, $t$, $s\ge0$, form a commuting family. \end{definition} \section{Commutative models} Let $T_t$, $t\ge0$ be a centered one-parameter group. For every $t$ consider a polar decomposition $T_t = U_t C_t$, where $C$ is self-adjoint non-negative operator, $U_t$ is a partial isometry, and $\ker U_t = \ker C_t$. \begin{proposition} $U_t$ is a strongly continuous centered semigroup of partial isometries. \end{proposition} \begin{proof} First we show that $U_t$ is a semigroup. First observe that the operator $U_tC_tU^*_t$ commutes with the family. Indeed, it is a non-negative square root of $T_tT_t^*$. Then, since $U_s^*U_s$ is a projection onto the co-kernel of $C_s$, we have \begin{align*} T_tT_s & = U_t C_t U_s C_s = U_t C_t U_s C_s (U_s^*U_s) = U_t C_t (U_sC_s U_s^*) U_s \\ &= U_t (U_s C_s U_s^*) C_t U_s = U_t U_s (C_s U_s^* C_t U_s). \end{align*} From the relation \[ C_s (U_s^* C_t U_s) = U_s^* U_s C_s U_s^* C_t U_s = U_s^* C_t U_s C_sU_s^* U_s = (U_s^* C_tU_s) C_s \] we see that $C_s U_s^* C_t U_s$ is a non-negative self-adjoint operator, its kernel coincides with $\ker U_tU_s$, and its square is \[ C_s U_s^* C_t U_s C_s U_s^* C_t U_s = T_s^*T_t^*T_tT_s = C_{t+s}^2. \] Therefore, $U_{t+s} = U_tU_s$, $t,s \ge 0$. For $t \to 0$, the operator $C_t = (T^*_t T_t)^{1/2}$ converges strongly to the identity; from this, one can easily deduce that $U_t$ is strongly continuous. Finally, we show that $U_t$ is a centered semigroup. But since $P_t = U_t^* U_t$ is a projection onto the co-kernel of $T_t^* T_t$, and $Q_t = U_t U_t^*$ is a projection onto the co-kernel of $T_t T_t^*$, these operators commute since $T_t$ is a centered semigroup. \end{proof} \section{Wold decomposition} \section{Infinitesimal generators} \section{Centered semigroups of partial isometries} \bibliography{ref,2,new} \bibliographystyle{amsplain} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: