% September, 1999
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\begin{document}
\title[Monotonicity and Concavity]{Monotonicity and Concavity
Properties of The Spectral Shift Function}
%\dedicatory{}
% Information for author
\author[]{Fritz Gesztesy,
Konstantin A.~Makarov, and Alexander~K.~Motovilov }
% Information for author
%\author{Fritz Gesztesy}
\address{Department of Mathematics,
University of
Missouri, Columbia, MO
65211, USA}
\email{fritz@math.missouri.edu\newline
\indent{\it URL:}
http://www.math.missouri.edu/people/fgesztesy.html}
% Information for author
%\author{Konstantin A.~Makarov}
\address{Department of Mathematics, University of
Missouri, Columbia, MO
65211, USA}
\email{makarov@azure.math.missouri.edu}
% Information for author
%\author{Alexander K.~Motovilov}
\address{
Physikalisches Institut, Universit\"at Bonn, D-53115~Bonn, Germany
}
\email{motovilov@physik.uni-bonn.de}
\address{On leave of absence from the
Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
}
\email{motovilv@thsun1.jinr.ru}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\dedicatory{Dedicated with great pleasure to Sergio
Albeverio on the occasion of his 60th birthday}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
\date{September, 1999}
\subjclass{Primary 47B44, 47A10; Secondary 47A20, 47A40}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
Let $H_0$ and $V(s)$ be self-adjoint, $V,V'$ continuously
differentiable in trace norm with $V''(s)\geq 0$ for
$s\in (s_1,s_2)$, and denote by
$\{E_{H(s)}(\lambda)\}_{\lambda\in\bbR}$ the family of spectral
projections of $H(s)=H_0+V(s)$. Then we prove for given
$\mu\in\bbR$, that $s\longmapsto
\tr\big (V'(s)E_{H(s)}((-\infty, \mu))\big ) $ is a
nonincreasing function with respect to $s$, extending a
result of Birman and Solomyak. Moreover, denoting by
$\zeta (\mu,s)=\int_{-\infty}^\mu d\lambda \,
\xi(\lambda,H_0,H(s))$ the integrated spectral shift
function for the pair $(H_0,H(s))$, we prove concavity of
$\zeta (\mu,s)$ with respect to $s$, extending previous
results by
Geisler, Kostrykin, and Schrader. Our proofs employ
operator-valued Herglotz functions and establish the latter
as an effective tool in this context.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction and principal results} \label{s1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the following $\calH$ denotes a complex separable Hilbert
space with scalar product $(\,\cdot, \, \cdot)_{\calH}$ (linear
in the second factor) and norm $\|\cdot\|_\calH$, $\calB(\calH)$
represents the Banach space of bounded linear operators
defined on
$\calH$,
$\calB_p(\calH), \,\, p\ge 1$ the standard Schatten-von Neumann
ideals of $\calB(\calH)$ (cf., e.g., \cite{GK69}, \cite{Si79})
and $\bbC_+ $ (resp., $\bbC_-$) the open complex upper
(resp., lower)
half-plane. Moreover, real and imaginary parts of a bounded
operator $T\in
\calB(\calH)$ are defined as usual by
$\Re (T)=(T+T^*)/2$, $\Im (T)=(T-T^*)/(2i)$.
The spectral shift function $\xi(\lambda, H_0,H)$ associated
with a pair of self-adjoint operators $(H_0,H)$, $H=H_0+V$,
$\dom(H_0)=\dom(H)$, where
\begin{equation}
\lb{nucl}
%
V=V^*\in \calB_1(\calH),
%
\end{equation}
is one of the fundamental spectral characteristics in the
perturbation theory of self-adjoint operators. It is
well-known
(see \cite{Kr53}, \cite{Kr62}, \cite{Kr83}, \cite{Kr89},
\cite{Li52}) that
for a wide function class ${\mathfrak K}(H_0,H)$, the
Lifshits-Krein trace
formula holds, that is,
\begin{equation}
\label{5.1}
%
\tr(\varphi(H)-\varphi(H_0) )=\int_{\bbR}
d\lambda\, \varphi'(\lambda)\,\xi(\lambda, H_0,H)\,,
\quad \varphi\in
{\mathfrak K}(H_0,H).
%
\end{equation}
In the case of trace class perturbations \eqref{nucl},
the spectral shift function is integrable, that is,
\begin{equation}\lb{L1}
%
\xi(\cdot, H_0, H)\in L^1(\bbR),
%
\end{equation}
and the following relations hold
\begin{eqnarray}
\label{ksi1}
%
\|\xi(\cdot, H_0, H)\|_{L^1(\bbR)}
&\le& \|V\|_{\calB_1(\calH)},\\
%
\label{trV}
\int_\bbR d\lambda\,\,\xi(\lambda, H_0, H)&=&\tr(V)\,.
%
\end{eqnarray}
The precise characterization of the class
\begin{equation}
{\mathfrak K}=\bigcap_{H_0, H}
{\mathfrak K}(H_0,H) \lb{1.6}
\end{equation}
of all those $\varphi$ for which \eqref{5.1} holds for any pair
of self-adjoint operators $H_0$ and $H=H_0+V$ with a trace class
difference \eqref{nucl}, is still unknown. In particular,
there are
functions $\varphi\in C^{1}_0(\bbR)$ for which \eqref{5.1} fails
(cf.~\cite{Fa80}, \cite{Pe94}). Necessary
conditions very close to sufficient ones for $\varphi$ belonging
to the
class ${\mathfrak K}$ have been found by Peller \cite{Pe85},
\cite{Pe90}. Here we only note that \eqref{L1} and
$(\varphi(H)-\varphi(H_0))\in\calB_1(\calH)$ hold, and
\eqref{5.1} is valid, if $\varphi'$ is the Fourier transform
of a finite Borel measure,
\begin{equation}
\label{5.2}
%
\varphi'(\lambda)=\int_\bbR d \nu(t)\,e^{-it\lambda}, \quad
\varphi\in C^1(\bbR), \,\,\,\int_\bbR d |\nu(t)|<\infty.
%
\end{equation}
We denote the function class \eqref{5.2} by $\calW_1(\bbR)$.
Different representations for the spectral shift function and
their interrelationships can be found in \cite{BP98}, for
further information we refer to \cite[Ch.~19]{BW83},
\cite{BY93}, \cite{BY93a}, \cite{GMN99}, \cite[Ch.~8]{Ya92} and
the references therein).
In the present short note we will focus on two particular
results: one, a monoto\-ni\-ci\-ty result obtained by Birman
and Solomyak \cite{BS75}, the other, a concavity result obtained
by Geisler, Kostrykin, and Schrader \cite{GKS95}, \cite{Ko99}.
We also present some extensions and new proofs that we
hope might give additional insights into the subject.
We start by recalling pertinent results discovered by Birman and
Solomyak \cite{BS75} in connection with the spectral averaging
formula (providing a representation for the spectral shift
function via an integral over the coupling constant) and a
monotonicity result of a certain trace with respect to the
coupling constant parameter.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}[\cite{BS75}]\label{t1.1}
Let $H_0$ and $V$ be self-adjoint in $\calH$,
$V\in\calB_1(\calH)$, and define
\begin{equation}
H_s=H_0+sV, \quad \dom(H_s)=\dom(H_0), \,\, s\in\bbR, \lb{1.8}
\end{equation}
with $\{E_{H_s}(\lambda)\}_{\lambda\in\bbR}$ the family of
orthogonal spectral projections of $H_s$. Moreover, denote by
$\xi(\cdot, H_0, H_1)$ the spectral shift function for the pair
$(H_0,H_1)$. Then for any Borel set $\Delta \subset \bbR$,
\begin{equation} \label{1.1}
\int_{\Delta} d\lambda\, \xi (\lambda , H_0, H_1)=
\int_0^1 ds\,\tr(VE_{H_s}(\Delta)).
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the same paper \cite{BS75}, Birman and Solomyak proved another
remarkable statement concerning the monotonicity of the
integrand in the
right-hand side of (\ref{1.1}) with respect to $s$ for
semi-infinite
intervals $\Delta =(-\infty,\lambda)$, $\lambda\in\bbR$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}[\cite{BS75}]\label{t1.2}
Assume the hypotheses in Theorem~\ref{t1.1}. Given $\mu\in
\bbR$, the
function
\begin{equation}\label{1.2}
%
s\longmapsto \tr\big (VE_{H_s}((-\infty,\mu))\big ),
\quad s\in\bbR,
%
\end{equation}
is a nonincreasing function with respect to $s\in\bbR$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The spectral averaging formula \eqref{1.1} combined with the
monotonicity result \eqref{1.2} is a convenient tool for
producing
estimates for the spectral shift function \cite{BS75}. For
instance,
\begin{equation}
\label{1.3}
%
\tr \big (VE_{H_1}((-\infty, \mu)) \big )\le
\int_{-\infty}^\mu d\lambda\, \xi (\lambda , H_0, H_1)
\le
\tr \big (VE_{H_0}((-\infty, \mu)) \big ).
%
\end{equation}
In particular, passing to the limit $\mu\to \infty$ in
\eqref{1.3} one obtains \eqref{trV} (see \cite{BS75}
for more details).
Another application of the pair of results
\eqref{1.1} and \eqref{1.2} leads to the proof of concavity
properties of the integrated spectral shift function with
respect to the coupling constant, originally discovered in the
case of Schr\"odinger operators by Geisler, Kostrykin, and
Schrader \cite{GKS95} and extended by Kostrykin \cite{Ko99},
\cite{Ko99a} to the general case presented next.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}[\cite{Ko99}, \cite{Ko99a}] \label{t1.3}
Let $\xi(\cdot, H_0, H_s)$ be the spectral shift function
in Theorem~\ref{t1.1}. Given $\mu\in \bbR$, the integrated
spectral shift function
\begin{equation}
%
\zeta_s(\mu)=
\int_{-\infty}^\mu d\lambda\, \xi (\lambda , H_0, H_s),
\quad s\in\bbR
%
\end{equation}
is a concave function with respect to the coupling constant
$s\in\bbR$.
More precisely, for any $s, t \in \bbR$, and for all
$\alpha\in[0,1]$,
the following inequality
\begin{equation}
%
\zeta_{\alpha s+ (1-\alpha)t}(\mu)\ge \alpha
\,\zeta_s(\mu) +(1-\alpha) \,\zeta_t(\mu)
%
\end{equation}
holds. Moreover, $\zeta_s(\mu)$ is subadditive with respect to
$s\in (0,\infty)$ in the sense that for any $s, t\ge 0$,
\begin{equation}
%
\zeta_{s+t}(\mu)\le\zeta_s(\mu)+\zeta_t(\mu).
%
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
While Theorem~\ref{t1.3} focuses on a linear coupling constant
dependence in $H_s=H_0+sV$, Kostrykin~\cite{Ko99} also
discusses the case of a nonlinear dependence on $s$ for
operators of the form $H(s)=H_0+ V(s)$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}[\cite{Ko99}, \cite{Ko99a}] \label{t1.3a}
Suppose $f:\bbR\to\bbR$ is an nonincreasing function of bounded
variation and $\{V(s)\}_{s\in\bbR}\in\calB_1(\calH)$ is operator
concave (i.e., $V(\alpha s+(1-\alpha)t)\geq \alpha V(s)
+(1-\alpha)V(t)$ for all $\alpha\in [0,1]$, $s,t\in\bbR$). Then
\begin{equation}
s \longmapsto g(V(s))=\int_\bbR d\lambda\,f(\lambda)
\xi(\lambda,H_0,H_0+V(s)),
\end{equation}
is concave in $s\in\bbR$. More precisely, for
all $0\le\alpha\le1$ and all $s,t\in\bbR$, the following
inequality
\begin{equation} \label{Concavity}
g(V(\alpha s+(1-\alpha)t))\geq \alpha g(V(s))+(1-\alpha)g(V(t))
\end{equation}
holds.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Actually, Kostrykin considered the general case of relative
trace class
perturbations in \cite{Ko99} but we omit further details in
this note.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r1.4}
The results of Theorems~\ref{t1.1} and \ref{t1.2} in \cite{BS75}
have
been obtained using the approach of Stieltjes' double operator
integrals \cite{BS66}--\cite{BS73}. Birman and Solomyak treated
the case $V(s)=sV$, $V\in\calB_1(\calH)$, that is, they
discussed
the case of a
linear dependence of the perturbation $V(s)$ with respect to the
coupling constant parameter $s$. The general case of a
nonlinear dependence $V(s)$ of $s$, assuming
$V'(s)\ge0$, in the context of the spectral averaging result
\eqref{1.1} has recently been treated by Simon \cite{Si98}.
In Theorem~\ref{t1.6} below we cite the most recent result of
this type obtained in \cite{GMN99}.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is convenient to introduce the following hypothesis.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{hypothesis}\label{h1.5}
Let $H_0$ be a self-adjoint operator in $\calH$ with domain
$\dom(H_0)$, and assume $\{V(s)\}_{s\in \Omega}\subset
\calB_1(\calH)$ to be a family of self-adjoint trace class
operators in $\calH$, where $\Omega \subseteq \bbR$ denotes
an open interval with $0\in\Omega$. Moreover, suppose that
$V(s)$ is
continuously differentiable in
$\calB_1(\calH)$-norm with respect to $s\in\Omega$.
For convenience (and without loss of generality) we may assume
that $V(0)=0$ in the following.
\end{hypothesis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the rest of the paper we will frequently use the notation
$(s_1,s_2)\subset\subset \Omega$ to denote an open interval
that is strictly contained in the interval $\Omega=(a,b)$
(i.e., $a0,
%
\end{equation}
one concludes that
\begin{equation}
\label{3.11}
%
\lim_{\varepsilon \downarrow 0}\varphi_{\mu,
\varepsilon}(\lambda)=
\chi_{(-\infty, \mu)}(\lambda),
%
\end{equation}
where $\chi_\Delta(\cdot)$ denotes the characteristic function
of the set $\Delta$.
Since
\begin{equation}
\label{3.12}
%
\sup_{\varepsilon>0}\|\varphi_{\mu,
\varepsilon}\|_{L^\infty(\bbR)} <\infty,
%
\end{equation}
\eqref{3.11} implies the strong convergence
\begin{equation}
\label{3.13}
%
\slim_{\varepsilon \downarrow 0}\varphi_{\mu,
\varepsilon}(H(s))=
E_{H(s)}((-\infty,\mu))
%
\end{equation}
by Theorem VIII.5 in \cite{RS80}. Combining
\eqref{3.11}--\eqref{3.13} with Theorem~1 in \cite{Gr73},
one infers
\begin{equation}
%
\lim_{\varepsilon \downarrow 0}\,\tr\bigl(V'(s)
\,\varphi_{\mu, \varepsilon}(H(s))\bigr)=
\tr\bigl(V'(s)E_{H(s)}((-\infty,\mu))\bigr).
%
\end{equation}
By Theorem~\ref{t1.7}, the function
\begin{equation}
%
s\longmapsto \tr\bigl(V'(s)\,\varphi_{\mu,
\varepsilon}(H(s))\bigr),\quad s\in (s_1,s_2),
%
\end{equation}
is nonincreasing, proving the assertion, since the
pointwise limit of nonincreasing functions is nonincreasing.
\hfill $\square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next we prove Corollary~\ref{c1.8}.
\smallskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\it Proof of Corollary~\ref{c1.8}.}\,
By Theorem~\ref{t1.6}
\begin{equation}
\label{3.17}
%
\int_{-\infty}^\mu d\lambda\, \xi (\lambda , H_0, H(s))=
\int_{0}^{s} dt\,\tr\bigl(V'(t)E_{H(t)}((-\infty, \mu))\bigr),
\quad s\in (s_1,s_2), \,\, \mu \in \bbR.
%
\end{equation}
By Theorem~\ref{t1.7} the integrand on the right-hand side
of \eqref{3.17}
is a nonincreasing function of $t$ and hence the
left-hand side of \eqref{3.17} is a concave function of
$s$. Thus
\eqref{Concav} holds.
In order to prove \eqref{3.16a} one notes that
$\zeta(\mu,0+)=0$ and that a necessary and sufficient
condition for a measurable concave function $f(t)$ to be
subadditive on $(0,\infty)$ is that $f(0+)\ge 0$ (see, e.g.,
\cite[Theorem~7.2.5]{Hi48}).
\hfill $\square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the case of semibounded operators the following statement
might
be useful. We recall that $\calW_1(\bbR)$ denotes the
function class
of $\varphi$ with $\varphi'$ the Fourier transform of a
finite
(complex) Borel measure (cf.~\eqref{5.2}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{t3.6}
Let $H_0$ be a self-adjoint operator in $\mathcal H$, bounded
from below, and assume the hypotheses of Theorem~\ref{t1.7}.
Denote by $\Lambda$ the smallest semi-infinite interval
containing the spectra of the family $H(s)$, $s\in (s_1, s_2)$,
\begin{equation}
\label{5.9}
%
\Lambda= \bigg[\inf_{s\in(s_1,s_2)}\spec(H(s)),\infty\bigg).
%
\end{equation}
Let $\varphi\in \calW_1(\bbR)\cap C^2(\bbR)$ be concave
on $\Lambda$ in the sense that
\begin{equation}
\label{5.10}
%
\varphi''\bigl|_{\Lambda} \bigr. \le 0,
%
\end{equation}
and
\begin{equation}
\label{5.11}
%
\varphi'(\lambda)=o(1) \text{ as }\lambda \to +\infty\,.
%
\end{equation}
Then the function
\begin{equation}
\label{vexx}
%
s\longmapsto\tr\bigl[\varphi\bigl(H(s)\bigr)-
\varphi(H_0)\bigr],
\quad s\in (s_1,s_2),
%
\end{equation}
is concave in the sense that for any $s,t\in (s_1,s_2)$ and for
any $0\le \alpha \le 1$,
\begin{align}
%
&\lefteqn{\tr\bigl[\varphi(H(\alpha s+(1-\alpha)t))-
\varphi(H_0)\bigr]}
\no \\
%
&\geq\alpha \,\tr\bigl[\varphi(H(s))-\varphi(H_0)\bigr]+
(1-\alpha)\,\tr\bigl[\varphi(H(t))-\varphi(H_0)\bigr]\,.
%
\end{align}
In particular,
\begin{equation}
\big[\varphi(H(\alpha s+(1-\alpha)t))- \alpha\,\varphi(H(s))-
(1-\alpha)\,\varphi(H_{t})\big]\in \calB_1(\calH)
\end{equation}
and
\begin{equation}
\tr \bigl[\varphi(H(\alpha s+(1-\alpha)t))- \alpha
\,\varphi(H(s))- (1-\alpha)\,\varphi(H(s))\bigr]\ge 0.
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
First, one observes that by \eqref{ksi1} the integrated spectral
shift function $\zeta(\lambda, s)$ given by \eqref{ssf} is
uniformly bounded, that is,
\begin{equation}
%
|\zeta(\lambda, s)|\le \|V(s)\|_{\calB_1(\calH)},
\quad
\lambda\in \bbR, \,\,\, s\in (s_1,s_2).
%
\end{equation}
Moreover, since $H_0$ is semibounded, one concludes that
$\bbR\backslash\Lambda\ne\emptyset$ by definition \eqref{5.9}
of the set $\Lambda$ and hence for all $s\in (s_1,s_2)$,
\begin{equation}
\label{5.15}
%
\zeta(\lambda,s)=0, \quad \lambda\in \bbR\backslash \Lambda.
%
\end{equation}
Thus, using \eqref{5.11} and \eqref{5.15}, one infers
\begin{equation}
\label{5.16}
%
\lim_{\lambda\to \pm \infty}
\,\varphi'(\lambda)\,\zeta(\lambda,s)=0,\quad s\in (s_1, s_2)\,.
%
\end{equation}
Next, combining \eqref{5.15} and \eqref{5.16}, an integration
by parts in the trace formula \eqref{5.1} yields
\begin{equation}
\label{5.13}
%
\tr\bigl[\varphi(H(s))-\varphi(H_0)\bigr]=-\int_{\Lambda}
d\lambda\, \varphi''(\lambda)\,\zeta(\lambda,s).
%
\end{equation}
Given $\lambda \in \bbR$, the integrated spectral shift function
$\zeta(\lambda, s)$ is concave with respect to $s\in (s_1,s_2)$ by
Corollary~\ref{c1.8} and hence the left-hand side of \eqref{5.13} is
also a concave function of $s$ by \eqref{5.10} (as a weighted
mean of concave functions with a positive weight).
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r3.7}
(i) If the measure $\nu$ in representation \eqref{5.2} is
absolutely continuous, then condition \eqref{5.11} holds
automatically by the Riemann--Lebesgue Lemma.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\smallskip
\noindent (ii) If $\varphi$ is convex on $\Lambda$, that is,
\begin{equation}
\varphi''\bigl|_\Lambda \bigr. \ge 0,
\end{equation}
then the function given by \eqref{vexx} is convex.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example} \lb{e3.8}
Under assumptions of Theorem~\ref{t3.6}, choosing
$\varphi\in\calW_1(\bbR)$ as
\begin{equation}
%
\varphi(\lambda)=\exp\big({-\lambda t}\big), \quad \lambda\in
\Lambda, \,\,t\ge0,
%
\end{equation}
one concludes that for any $t>0$,
\begin{equation}
%
s\longrightarrow\tr
%
\bigl[\exp\bigl(-tH(s)\bigr)-\exp\bigl(-tH_0\bigr)\bigr]
%
\end{equation}
is a convex function of $s\in (s_1,s_2)$.
\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\noindent {\bf Acknowledgments.} We are indebted to Vadim
Kostrykin for kindly making available to us
reference~\cite{Ko99} prior to its publication.
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