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\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 strongly continuous one-parameter semigroup. 
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}
If $T_t$ is a centered semigroup, then $U_t$ is a strongly continuous 
centered semigroup of partial isometries.
Conversely, if $U_t$ is a centered semigroup of partial isometries,
then the semigroup $T_t$ is centered. 
\end{proposition}

\begin{proof}
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.

Now let $U_t$ be a centered semigroup. From $T_s T_t = T_{t+s}$ we have
\begin{equation}\label{eq:utc}
U_tC_t U_sC_s = U_{t+s}C_{t+s},
\end{equation}
and multiplying by $U_s^*U_t^*$ from the right, we get
\[
U_s^* C_t U_s C_s = C_{t+s}.
\]
Since $C_{t+s}$ is self-adjoint, taking the adjoints in the latter
equality, we conclude that the operators $U_s^* C_t U_s$ and $C_s$
commute. Then for $t>s$
\[
 C_t C_s = C_{s+(t-s)}C_s = U^*_s C_{t-s} U_s C_s C_s = C_s U^*_s
 C_{t-s} U_s C_s = C_s C_t,
\]
or $[T_t^*T_t, T_s^*T_s]=0$.

Again, multiplying \eqref{eq:utc} by $U_t^*$ from the right, and by
$U_s^*$ from the left, we get 
\begin{equation}\label{eq:ucu}
C_t U_s C_s U_s^* = U_s C_{t+s} U_s^*.
\end{equation}
Taking adjoints and observing that the right-hand side is
self-adjoint, we conclude that $C_t$ and $U_sC_s U_s^*$ commute, or
$[T_t^*T_t, T_sT_s^*]=0$.

Finally, applying \eqref{eq:ucu}, we have for $t>s$
\begin{multline*}
U_sC_s U_s^* U_tC_tU_t^*  = U_s C_s U_{t-s} C_t U_t^* =
U_sC_s(U_{t-s} C_{(t-s)+s} U_{t-s}^*) U_s^* 
\\
=U_sC_s(U_{t-s}C_{t-s}U^*_{t-s} C_s) U_s^* = U_s (C_s U_{t-s} C_{t-s}
U^*_{t-s}) C_s U^*_s 
\\
= U_s U_{t-s} C_t U_{t-s}^* C_s U^*_s 
= U_tC_tU_t^* U_sC_sU^*_s, 
\end{multline*}
or $[T_sT^*_s, T_tT^*_t]=0$. Thus, the semigroup $T_t$ is centered.
\end{proof}



\section{Wold decomposition}

\section{Infinitesimal generators}

\section{Centered semigroups of partial isometries}



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