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\documentclass{article}
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\author{Vasyl Ostrovskyi}
\title{Centered one-parameter semigroups}


\newtheorem{definition}{Definition}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{remark}{Remark}
\theoremstyle{remark}
\newtheorem{example}{Example}

\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{Centered semigroups}

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{theorem}
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{theorem}

\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}

\begin{theorem}
Let $T_t$, $t\ge0$ be a centered semiroup in a Hilbert space $H$. The
space $H$ can be decomposed into a direct sum of invariant with
respect to $T_t$, $T_t^*$, $t\ge0$, subspaces, $H=H_0\oplus H_+ \oplus
H_-\oplus H_1$ such that
\begin{itemize}

\item[\textup {i)}]
in $H_1$ the operators $T_t$ are invertible, $\ker T_t = \ker 
T_t^*=\{0\}$, $t\ge0$\textup;

\item[\textup{ii)}]
in $H_+$ the semigroup is such that $\ker T_t = \{0\}$, $t\ge0$,
$\bigcup_{t\ge0} \ker T_t^* = H$\textup;  

\item[\textup{iii)}]
in $H_-$ the semigroup is such that
$\ker T_t^* = \{0\}$, $t\ge0$, $\bigcup_{t\ge0} \ker T_t = H$\textup; 

\item[\textup{iv)}]
in $H_0$ the semigroup is such that
$\bigcup_{t\ge0} \ker T_t = H$, and $\bigcup_{t\ge0} \ker T_t^* = H$.  
\end{itemize}
\end{theorem}

\begin{proof}
We start with one-parameter centered semigroups of partial isometries,
and then apply the polar decomposition.

Let $U_t$ be a centered one-parameter semigroup of partial
isometries. Introduce the projections $P_t = U_t^*U_t$ and $P_{-t} =
U_tU_t^*$, $t\ge0$. 
\end{proof}


\section{Commutative models}
Introduce the projections $P_t = U_t^*U_t$ and $P_{-t} = U_tU^*_t$,
$t\ge0$, and the self-adjoint operators $A_t = C_t$, $A_{-t} =
U_tC_tU^*_t$, $t\ge0$.

\begin{proposition}
The following relations hold $(s \ge0)$ for the projections $P_t$,
$t\in \mathbb{R}$
\begin{align}
P_t U_s &= U_s P_{t+s}, \quad t\ge 0,
& P_t U_s^* & = U_s^* P_{t-s}, \quad t\ge s,\notag
\\
P_t U_s &= U_s P_{t+s}, \quad t \le 0, t+s \le 0,
& P_tU_s^* & = U_s^*, \quad 0\le t\le s,\notag
\\
P_t U_s &= U_s, \quad t\le 0, t+s \ge 0,
& P_tU_s^* & = U_s^* P_{t-s}, \quad t\le 0.\label{pu}
\end{align}
and for the operators $A_t$, $t \in \mathbb{R}$:
\begin{align}
A_t U_s A_s & = U_s A_{t+s}, \quad t\ge 0,\notag
\\
A_t U_s & = U_s A_s A_{t+s}, \quad t\le 0, t+s \le 0,\notag
\\
A_t U_s A_{t+s} & = U_s A_s, \quad t\le 0, t+s \ge0.\label{aua}
\end{align} 
\end{proposition}

\begin{proof}
Just calculations.
\end{proof}

We rewrite the last relations in a slightly different form. Observe
that on $\ker A_s^\perp$ the operator $A_s$ has (possibly unbounded)
inverse $A_s^{-1}$. We define it on the whole space putting it zero on
$\ker A_s$. Then we rewrite  equalities  \eqref{aua} as
\begin{align}
A_t U_s & = U_s A_{t+s}A_s^{-1},\quad t\ge 0,\notag
\\
A_tU_s & = U_s A_{t+s} A_s, \quad t \le -s,\notag
\\
A_t U_s & = U_s A_{t+s}^{-1} A_s, \quad -s \le t \le 0. \label{au}
\end{align}
Notice that since the left-hand side is bounded, such is the right
one. Also we eliminated projections $P_s$ and $P_{t+s}$ on the kernels
of $A_s$ and $A_{t+s}$, which is possible due to \eqref{pu}.

On the space $\mathbb R^{\mathbb R}$ of all real functions introduce
the mappings 
\[
F_s  \lambda (t) = \begin{cases}\lambda(t+s)/\lambda(s),& t\ge0,\\
  \lambda(s)/\lambda(t +s), 
  & -s \le t \le 0, \\ \lambda(t +s)\lambda(s),& t \le -s.
\end{cases}
\]

\textbf{Newtext}
Consider the case when $C_t >0$, i.e., $\ker C_t = 0 $ for all $t$.
Write $A_t = C_t$, $t\ge0$, $A_{-t}= U_t C_tU_t^*$.

\begin{proposition}
The following relations hold
\[
A_t U_s = \begin{cases} U_s A_{t+s} A_s^{-1},& t\ge0,\\ U_s
  A_{t+s}^{-1} A_s, &-s \le t \le 0,\\ U_s A_{t+s} A_s, &t\le
  -s. \end{cases}
\]
\end{proposition}

The operators $A_t$ form a commuting family; therefore (check?!), the
following spectral decomposition holds:
\begin{align*}
H& = \int_\sigma^\oplus H_{\lambda(\cdot)} \, d\mu(\lambda(\cdot)),
\\
(A_t f(\lambda))(\cdot) & = \lambda(t) \,
f(\lambda(\cdot)),
\end{align*}
where $\sigma = \{ \lambda (\cdot) \colon \mathbb{R} \to \mathbb{R}
\mid \lambda(0) =1, \, \lambda(t) >0\}$.
In particular, if the joint spectrum of the commutiong family is
simple,
\begin{align*}
H& = L_2(\sigma, d\mu(\lambda(\cdot))),
\\
(A_t f(\lambda))(\cdot) & = \lambda(t) \, f(\lambda(\cdot)).
\end{align*}

For any $s \ge0$ introduce the mappings $F_s \colon \sigma \to
\sigma$, and  $G_s = F_s^{-1}$ as follows:
\begin{align*}
(F_s(\lambda(\cdot)))(t) & = 
  \begin{cases} \lambda(t+s)/\lambda(s), & t\ge0,\\
    \lambda(s)/\lambda(t+s), & -s\le t \le 0, \\ \lambda(t+s)\,
    \lambda(s),& t\le s,\end{cases}
\\
(G_s(\lambda(\cdot)))(t) & = 
  \begin{cases} \lambda(t-s)\,\lambda(-s), & t\ge s,\\
    \lambda(-s)/\lambda(t-s), & 0\le t \le s, \\ \lambda(t-s)/
    \lambda(-s),& t\le 0,\end{cases}
\end{align*}

According to the theorem on commutative models, for $U_s$ we have the
following representation:
\[
(U_sf(\lambda))(\cdot) = \alpha_s(\lambda(\cdot))\,
\rho_s(\lambda(\cdot)) \, f(G_s(\lambda)(\cdot)),
\]
where $\alpha_s$ is a unitary operator-valued function, $\rho$ is a
norming factor.

We consider some examples of quasi-invariant ergodic measures and the
corresponding examples of centered semigroups.

\begin{example}
Stationary points. The simples case of an ergodic quasi-invariant
measure is the measure concentrated at a stationary point. 
The only function invariant with respect to $F_s$,
$G_s$, is the identity constant. Then $C_t = I$ for all $t$, and we
have a strongly continuous unitary semigroup.
\end{example}

\begin{example}
Measures concentrated on a single orbit. Take a function
$\lambda(\cdot)\colon \mathbb{R} \to \mathbb{R}_+$, $\lambda(0)=1$,
and consider its orbit 
\[
\mathcal  O = \{\lambda_s(\cdot) , s \in \mathbb R\}, \quad
\lambda_s(t) = \begin{cases} (F_s \lambda(\cdot))(t), & s\ge 0,\\
  (G_{-s}\lambda(\cdot))(t), & s \le 0.\end{cases}
\]
The mapping $\mathbb{R} \ni s \mapsto \lambda_s(\cdot)$ establishes  a
one-to-one correspondence between points of a line and $\mathcal
O$. Take the Lebesgue measure on $\mathbb{R}$, and denote by $\mu$ its image
under this  mapping. Then $\mu$  is quasi-invariant measure on
$\sigma$ concentrated on $\mathcal O$. The inverse mapping establishes
an isomorphism between $L_2(\sigma, d\mu)$, and $L_2(\mathbb R, d
s)$. Under this isomorphism we have
\[
(A_t f) (s) = \lambda_s(t) \, f(s).
\]

\end{example}



\section{Infinitesimal generators}

\section{Centered semigroups of partial isometries}



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