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\begin{center}
\vspace{2 cm}
{\fifteen OPERATOR-NORM CONVERGENCE OF THE TROTTER PRODUCT FORMULA FOR SECTORIAL
GENERATORS}
\bigskip
\bigskip
Vincent CACHIA and Valentin A. ZAGREBNOV
\end{center}
\noindent{\footnotesize Centre de Physique Th\'eorique, CNRS Luminy case 907,
F-13288 Marseille cedex 9, and Universit\'e de la M\'editerran\'ee
(Aix-Marseille II).\\ Email: cachia@cpt.univ-mrs.fr and
zagrebnov@cpt.univ-mrs.fr}
\bigskip
\bigskip
\noindent {\bf Abstract}.
The operator-norm convergence of the Trotter product formula is known for
self-adjoint semigroups with compactness or smallness conditions on the
generators involved in this formula. We generalize these two types of results to
sectorial generators.
\medskip
\noindent {\bf Mathematics Subject Classifications:} 47D03, 47B25, 35K22, 41A80.
\medskip
\noindent {\bf Key words}: Trotter product formula, operator-norm convergence,
holomorphic semigroups, sectorial operators.
\bigskip
\section{Introduction and main result}
Nowadays there are two strategies to establish the operator-norm convergence of
the Trotter product formula: with error bounds \cite{R}, [1]-[6], [11], [12];
and without them \cite{NZ0}, \cite{NZ3}. The present letter belongs to the
second class. Among the preceding results, all (except \cite{CZ}) concern
self-adjoint semigroups. The aim of this letter is to generalize to sectorial
generators the following results of \cite{NZ3}, \cite{NZ1}, \cite{NZ2} (where,
in addition, the error bounds are given): the first one uses compactness
conditions, and the second one smallness conditions.
\begin{prop}\label{P1}
Let $A$ and $B$ be non-negative densely defined self-adjoint operators on a
Hilbert space $\cal H$. If $(I+A)^{-1}$ or $(I+A)^{-1}(I+B)^{-1}$ are compact,
then the sequence $\left\{\left(e^{-tA/n}e^{-tB/n}\right)^n\right\}_{n\geq 1}$
converges in the operator-norm to $e^{-tH}P$ locally uniformly for $t>0$. Here
$H=A\dot{+}B$ is the form-sum of $A$ and $B$, and $P$ is the orthogonal
projector on the closed subspace $\overline{{\cal D}(A^{1/2})\cap{\cal
D}(B^{1/2})}$.
\end{prop}
\begin{prop}\label{P2}
Let $A$ and $B$ be non-negative densely defined self-adjoint operators on a
Hilbert space $\cal H$, with $A\geq I$, $B\geq 0$, and ${\cal
D}(A^\alpha)\subset {\cal D}(B^\alpha)$ for some $1/2<\alpha\leq 1$, $\|B^\alpha
u\|\leq a\|A^\alpha u\|$, $u\in{\cal D}(A^\alpha)$ for $a<1$. Then the sequence
$\left\{\left(e^{-tA/n}e^{-tB/n}\right)^n\right\}_{n\geq 1}$ converges in the
operator-norm to $e^{-tH}$ uniformly in $t\geq 0$. Here $H=A\dot{+}B$ is the
form-sum of $A$ and $B$.
\end{prop}
Now let $A$ and $B$ be m-sectorial operators on a Hilbert space $\cal H$ with
domains ${\cal D}(A)$ and ${\cal D}(B)$. Let $a$ and $b$ be associated closed
sectorial forms with positive vertices and semi-angles $\theta_A$ and $\theta_B$
(here and below we follow notations of \cite{Kato}). Let $S_\theta$ denote the
open sector $\{z\in{\bb C}, z\neq 0, |\arg z|<\theta\}$. In the present paper we
obtain the following generalization of Proposition 1:
\begin{thm}
If $A$ and $B$ are m-sectorial operators on a Hilbert space $\cal H$ and if
$(I+A)^{-1}$ or $(I+{\mathrm Re}A)^{-1/2}(I+{\mathrm Re}B)^{-1/2}$ are compact,
then the sequence $\left\{\left(e^{-tA/n}e^{-tB/n}\right)^n\right\}_{n\geq 1}$
converges in operator-norm to $e^{-tH}P$ for any $t\in S_\theta$. Here
$H=A\dot{+}B$ is the form-sum of $A$ and $B$, $P$ is the orthogonal projector on
$\overline{{\cal D}(a)\cap{\cal D}(b)}$, and
$\theta=\pi/2-\max\{\theta_A,\theta_B\}$. The convergence is uniform on the
compact subsets of the sector $S_\theta$.
\end{thm}
On the other hand, we generalize Proposition 2 under the assumptions reflecting
the smallness of $B$ with respect to $A$:
\begin{eqnarray*}
& (i) & \ \mbox{Re}A, \mbox{Re}B\geq\gamma>0 ;\\
& (ii) & \ {\cal D}(\mbox{Re}A)\subset{\cal D}(\mbox{Im}A); \\
& (iii) & \ {\cal D}(\mbox{Re}B)\subset{\cal D}(\mbox{Im}B); \\
& (iv) & \ {\cal D}((\mbox{Re}A)^\alpha)\subset{\cal D}((\mbox{Re}B)^\alpha)
\mbox{ for some } \alpha\in (1/2,1]; \\
& (v) & \|(\mbox{Re}B)^\alpha u\|\leq c \|(\mbox{Re}A)^\alpha u\|,\ u\in{\cal
D}((\mbox{Re}A)^\alpha),\ 0From $(iv)$ it follows that ${\cal D}(a)\subset{\cal D}(b)$, thus
$\overline{{\cal D}(a)\cap{\cal D}(b)} = {\cal H}$ and the form-sum $A\dot{+}B$
is densely defined.
To clarify assumption $(ii)$ we give a sufficient condition which implies
$(ii)$. Let us consider the standard representation $A=G(I+iC)G$ of the
sectorial operator $A$. Here $G=(\mbox{Re}A)^{1/2}$ and $C$ is a self-adjoint
operator bounded by $\tan\theta_A$ (see \cite[Ch.VI Theorem 3.2]{Kato}). If
$C({\cal D}(G)\cap{\cal R}(G))\subset{\cal D}(G)$, where ${\cal R}(G)$ is range
of $G$, then $(ii)$ holds. A similar condition provides $(iii)$.
\section{Proofs}
We start with an abstract couple of m-sectorial operators $A$ and $B$ on a
Hilbert space $\cal H$. Following idea formulated in Addendum of \cite{K}, we
consider two families of closed forms:
\begin{eqnarray}
a(z) & = & \mbox{Re }a + z\mbox{Im }a ,\ {\cal D}(a(z))={\cal
D}(a);\label{defaz}\\
b(z) & = & \mbox{Re }b + z\mbox{Im }b ,\ {\cal D}(b(z))={\cal
D}(b);\label{defbz}
\end{eqnarray}
where $a$ and $b$ are associated with the m-sectorial operators $A$ and $B$
defined above, and $z\in{\bb C}$.
\begin{lem}
There exists $\epsilon>0$ such that the forms $a(z)$ and $b(z)$ are sectorial
for any $z$ in the strip $D=\{z\in{\bb C}: |\mbox{\rm Re }z|<\epsilon\}$. The
families of forms are also holomorphic of type (a) for $z\in\bb C$.
\end{lem}
\begin{preuve}
Let $z=x+iy$ with $x,y\in{\bb R}$, and $u\in{\cal D}(a)$. Since $a$ is
sectorial, we have:
\begin{equation}\label{in1}
|x\mbox{Im }a[u]| \leq |x| \tan\theta_A\mbox{Re }a[u];
\end{equation}
hence $\mbox{Re }a(z) = \mbox{Re }a + x\mbox{Im }a$ is positive for
$|x|\tan\theta_A<1$. Moreover, for such $x$:
\begin{equation}\label{in2}
\mbox{Re }a(z)[u] \geq \mbox{Re }a[u] - |x\mbox{Im }a[u]| \geq
(1-|x|\tan\theta_A)\mbox{Re }a[u].
\end{equation}
By (\ref{in1}) and (\ref{in2}) we obtain:
\begin{eqnarray}
|\mbox{Im }a(z)[u]|=|y\mbox{Im }a[u]| & \leq & |y|\tan\theta_A\mbox{Re }a[u]
\nonumber\\
& \leq & {|y|\tan\theta_A \over 1-|x|\tan\theta_A}\mbox{Re }a(z)[u],
\end{eqnarray}
which proves that $a(z)$ is sectorial for any $z$ in the strip $|\mbox{Re
}z|<\cot\theta_A$. Similarly, $b(z)$ is sectorial for $|\mbox{Re
}z|<\cot\theta_B$, and we can set $\epsilon = \min\{\cot\theta_A,
\cot\theta_B\}$.
On the other hand, since by (\ref{defaz}) and (\ref{defbz}) one has:
\begin{eqnarray}
{d\over dz} a(z)[u] & = & \mbox{Im }a[u],\ u\in{\cal D}(a),\\
{d\over dz} b(z)[u] & = & \mbox{Im }b[u],\ u\in{\cal D}(b),
\end{eqnarray}
the forms $a(z)$, $b(z)$ are holomorphic families of type (a) for $z\in\bb C$
(cf \cite[Ch.VII \S 4]{Kato}).
\end{preuve}
\begin{cor}
By the first representation theorem \cite[Ch.VI, Theorem 2.1]{Kato}, there exist
m-sectorial operators $A(z)$ and $B(z)$ associated with the closed sectorial
forms $a(z)$ and $b(z)$ ($|{\mathrm Re }z|<\epsilon$) such that these operators
form holomorphic families and are locally uniformly sectorial \cite[Ch.VII,
Theorem 4.2]{Kato}.
\end{cor}
If $A$ is an m-sectorial operator with positive vertex and semi-angle
$\theta_A$, then $-A$ generates a holomorphic contraction semigroup (we denote
it by $e^{-tA}$, where $t$ is a complex number with $|\arg t|< \pi/2-\theta_A$),
moreover we have the following result \cite[Ch.IX, Theorem 2.6 and
footnote]{Kato}:
\begin{prop}\label{PK}
Let $T(z)$ be a holomorphic family of closed operators in an open domain
$D\subset{\bb C}$, and for any $z\in D$ operator $-T(z)$ generates a holomorphic
semigroup of angle $\theta(z)$. Then the semigroup $e^{-tT(z)}$ is holomorphic
in $z\in D$ for $t$ in some open sector containing $t>0$.
\end{prop}
\begin{preuve}
The holomorphic semigroup $e^{-tT(z)}$ can be defined by the Dunford-Taylor
integral
\begin{equation}\label{dfsgh}
e^{-tT(z)} = {1\over 2\pi i} \int_\Gamma e^{\zeta t}(T(z)+\zeta)^{-1}d\zeta,
\end{equation}
where $\Gamma$ is a curve running from $\infty e^{-i(\theta(z)+\pi/2-\delta)}$
to $\infty e^{i(\theta(z)+\pi/2-\delta)}$ and enclosing the spectrum of $-T(z)$.
Since $T(z)$ is a holomorphic family, $(T(z)+\zeta)^{-1}$ is holomorphic in $z$
for $\zeta$ in the resolvent set of $-T(z)$. Moreover, in virtue of the
estimate:
\begin{equation}\label{esder}
\left\|{d\over dz}(T(z)+\zeta)^{-1}\right\|= \left\|{1\over 2\pi i} \int_C
(T(z')+\zeta)^{-1}(z'-z)^{-2}dz'\right\| \leq {M_\delta\over|\zeta|r}
\end{equation}
for $|\arg \zeta|\leq \pi/2 + \theta(z)-\delta$, the representation
(\ref{dfsgh}) can be differentiated under the integral sign. Here $C$ is a small
circle around $z$ of radius $r$, and $$M_\delta = \sup_{|\arg \zeta|\leq \pi/2 +
\theta(z)-\delta \atop z'\in C}\|\zeta(T(z')+\zeta)^{-1}\|.$$
Therefore, (\ref{dfsgh}) and (\ref{esder}) imply the assertion.
\end{preuve}
Applying this result to our case, we find that the semigroups $e^{-tA(z)}$,
$e^{-tB(z)}$ are holomorphic in $z$ with $|\mbox{Re }z|<\epsilon$ and in $t$ in
some (depending on $z$) open sector containing $t>0$. For real $t\geq 0$, these
semigroups are contractive.
Notice that for holomorphic functions with bounded operator-values there is no
distinction between uniform, strong and weak analyticity \cite[Ch.III, Theorem
3.12]{Kato}. For that reason the following result, known as the theorem of
Vitali \cite[Theorem 3.14.1]{HP}, will achieve the proof of our Theorems:
\begin{prop}\label{Vitali}
Let $f_n$ be a family of holomorphic functions on an open $D\subset\bb C$, with
values in a Banach space $\cal X$. Suppose that $\|f_n(z)\|\leq M$ for any
$n\in\bb N$ and $z\in D$, and $f_n(z)$ converges for $z$ in a subset of $D$
having a limit point in $D$. Then $f_n(z)$ converges for any $z\in D$, uniformly
on compact subsets of $D$, and the limit function is holomorphic in $D$.
\end{prop}
Our method of generalization of results \cite{NZ1}, \cite{NZ2}, and \cite{NZ3},
consists in applying this statement for some fixed $t>0$ to the family of
functions $$f_n(z) = \left(e^{-tA(z)/n}e^{-tB(z)/n}\right)^n,\ n=1,2,\dots\,,$$
with values in the Banach space of bounded operators on $\cal H$. By Lemma 1,
Corollary 1, and Proposition \ref{PK}, these functions are holomorphic in
$D=\{z\in{\bb C}: |\mbox{Re }z|<\epsilon\}$, and $\|f_n(z)\|\leq 1$ for any
$z\in D$. We will show that, under conditions formulated in Theorems 1 (or 2),
Proposition 1 (or 2) can be applied to the self-adjoint operators $A(x)$ and
$B(x)$ for real and small $x$. Thus the operator-norm convergence will be
extended to $D$. In particular, for $z=i\in D$ we find the Trotter product
formula for initial operators $A\equiv A(z=i)$ and $B\equiv B(z=i)$, and for
$t>0$.
\noindent{\it Proof of Theorem 1}
Let $(I+A)^{-1}$ be compact. Since by Lemma 1 and Corollary 1, $I+A(z)$ is a
holomorphic family of operators invertible for $z\in D$, the operators
$(I+A(z))^{-1}$ are also compact for any $z\in D$ by \cite[Ch.VII, Theorem
4.3]{Kato}. In particular it is true for $z=x$ from the real interval
$(-\epsilon, \epsilon)$. Then combining Proposition 1 for the self-adjoint
families $A(x)$ and $B(x)$, $x\in(-\epsilon,\epsilon)$ and Proposition
\ref{Vitali}, one gets the first part of Theorem 1.
Now let the product $(I+{\mathrm Re}A)^{-1/2}(I+{\mathrm Re}B)^{-1/2}$ be
compact. Consider holomorphic families of invertible m-sectorial operators
$I+A(z)$ and $I+B(z)$, where $z\in D$, cf Corollary 1. By the standard
representation \cite[Ch.VI Theorem 3.2]{Kato} one gets:
\begin{eqnarray}
I+A(z) & = & (I+\mbox{Re}A)^{1/2}(I+z\tilde{C}_A)(I+\mbox{Re}A)^{1/2}, \\
I+B(z) & = & (I+\mbox{Re}B)^{1/2}(I+z\tilde{C}_B)(I+\mbox{Re}B)^{1/2},
\end{eqnarray}
where $\tilde{C}_A$ and $\tilde{C}_B$ are self-adjoint operators bounded
respectively by $\tan\theta_A$ and $\tan\theta_B$. Since for real $x\in D$ we
obtain:
\begin{eqnarray*}
& & (I+A(x))^{-1}(I+B(x))^{-1} = (I+\mbox{Re}A)^{-1/2}(I+x\tilde{C}_A)^{-1}\\
& & \hspace{1cm} (I+\mbox{Re}A)^{-1/2}
(I+\mbox{Re}B)^{-1/2}(I+x\tilde{C}_B)^{-1}(I+\mbox{Re}B)^{-1/2},
\end{eqnarray*}
the family $(I+A(x))^{-1}(I+B(x))^{-1}$ is compact. Combining as above
Proposition 1 for $A(x)$ and $B(x)$, $x\in(-\epsilon,\epsilon)$ with Proposition
\ref{Vitali}, one finishes Theorem~1.
\noindent{\it Proof of Theorem 2}
We begin by proving a technical Lemma \ref{l2} which is based on the following
result known as the Heinz-Kato Theorem \cite[\S 2.3.3]{Tanabe}:
\begin{prop}
Let $A$ and $B$ be positive self-adjoint operators in a Hilbert space $\cal H$,
and suppose ${\cal D}(A)\subset{\cal D}(B)$. For all $\alpha$ satisfying
$0<\alpha<1$, we have ${\cal D}(A^\alpha)\subset{\cal D}(B^\alpha)$. If, in
addition, there is a certain number $M$ such that $\|Bu\|\leq M\|Au\|$ for all
$u\in{\cal D}(A)$, then $\|B^\alpha u\|\leq M^\alpha\|A^\alpha u\|$ for all
$u\in{\cal D}(A^\alpha)$.
\end{prop}
\begin{lem}\label{l2}
If $A$ and $B$ satisfy the assumptions (i)-(v), then for any real $x$ with
$|x|<\min\{\epsilon, 1/\lambda\}$ one has:
\begin{equation}\label{esl2}
\|B(x)^\alpha u\| \leq \left({1+\mu|x|\over 1-\lambda|x|}\right)^\alpha
c\|A(x)^\alpha u\|,\ u\in{\cal D}(A(x)^\alpha).
\end{equation}
Here $\epsilon$ is as in Lemma 1, $\lambda$ and $\mu$ are defined by (\ref{2i})
and (\ref{3i}), whereas $c$ and $0<\alpha\leq 1$ are the same as in (v).
\end{lem}
\begin{preuve}
By (\ref{3i}), which follows from $(iii)$, we have
\begin{equation}
\|B(x)u\| \leq \|\mbox{Re}Bu\| + |x| \|\mbox{Im}Bu\| \leq (1+\mu|x|)
\|\mbox{Re}Bu\|,
\end{equation}
for any $u\in{\cal D}(\mbox{Re}B)\subset{\cal D}(B(x))$. Since $B(x)$ is
self-adjoint, this inequality implies by the Heinz-Kato Theorem that
\begin{equation}\label{ineq1}
\|B(x)^\alpha u\| \leq (1+\mu|x|)^\alpha \|(\mbox{Re}B)^\alpha u\|,\ u\in{\cal
D}((\mbox{Re}B)^\alpha).
\end{equation}
In virtue of (\ref{2i}) and perturbation theory of self-adjoint operators, the
sum $\mbox{Re}(A) + x\mbox{Im}(A)$ is self-adjoint on the domain ${\cal
D}(\mbox{Re}A)$ for $|x|<1/\lambda$. Since by construction (Corollary 1)
operator $A(x)$ is a self-adjoint extension of this sum, we have ${\cal D}(A(x))
= {\cal D}(\mbox{Re}A)$. By (\ref{2i}) for any $u$ in this domain we obtain:
\begin{equation}
\|\mbox{Re}Au\| \leq \|A(x)u\| + |x| \|\mbox{Im}Au\| \leq \|A(x)u\| + \lambda|x|
\|\mbox{Re}Au\|.
\end{equation}
Since $|x|<1/\lambda$, one gets:
\begin{equation}
\|\mbox{Re}Au\| \leq {1\over 1-\lambda|x|}\|A(x)u\|,
\end{equation}
and:
\begin{equation}\label{ineq2}
\|(\mbox{Re}A)^\alpha u\| \leq {1\over (1-\lambda|x|)^\alpha} \|A(x)^\alpha
u\|,\ u\in{\cal D}(A(x)^\alpha)
\end{equation} by the Heinz-Kato Theorem.
Finally the inequalities (\ref{ineq1}), (\ref{ineq2}), together with assumption
$(v)$ give the annonced estimate (\ref{esl2}).
\end{preuve}
Since $c<1$, there exists $0<\eta<\min\{\epsilon,1/\lambda\}$ such that for
$|x|<\eta$,
\begin{equation}
\left({1+\mu|x|\over 1-\lambda|x|}\right)^\alpha c<1.
\end{equation}
Thus the conditions of Proposition \ref{P2} are satisfied for families of
self-adjoint operators $A(x)$ and $B(x)$ for any $x\in (-\eta,\eta)$. Since the
interval $(-\eta, \eta)$ has a limit point in the domain $D=\{z\in{\bb C},
|\mbox{Re }z|<\epsilon\}$, the theorem of Vitali accomplishes the proof of
Theorem 2 for real $t>0$ and $z=i$.
To extend the result to $t\in S_\theta$ we use the theorem of Vitali once again.
Since $A=A(z=i)$ and $B=B(z=i)$ are sectorial of semi-angles $\theta_A$,
$\theta_B$, we put $\theta=\pi/2-\max\{\theta_A,\theta_B\}$. Then semigroups
$e^{-tA}$ and $e^{-tB}$ are holomorphic and bounded by $1$ on the open sector
$S_{\theta}=\{z\in{\bb C}, z\neq 0, |\arg z|<\theta\}$. Thus by the theorem of
Vitali the operator-norm convergence of the Trotter formula for $t>0$ implies
the convergence in the same topology for any $z\in S_\theta$, uniformly on
compact subsets. Since the limit function is holomorphic, it is the analytic
continuation of the semigroup $e^{-tH}P$ from the real semi-axis to the sector
$S_\theta$.
\section{Conclusion}
Since the first annonced result \cite{R} about the operator-norm convergence of
the Trotter product formula, it appeared number of papers considered
self-adjoint contraction semigroups: first for Schr\"odinger semigroups
\cite{DIT}, \cite{H}, \cite{IT1}, \cite{IT2}; then on the abstract level
\cite{IT}, \cite{NZ0}, \cite{NZ1}, \cite{NZ2}, \cite{NZ3}, \cite{NZ4}. Recently
it was proven \cite{T} that the result of \cite{NZ2} is in fact optimal in the
following sense: the error bound of \cite{NZ2} is the best possible for
$\alpha\in(1/2,1)$, and for fractional powers $\alpha\leq 1/2$, the
operator-norm convergence does not hold without supplementary conditions (see
Proposition \ref{P2} and discussion in \cite{NZ3}).
In this context, the present letter shows that self-adjointness is not an
essential ingredient for the operator-norm convergence of the Trotter formula.
It takes place for sectorial generators (insuring analyticity of such
semigroups) under compactness or smallness conditions similar to \cite{NZ1},
\cite{NZ2}, and \cite{NZ3}. Notice that similar to \cite{NZ3} we establish the
operator-norm convergence of the Trotter formula without error estimates known
for the self-adjoint semigroups, see \cite{NZ1}, \cite{NZ2}, and \cite{NZ4}.
Finally, smallness conditions and corresponding results of this letter are
different from those of \cite{CZ}. We consider here a certain class of sectorial
generators on a Hilbert space (see $(i)$-$(v)$), instead of general holomorphic
semigroups on a Banach space as in \cite{CZ}. In \cite{CZ} we proved the Trotter
formula for a holomorphic contraction semigroup generated by $A$ via an error
bound estimate, but only for those $B$ which have relative bound zero with
respect to the operator $A$.
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\end{document}