# The Hypocoercivity Index for the short time behavior of linear time-
invariant ODE systems

Franz Achleitner, Anton Arnold, and Eric A. Carlen TU Wien, Institute of
Analysis and Scientific Computing, Wiedner Hauptstr. 8-10, A-1040 Wien,
Austria<EMAIL_ADDRESS>Wien, Institute of Analysis and
Scientific Computing, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria,
<EMAIL_ADDRESS>of Mathematics, Rutgers University, 110
Frelinghuysen Rd., Piscataway NJ 08854, USA<EMAIL_ADDRESS>
###### Abstract

We consider the class of conservative-dissipative ODE systems, which is a
subclass of Lyapunov stable, linear time-invariant ODE systems. We
characterize asymptotically stable, conservative-dissipative ODE systems via
the hypocoercivity (theory) of their system matrices. Our main result is a
concise characterization of the hypocoercivity index (an algebraic structural
property of matrices with positive semi-definite Hermitian part introduced in
Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of
the norm of the matrix exponential for the associated conservative-dissipative
ODE system.

Keywords: semi-dissipative ODE systems, hypocoercivity (index)

MSC: 34D05, 34A30, 34Exx

##  1 Introduction

In this paper we shall use hypocoercivity techniques to characterize the short
time behavior of linear time-invariant ODE systems of the form

${\mathbf{x}}^{\prime}(t)=-B{\mathbf{x}}(t)\,,$ (1.1)

with matrices $B\in{\mathord{\mathbb{C}}}^{n\times n}$ whose Hermitian part
$B_{H}:=(B+B^{*})/2$ is positive semi-definite111We use the following
notation: The conjugate transpose of a matrix
$B\in{\mathord{\mathbb{C}}}^{n\times n}$ is denoted by $B^{*}$. Positive
definiteness (resp. semi-definiteness) of Hermitian matrices is denoted by
$B>0$ (resp. $B\geq 0$)., such that $-B$ is a conservative-dissipative (or
semi-dissipative) matrix, see Definition 1.1 below. An extension to  _stable
systems_ of the form (1.1), i.e. matrices $B$ having their spectrum in the
closed right half plane where, additionally, purely imaginary eigenvalues are
non-defective222An eigenvalue is non-defective if its algebraic and geometric
multiplicities coincide., are discussed in the parallel article [7].

Concerning the short time behavior of the propagator $P(t)$ of (1.1), given by
the fundamental matrix $P(t):=e^{-Bt}$, we shall be interested in estimates on
the spectral norm of the matrix exponential $e^{-Bt}$ of the form

$\|P(t)\|_{2}=1-c\,t^{a}+{\mathcal{O}}(t^{a+1})\quad\mbox{ for }t\to 0^{+}\,,$
(1.2)

where $c>0$ and $a\in{\mathord{\mathbb{N}}}$. For practical reason, we shall
refer to the spectral norm of the matrix exponential $e^{-Bt}$ as the
_propagator norm_. The large time behavior, including conditions guaranteeing
exponential decay estimates of the form

$\|P(t)\|_{2}\leq c\,e^{-\mu t},\qquad t\geq 0\,,$ (1.3)

with some $c\geq 1$, and $\mu>0$ will be discussed in the forthcoming article
[5].

Following [20], the matrix $B$ is called _hypocoercive_ if (1.3) holds, and
$B$ is _coercive_ if and only if $\|P(t)\|_{2}\leq e^{-\mu t}$ for some
$\mu>0$ and all $t\geq 0$. For the intermediate time regime we shall be
interested in a _waiting time_ $t_{0}$ such that all solutions will have
decayed (at least) by a factor $1/e$, i.e.

$\|P(t)\|_{2}\leq\frac{1}{e}\,,\qquad t\geq t_{0}\,.$ (1.4)

Here we shall include numerical evidence on this intermediate time regime; a
detailed analysis will be provided in [5].

Our interest in systems (1.1) is motivated by non-equilibrium statistical
physics and, in particular, kinetic theory. There, a frequently encountered
class of linear equations has the form

$\frac{\partial}{\partial t}f(t,x,v)=-v\cdot\nabla_{x}f(t,x,v)+Qf(t,x,v)\ ,$

where $Q$ is a linear operator, the so-called linearized collision operator,
describing changes in velocity resulting from binary collisions. Take the
domain of the $x$ variable to be a $d$–dimensional torus $\mathcal{T}$ of
side-length $L$. The operator $v\cdot\nabla_{x}$, called the streaming
operator is anti-Hermitian on a weighted $L^{2}$-space on
$\mathcal{T}\times{\mathord{\mathbb{R}}}^{d}$, whereas $Q$ is negative semi-
definite on the same weighted $L^{2}$-space. Concerning examples we refer to
[12, §1.4], [3, 4] where a modal decomposition (in $x$) of kinetic BGK-type
equations333named after the physicists Bhatnagar–Gross–Krook. led to ODE
systems like (1.1); in the case of continuous velocities it actually led to
“infinite matrices” $B$. In the follow-up paper [11] a spectral decomposition
of Fokker–Planck equations allowed for a precise analysis of their short time
behavior in the spirit of (1.2).

Very often, one is interested in initial data $f_{0}(x,v)$ for the equation
that are nearly constant on spatial scales much smaller than $L$, where $L$ is
large, so that $v\cdot\nabla_{x}f(t,x,v)$ will be of order $1/L$. Finite
dimensional approximations of such systems result in systems of ODEs of the
form (1.1), where its Hermitian part $B_{H}$ is positive semi-definite. Hence
it is often natural to write (1.1) in the form

${\mathbf{x}}^{\prime}(t)=-\epsilon A{\mathbf{x}}(t)-C{\mathbf{x}}(t)\,,$
(1.5)

where $\epsilon\in{\mathord{\mathbb{R}}}$,
$A\in{\mathord{\mathbb{C}}}^{n\times n}$ is anti-Hermitian, and
$C\in{\mathord{\mathbb{C}}}^{n\times n}$ is positive semi-definite Hermitian.
The systems (1.1) and (1.5) are related via $B_{A}=\epsilon A$ and $B_{H}=C$,
where $B_{A}:=(B-B^{*})/2$ denotes the anti-Hermitian part of $B$. One is then
led to study the asymptotics of the solution as $\epsilon\to 0$. This is often
referred to as _asymptotic limit_.

The main goal of this paper is to give a concise interpretation of the
_hypocoercivity index_ (an algebraic structural property of the matrix $B$,
introduced in [4]) in terms of the exponent $a$ in short time estimate (1.2).

The rest of this paper is organized as follows: In the remainder of §1 we
define and characterize _conservative-dissipative ODE systems_ and
_hypocoercive matrices_. In §2.1 we give three equivalent definitions of the
_hypocoercivity index_ of a matrix and three corresponding _Kalman rank
conditions_. Our main result, Theorem 2.7, gives a sharp characterization of
the hypocoercivity index of a matrix $B$ in terms of the short time decay of
the spectral norm of the associated matrix exponential $e^{-Bt}$. It is
presented in §2.2, while technical parts of the proof are deferred to A.
Finally, §2.3 gives a numerical illustration of Theorem 2.7.

###  1.1 Conservative-dissipative systems of ODEs

It is well known that the null solution ${\mathbf{x}}(t)\equiv 0$ of a linear
system (1.1) is _(Lyapunov) stable_ if all eigenvalues of $-B$ have non-
positive real part and the eigenvalues on the imaginary axis are non-
defective, and the null solution ${\mathbf{x}}(t)\equiv 0$ of (1.1) is
_asymptotically stable_ if all eigenvalues of $-B$ have negative real part.
For practical reasons, if the null solution ${\mathbf{x}}(t)\equiv 0$ of a
linear system (1.1) is (asymptotically) stable then we will call (1.1) an
(asymptotically) stable system.

Consider a linear system of ODEs (1.1) with matrix
$B\in{\mathord{\mathbb{C}}}^{n\times n}$. Then the derivative of the squared
Euclidean norm of a solution ${\mathbf{x}}(t)$ satisfies

$\frac{\,\operatorname{d}\\!missing}{\,\operatorname{d}\\!t}\|{\mathbf{x}}(t)\|_{2}^{2}=\langle-B{\mathbf{x}}(t),{\mathbf{x}}(t)\rangle+\langle{\mathbf{x}}(t),-B{\mathbf{x}}(t)\rangle=-2\langle{\mathbf{x}}(t),B_{H}{\mathbf{x}}(t)\rangle.$
(1.6)

Therefore a sufficient condition for $B$ to generate a stable system (1.1) is
that its Hermitian part is positive semi-definite444However, the Hermitian
part of a matrix $B$ pertaining to a stable system does not have to be
positive semi-definite; a different equivalent norm will generally yield a
different sufficient condition.. This fact and the importance of this subclass
of stable systems in kinetic theory inspires the following definition:

######  Definition 1.1.

A matrix $-B\in{\mathord{\mathbb{C}}}^{n\times n}$ is called _dissipative_
(resp. _conservative-dissipative_ or _semi-dissipative_) if the Hermitian part
of $-B$ is negative definite (resp. negative semi-definite).

For a (conservative-)dissipative matrix $-B\in{\mathord{\mathbb{C}}}^{n\times
n}$, the associated system of ODEs (1.1) is called a
_(conservative-)dissipative ODE system_.

For practical reasons, a matrix $B\in{\mathord{\mathbb{C}}}^{n\times n}$ is
called _positive conservative-dissipative_ if the Hermitian part of $B$ is
positive semi-definite.

This conservative-dissipative property of a matrix $B$ is invariant under
unitary transformations, but it is not invariant under similarity
transformations (see [7] for details). Let $\lambda^{B_{H}}_{min}$ and
$\lambda^{B_{H}}_{max}$ denote the least and the greatest eigenvalues of
$B_{H}$, respectively. By the Rayleigh-Ritz variational principle

$\lambda^{B_{H}}_{min}=\min_{\|{\mathbf{x}}\|_{2}=1}\langle{{\mathbf{x}}},{B_{H}{\mathbf{x}}}\rangle\qquad{\rm
and}\qquad\lambda^{B_{H}}_{max}=\max_{\|{\mathbf{x}}\|_{2}=1}\langle{{\mathbf{x}}},{B_{H}{\mathbf{x}}}\rangle\
.$

It follows immediately that

$\lambda^{B_{H}}_{min}\leq\min\\{\Re\lambda\,:\,\lambda\in\sigma(B)\\}\leq\max\\{\Re\lambda\,:\,\lambda\in\sigma(B)\\}\leq\lambda^{B_{H}}_{max}\,.$
(1.7)

###  1.2 Hypocoercive matrices.

In the introduction, matrices $B\in{\mathord{\mathbb{C}}}^{n\times n}$ are
called _hypocoercive_ if the associated propagator $P(t):=e^{-tB}$ satisfies
(1.3) for some $c\geq 1$ and $\mu>0$. In other words, $B$ is hypocoercive if
(the null-solution ${\mathbf{x}}(t)\equiv 0$ of) the associated linear system
(1.1) is exponentially stable. For linear time-invariant systems (1.1),
exponential stability is equivalent to asymptotic stability (i.e. all
solutions approach the origin in the large time limit), which is equivalent to
the condition that all eigenvalues of $-B$ have negative real part. Starting
with the classical notion of a coercive operator/matrix, we characterize
hypocoercive matrices as follows (see e.g. [4]):

######  Definition 1.2.

A matrix $B\in{\mathord{\mathbb{C}}}^{n\times n}$ is called _coercive_ if its
Hermitian part $B_{H}$ is positive definite, and it is called _hypocoercive_
if the spectrum of $B$ lies in the _open_ right half plane.

Hypocoercive matrices are often called _positive stable_. We use the notion of
hypocoercivity to emphasize the link to the analogous situation in partial
differential equations, see [4, 10, 20].

It is well-known, see (1.7), that for positive conservative-dissipative
matrices $B\in{\mathord{\mathbb{C}}}^{n\times n}$, the spectrum of $B$ lies in
the closed right half plane, but there may be purely imaginary eigenvalues. In
this case, the existence of purely imaginary eigenvalues can be characterized
as follows:

######  Proposition 1.3 ([18, Lemma 3.1], [4, Lemma 2.4 with Proposition
1(B2), (B4)]).

Let $B\in{\mathord{\mathbb{C}}}^{n\times n}$ be (positive) conservative-
dissipative. Then, $B$ has an eigenvalue on the imaginary axis if and only if
$B_{H}v=0$ for some eigenvector $v$ of $B_{A}$.

Hence, a positive conservative-dissipative matrix $B$ is hypocoercive if and
only if no eigenvector of the anti-Hermitian part lies in the kernel of the
Hermitian part. The latter condition is well known in control theory, and
there exists a range of equivalent characterizations, see e.g. [4, Proposition
1] and [18, Lemma 3.1]: For example, positive conservative-dissipative
matrices $B\in{\mathord{\mathbb{C}}}^{n\times n}$ are hypocoercive if and only
if

“No non-trivial subspace of $\ker(B_{H})$ is invariant under $B_{A}$.”  (1.8)

The characterization (1.8) implies the following result:

######  Lemma 1.4.

Let $B\in{\mathord{\mathbb{C}}}^{n\times n}$ be positive conservative-
dissipative. Then $B=B_{A}+B_{H}$ is hypocoercive if and only if $\epsilon
B_{A}+B_{H}$ is hypocoercive for all $\epsilon\neq 0$.

But even if hypocoercivity of $B$ is known, it is not trivial to obtain an
exponential decay estimate (1.3) with a quantitative (or even optimal) decay
rate $\mu$. Indeed, a simple energy estimate (i.e. pre-multiplying (1.1) by
${\mathbf{x}}^{*}$) or using Trotter’s product formula only yields
conservative-dissipativity of the system, but no decay: So, its propagator
$P(t)=e^{-Bt}$ satisfies at least $\|P(t)\|_{2}\leq 1$ for all $t\geq 0$.

Let us also comment on the short time behavior. If a decay estimate (1.2)
holds with some exponent $a>1$, then an estimate of the form $\|P(t)\|_{2}\leq
e^{-\mu t}$ (as it is typical for coercive matrices $B$) is impossible
(consider the Taylor expansion of $\|P(t)\|_{2}$ around $t=0$). In such cases
the system can only be hypocoercive, along with an estimate (1.3) with $c>1$.

Our conditions for hypocoercivity lead naturally to the notion of an index of
hypocoercivity, which is a non-negative integer whenever $B$ is hypocoercive,
and is zero if and only if $B$ is coercive (see §2.1).

##  2 Hypocoercivity index and the short time decay of conservative-
dissipative ODE systems

In this section we shall present our main result, i.e. a concise
characterization of the hypocoercivity index in terms of the short time
behavior of conservative-dissipative ODE systems (1.1).

###  2.1 Hypocoercivity index

First we recall from [4, §2.2] the definition of the hypocoercivity index:

######  Definition 2.1.

Let $B\in{\mathord{\mathbb{C}}}^{n\times n}$ be positive conservative-
dissipative. Its _hypocoercivity index (HC-index)_ ${m_{HC}}$ is defined as
the smallest integer $m\in{\mathord{\mathbb{N}}}_{0}$ (if it exists) such that
the matrix

$T_{m}:=\sum_{j=0}^{m}B_{A}^{j}B_{H}(B_{A}^{*})^{j}$ (2.1)

is positive definite. The matrix $B$ is coercive iff ${m_{HC}}=0$,
hypocoercive iff ${m_{HC}}\in{\mathord{\mathbb{N}}}_{0}$ (due to [4, Lemma
2.4]), and for non-hypocoercive matrices $B$ we set ${m_{HC}}=\infty$.

The Hermitian matrix $T_{m}$ in Definition 2.1 readily shows that the
hypocoercivity index of a positive conservative-dissipative matrix $B$ is
invariant under unitary congruence transformations but, in general, not under
similarity transformations (see [6] for details).

In [9, Lemma 2.3] it was proven that this index equals the smallest integer
$m\in{\mathord{\mathbb{N}}}_{0}$ such that

$\operatorname{rank}\big{\\{}\sqrt{B_{H}},\,B_{A}\sqrt{B_{H}},\ldots,B_{A}^{m}\sqrt{B_{H}}\big{\\}}=n\,,$
(2.2)

which is often called Kalman rank condition (see also [4, Proposition 1(B1)]).

######  Remark 2.2.

When considering rather $\epsilon B_{A}+B_{H}$, its hypocoercivity property
and its index of hypocoercivity are independent of $\epsilon\neq 0$, which
follows trivially from (2.2). Due to the above mentioned equality of the
indices, the positive definiteness of $\sum_{j=0}^{{m_{HC}}}(\epsilon
B_{A})^{j}B_{H}(\epsilon B_{A}^{*})^{j}$ is hence also independent of
$\epsilon\neq 0$.

Next we shall present four (equivalent) variants of the Kalman rank condition:

######  Lemma 2.3.

Let $B\in{\mathord{\mathbb{C}}}^{n\times n}$ be positive conservative-
dissipative. Consider the following four Kalman rank conditions:

$\displaystyle\exists m\in{\mathord{\mathbb{N}}}_{0}:$
$\displaystyle\quad\operatorname{rank}\\{\sqrt{B_{H}},\ B_{A}\sqrt{B_{H}},\
\ldots,\ B_{A}^{m}\sqrt{B_{H}}\\}=n\ ,$ (2.3a) $\displaystyle\exists
m\in{\mathord{\mathbb{N}}}_{0}:$
$\displaystyle\quad\operatorname{rank}\\{\sqrt{B_{H}},\ B\sqrt{B_{H}},\
\ldots,\ B^{m}\sqrt{B_{H}}\\}=n\ ,$ (2.3b) $\displaystyle\exists
m\in{\mathord{\mathbb{N}}}_{0}:$
$\displaystyle\quad\operatorname{rank}\\{\sqrt{B_{H}},\ B^{*}\sqrt{B_{H}},\
\ldots,\ (B^{*})^{m}\sqrt{B_{H}}\\}=n\ ,$ (2.3c) $\displaystyle\exists
m\in{\mathord{\mathbb{N}}}_{0}:$
$\displaystyle\quad\operatorname{rank}\\{C_{0},C_{1},\ \ldots,\
C_{m}\\}=n\text{ with }C_{0}:=\sqrt{B_{H}};\
C_{j+1}:=[C_{j},B_{A}]=C_{j}B_{A}-B_{A}C_{j},\ j\in{\mathord{\mathbb{N}}}_{0}\
.$ (2.3d) The conditions (2.3a)–(2.3d) are equivalent in the sense that, if
there exists $m\in{\mathord{\mathbb{N}}}_{0}$ such that one condition holds,
then the other three conditions hold as well for the same $m$.

###### Proof.

In fact, we prove that for all $m\in{\mathord{\mathbb{N}}}_{0}$ the ranges of
all four matrices in (2.3) are equal,

$\displaystyle\operatorname{range}\\{\sqrt{B_{H}},\ B_{A}\sqrt{B_{H}},\
\ldots,\ B_{A}^{m}\sqrt{B_{H}}\\}$
$\displaystyle=\operatorname{range}\\{\sqrt{B_{H}},\ B\sqrt{B_{H}},\ \ldots,\
B^{m}\sqrt{B_{H}}\\}$ (2.4a)
$\displaystyle=\operatorname{range}\\{\sqrt{B_{H}},\ B^{*}\sqrt{B_{H}},\
\ldots,\ (B^{*})^{m}\sqrt{B_{H}}\\}$ (2.4b)
$\displaystyle=\operatorname{range}\\{C_{0},C_{1},\ \ldots,\ C_{m}\\},$ (2.4c)

which can be done inductively:

The base case $m=0$ is trivial, since all four matrices in (2.4) are equal to
$\sqrt{B_{H}}$. We start with the induction step for the equivalence of the
first identity in (2.4): Assume (2.4a) holds for some
$m\in{\mathord{\mathbb{N}}}_{0}$. Then, the following representations hold:
$(B^{m+1}-B_{A}^{m+1})\sqrt{B_{H}}=\sum_{j=0}^{m}B_{A}^{j}\sqrt{B_{H}}X_{j}$,
with appropriate matrices $X_{j}\in{\mathord{\mathbb{C}}}^{n\times n}$, and
e.g., $X_{m}=B_{H}$. Therefore

$\operatorname{range}\big{(}(B^{m+1}-B_{A}^{m+1})\sqrt{B_{H}}\big{)}\subset\operatorname{range}\\{\sqrt{B_{H}},\
B_{A}\sqrt{B_{H}},\ \ldots,\ B_{A}^{m}\sqrt{B_{H}}\\}$

which implies

$\displaystyle\operatorname{range}\\{\sqrt{B_{H}},\ B\sqrt{B_{H}},\ \ldots,\
B^{m}\sqrt{B_{H}},\ B^{m+1}\sqrt{B_{H}}\\}$
$\displaystyle=\operatorname{range}\\{\sqrt{B_{H}},\ B_{A}\sqrt{B_{H}},\
\ldots,\ B_{A}^{m}\sqrt{B_{H}},\ B^{m+1}\sqrt{B_{H}}\\}$
$\displaystyle=\operatorname{range}\\{\sqrt{B_{H}},\ B_{A}\sqrt{B_{H}},\
\ldots,\ B_{A}^{m}\sqrt{B_{H}},\ B_{A}^{m+1}\sqrt{B_{H}}\\}\,.$

Hence, $\operatorname{range}\\{\sqrt{B_{H}},\ B_{A}\sqrt{B_{H}},\ \ldots,\
B_{A}^{m+1}\sqrt{B_{H}}\\}=\operatorname{range}\\{\sqrt{B_{H}},\
B\sqrt{B_{H}},\ \ldots,\ B^{m+1}\sqrt{B_{H}}\\}$ follows.

In the same way, for $m\in{\mathord{\mathbb{N}}}_{0}$, the identity

$\operatorname{range}\\{\sqrt{B_{H}},\ B_{A}\sqrt{B_{H}},\ \ldots,\
B_{A}^{m}\sqrt{B_{H}}\\}=\operatorname{range}\\{\sqrt{B_{H}},\
B^{*}\sqrt{B_{H}},\ \ldots,\ (B^{*})^{m}\sqrt{B_{H}}\\}$

is proven.

Since the equivalence of $\operatorname{range}\\{\sqrt{B_{H}},\
B_{A}\sqrt{B_{H}},\ \ldots,\ B_{A}^{m}\sqrt{B_{H}}\\}$ and
$\operatorname{range}\\{C_{0},C_{1},\ \ldots,\ C_{m}\\}$ follows very
similarly, we only give the key steps: Note that the term of $C_{j+1}$ with
the leading maximum exponent of $B_{A}$ is of the form
$(-B_{A})^{j+1}\sqrt{B}_{H}$. This allows to compute
$C_{m+1}-(-B_{A})^{m+1}\sqrt{B_{H}}=\sum_{j=0}^{m}B_{A}^{j}\sqrt{B_{H}}X_{j}$,
with appropriate matrices $X_{j}\in{\mathord{\mathbb{C}}}^{n\times n}$ and,
e.g., $X_{m}=(-1)^{m}(m+1)B_{A}$. Therefore

$\operatorname{range}\big{(}C_{m+1}-(-B_{A})^{m+1}\sqrt{B_{H}}\big{)}\subset\operatorname{range}\\{\sqrt{B_{H}},\
B_{A}\sqrt{B_{H}},\ \ldots,\ B_{A}^{m}\sqrt{B_{H}}\\}$

and the rest of the proof is as before. ∎

Next we shall present two (equivalent) variants of Definition 2.1 for the
hypocoercivity index that are related to similar concepts in the literature:
In [10, §2], the definition of the hypocoercivity index of a degenerate
Fokker–Planck equation with linear drift involves a positive stable matrix $B$
and its Hermitian part $B_{H}\geq 0$ (rather than the Hermitian part and anti-
Hermitian part of $B$ as in Definition 2.1 here). In order to connect these
two situations we shall establish the equivalence of these two definitions in
the subsequent lemma.

The condition $T_{m}>0$ from Definition 2.1 can also be related to
_Hörmander’s “rank $r$” bracket condition_ for hypoellipticity, cf. [15]. In
particular, in [20] iterated commutators were used to establish hypocoercivity
of kinetic PDEs by constructing an appropriate Lyapunov functional. In Lemma
2.4, Equation (2.5d), below we shall mimic condition (3.5) of [20] for the
ODE-system (1.1).

######  Lemma 2.4.

Let $B\in{\mathord{\mathbb{C}}}^{n\times n}$ be positive conservative-
dissipative. Consider the following four hypocoercivity conditions:

$\displaystyle\exists m\in{\mathord{\mathbb{N}}}_{0}:$ $\displaystyle\quad
T_{m}:=\sum_{j=0}^{m}B_{A}^{j}B_{H}(B_{A}^{*})^{j}>0\ ,$ (2.5a)
$\displaystyle\exists m\in{\mathord{\mathbb{N}}}_{0}:$
$\displaystyle\quad\widetilde{T}_{m}:=\sum_{j=0}^{m}B^{j}B_{H}(B^{*})^{j}>0\
,$ (2.5b) $\displaystyle\exists m\in{\mathord{\mathbb{N}}}_{0}:$
$\displaystyle\quad\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{m}:=\sum_{j=0}^{m}(B^{*})^{j}B_{H}B^{j}>0\
,$ (2.5c) $\displaystyle\exists m\in{\mathord{\mathbb{N}}}_{0}:$
$\displaystyle\quad\widehat{T}_{m}:=\sum_{j=0}^{m}C^{*}_{j}C_{j}>0\text{ with
}C_{0}:=\sqrt{B_{H}};\ C_{j+1}:=[C_{j},B_{A}]=C_{j}B_{A}-B_{A}C_{j},\
j\in{\mathord{\mathbb{N}}}_{0}\ .$ (2.5d)

The conditions (2.5a)–(2.5d) are equivalent in the sense that, if there exists
$m\in{\mathord{\mathbb{N}}}_{0}$ such that one condition holds, then the other
three conditions hold as well for the same $m$.

###### Proof.

According to [9, Lemma 2.3], each of these four matrix inequalities
(2.5a)–(2.5d) is equivalent to the corresponding Kalman rank conditions
(2.3a)–(2.3d) where we used in the last case that $C_{j}^{*}=C_{j}$
(verifiable by a simple induction). Moreover, the Kalman rank conditions
(2.3a)–(2.3d) are equivalent due to Lemma 2.3. ∎

This lemma shows that the hypocoercivity index can equally be defined as the
smallest integer $m\in{\mathord{\mathbb{N}}}_{0}$ such that
$\widetilde{T}_{m}>0$,
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{m}>0$, or
$\widehat{T}_{m}>0$. Hence, it also gives the smallest necessary number of
iterated commutators of $B_{H}\geq 0$ with the matrix $B_{A}$ such that their
ranges span all of ${\mathord{\mathbb{C}}}^{n}$ — in the spirit of Hörmander’s
hypoellipticity theorem.

As an example we consider two matrices with the same Hermitian part
$B_{H}=\operatorname{diag}(0,0,1,1)$ such that
$\operatorname{rank}(\ker(B_{H}))=2$ but two different anti-Hermitian parts
such that one and, respectively, two iterated commutators are needed:

######  Example 2.5.

Consider

$\displaystyle B^{(1)}$ $\displaystyle=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\
0&-1&1&0\\\ -1&0&0&1\end{pmatrix}\text{ such that }$ $\displaystyle
B_{A}^{(1)}$ $\displaystyle=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&-1&0&0\\\
-1&0&0&0\end{pmatrix},\quad$
$\displaystyle\big{[}\sqrt{B_{H}^{(1)}},B_{A}^{(1)}\big{]}$
$\displaystyle=\begin{pmatrix}0&0&0&-1\\\ 0&0&-1&0\\\ 0&-1&0&0\\\
-1&0&0&0\end{pmatrix},$ and $\displaystyle B^{(2)}$
$\displaystyle=\begin{pmatrix}0&1&0&0\\\ -1&0&1&0\\\ 0&-1&0&0\\\
0&0&0&0\end{pmatrix}\text{ such that }$ $\displaystyle B_{A}^{(2)}$
$\displaystyle=\begin{pmatrix}0&1&0&0\\\ -1&0&1&0\\\ 0&-1&0&0\\\
0&0&0&0\end{pmatrix},$
$\displaystyle\big{[}\sqrt{B_{H}^{(2)}},B_{A}^{(2)}\big{]}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&-1&0\\\ 0&-1&0&0\\\
0&0&0&0\end{pmatrix}.$

Hence
$\operatorname{rank}\big{\\{}\sqrt{B_{H}^{(1)}},\,\big{[}\sqrt{B_{H}^{(1)}},B_{A}^{(1)}\big{]}\big{\\}}=4$,
but
$\operatorname{rank}\big{\\{}\sqrt{B_{H}^{(2)}},\,\big{[}\sqrt{B_{H}^{(2)}},B_{A}^{(2)}\big{]}\big{\\}}=3$.
Thus, by Lemma 2.3, ${m_{HC}}(B^{(1)})=1$ and ${m_{HC}}(B^{(2)})=2$.

While finite dimensional positive conservative-dissipative matrices are
hypocoercive if and only if they have a finite hypocoercivity index, this is
not true in the infinite dimensional case:

######  Example 2.6.

Consider a block-diagonal “ODE”, with each block of the form

$E_{k}=\begin{pmatrix}0&1&&&\\\ -1&\ddots&\ddots&&\\\ &\ddots&\ddots&1&\\\
&&-1&0&1\\\ &&&-1&1\end{pmatrix}\in{\mathord{\mathbb{R}}}^{k\times k}\
,\quad(E_{k})_{H}=\operatorname{diag}(0,0,\ldots,0,1)\
,\quad(E_{k})_{A}=\operatorname{toeplitz}(-1,0,1)\ .$ (2.6)

Then, the matrices $E_{k}$, $k\in{\mathord{\mathbb{N}}}$ are positive
conservative-dissipative. Evidently, $(E_{k})_{H}{\mathbf{v}}=0$ if and only
if ${\mathbf{v}}=(v_{1},\dots,v_{k-1},0)$. For such a vector,

$(E_{k})_{A}{\mathbf{v}}=(v_{2},v_{3}-v_{1},\dots,-v_{k-2},-v_{k-1})\ .$

A simple inductive argument shows that if
$(E_{k})_{A}{\mathbf{v}}=\lambda{\mathbf{v}}$ for any $\lambda\neq 0$, then
${\mathbf{v}}=0$. By Proposition 1.3, $E_{k}$ has no eigenvalue on
$i{\mathord{\mathbb{R}}}$. Hence, the matrices $E_{k}$,
$k\in{\mathord{\mathbb{N}}}$ are hypocoercive, satisfying the estimate

$\|e^{-E_{k}t}\|_{2}\leq c_{k}e^{-\lambda_{k}t}\ ,\qquad t\geq 0\ .$ (2.7)

for constants $c_{k}\geq 1$ and $\mu_{k}>0$, $k\in{\mathord{\mathbb{N}}}$. We
now make a simple rescaling, defining $\widetilde{E}_{k}:=r_{k}E_{k}$ for
$r_{k}>0$ to be chosen below, and consider the propagator $P(t)$ for
$\operatorname{diag}(\widetilde{E}_{1},\widetilde{E}_{2},\ldots)$, and let
$\|\cdot\|_{2}$ be the spectral norm on $\ell^{2}$. By (2.7),

$\|e^{-\widetilde{E}_{k}}\|_{2}=\|e^{-r_{k}E_{k}}\|_{2}\leq
c_{k}e^{-r_{k}\mu_{k}}\leq\frac{1}{e}\quad\text{for}\quad r_{k}=\frac{1+\log
c_{k}}{\mu_{k}}\ .$

Therefore, making this choice of $r_{k}$, $\|P(1)\|_{2}\leq 1/e$, so that
(1.4) is satisfied for $t_{0}=1$. Thus this infinite dimensional system is
hypocoercive.

We now show that the combined system has an infinite hypocoercivity index. For
this we can ignore the scaling and work with the original matrices $E_{k}$.
Let ${\mathbf{e}}_{j}$ denote the $j$-th standard basis vector in
${\mathord{\mathbb{C}}}^{k}$. Then,
$(E_{k})_{H}={\mathbf{e}}_{k}{\mathbf{e}}_{k}^{*}$ and $T_{m}$ as specified in
(2.1) is given by

$T_{m}=\sum_{j=0}^{m}(-1)^{j}(E_{k})_{A}^{j}{\mathbf{e}}_{k}{\mathbf{e}}_{k}^{*}(E_{k})_{A}^{j}\
.$

For $m=1,\ldots,k-1$, it is evident that

$|{\mathbf{e}}_{m}^{*}(E_{k})_{A}^{j}{\mathbf{e}}_{k}|^{2}=0\quad\text{for}\quad
j<k-m\quad\text{but}\quad|{\mathbf{e}}_{m}^{*}(E_{k})_{A}^{k-m}{\mathbf{e}}_{k}|^{2}=1\
.$

Hence, ${m_{HC}}(E_{k})=k-1$. Thus, the combined system
$\operatorname{diag}(E_{1},E_{2},\ldots)$ can not have a finite ${m_{HC}}$.

###  2.2 Short time decay of conservative-dissipative ODE systems

Here we shall prove that the hypocoercivity index of a conservative-
dissipative ODE system characterizes the decay of its propagator norm for
short time. We denote the solution semigroup pertaining to (1.1) by
$P(t):=e^{-Bt}\in{\mathord{\mathbb{C}}}^{n\times n}$, and its spectral norm by
$\|P(t)\|_{2}:=\sup\\{\|P(t){\bf x}\|_{2}\ :\ \|{\bf x}\|_{2}=1\\}$, which is
also the largest singular value of $P(t)$. Its short time decay is related to
the hypocoercivity index as follows:

######  Theorem 2.7.

Let the ODE system (1.1) be conservative-dissipative with (positive
conservative-dissipative) matrix $B\in{\mathord{\mathbb{C}}}^{n\times n}$.

  1. (a)

The (positive conservative-dissipative) matrix $B$ is hypocoercive (with
hypocoercivity index ${m_{HC}}\in{\mathord{\mathbb{N}}}_{0}$) if and only if

$\|e^{-Bt}\|_{2}=1-ct^{a}+{\mathcal{O}}(t^{a+1})\quad\text{ for
}t\in[0,\epsilon),$ (2.8)

for some $a,c,\epsilon>0$. In this case, necessarily $a=2m_{HC}+1$.

  2. (b)

Moreover, for ${m_{HC}}\geq 1$, the optimal multiplicative factor $c$ in (2.8)
is given by

$\begin{split}c&=\frac{1}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}\min_{{\mathbf{x}}\in\ker\big{(}\widetilde{\widetilde{T}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}\|=1}\langle{{\mathbf{x}}},{(B^{*})^{m_{HC}}B_{H}B^{m_{HC}}{\mathbf{x}}}\rangle\\\
&=\frac{1}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}\min_{{\mathbf{x}}\in\ker\big{(}\widetilde{\widetilde{T}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}\|=1}\langle{{\mathbf{x}}},{(B_{A}^{*})^{m_{HC}}B_{H}B_{A}^{m_{HC}}{\mathbf{x}}}\rangle,\end{split}$
(2.9)

and for ${m_{HC}}=0$ we have
$c=\min_{\|{\mathbf{x}}\|=1}\langle{{\mathbf{x}}},{B_{H}{\mathbf{x}}}\rangle$.

###### Proof.

Since the proof for the coercive case, i.e. ${m_{HC}}=0$, is trivial, we shall
confine ourselves now to ${m_{HC}}\geq 1$:

Part (a): For sufficiently small time $t_{0}>0$, there exists a real analytic
function $\Phi:[0,t_{0}]\to{\mathord{\mathbb{R}}}$ such that

$\|P(t)\|_{2}=\Phi(t)\quad\mbox{ for all }t\in[0,t_{0}]\,,$ (2.10)

e.g. see [17, Lemma 1]. Alternatively, the statement can be derived from [16,
Part II.§6] or [14, Theorem 4.3.17].

For the forward direction we assume that $B$ has a finite HC-index
$m_{HC}\in{\mathord{\mathbb{N}}}$. Hence we are left to prove that the Taylor
expansion of $\|P(t)\|_{2}$ has the form (2.8). This proof will be split into
two separate parts, the lower and the (technically more subtle) upper bound.

Lower bound: First, we shall prove that there exists $c_{1}>0$ such that

$\|P(t)\|_{2}\geq 1-c_{1}t^{a}+{\mathcal{O}}(t^{a+1})\quad\text{as }t\to
0^{+}\ $ (2.11a) or, equivalently, $\|P(t)\|_{2}^{2}\geq
1-2c_{1}t^{a}+{\mathcal{O}}(t^{a+1})\quad\text{as }t\to 0^{+}\ $ (2.11b)

holds with $a=2{m_{HC}}+1$. 555In inequalities, such as (2.11), the Landau
symbol ${\mathcal{O}}$ is used in the sense that there exist constants
$\epsilon,M>0$ such that $\|P(t)\|_{2}\geq 1-c_{1}t^{a}-M\ t^{a+1}$ for
$t\in[0,\epsilon)$. To this end suppose ${\mathbf{x}}_{0}$ is any unit vector
such that for some $m\in{\mathord{\mathbb{N}}}$ (_not_ necessarily
$m=m_{HC}$),

${\mathbf{x}}_{0}\in\ker\left(\sum_{j=0}^{m-1}(B^{*})^{j}B_{H}B^{j}\right)=\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{m-1}\big{)}\,,$
(2.12)

which is equivalent to

$\|\sqrt{B_{H}}B^{j}{\mathbf{x}}_{0}\|_{2}=0\quad{\rm for}\quad 0\leq j\leq
m-1\ .$ (2.13)

If such an ${\mathbf{x}}_{0}$ exists, we will show for the corresponding
trajectory that

$\|P(t){\mathbf{x}}_{0}\|_{2}^{2}=1-t^{2m+1}\frac{2}{(2m+1)!}\binom{2m}{m}\langle{{\mathbf{x}}_{0}},{(B^{*})^{m}B_{H}B^{m}{\mathbf{x}}_{0}}\rangle+\mathcal{O}(t^{2m+2})\
.$ (2.14)

Note that our hypotheses allow the possibility that
$\langle{\mathbf{x}}_{0},(B^{*})^{m}B_{H}B^{m}{\mathbf{x}}_{0}\rangle=0$ in
which case we have simply
$\|P(t){\mathbf{x}}_{0}\|_{2}^{2}=1+\mathcal{O}(t^{2m+2})$.

Now take ${\mathbf{x}}_{0}$ as in Equation (2.12) with
$\|{\mathbf{x}}_{0}\|=1$, and define $q(t):=\|P(t){\mathbf{x}}_{0}\|^{2}$.
Standard theory gives that (see also Equation (1.6))

${q}^{\prime}(t)=-2\|\sqrt{B_{H}}P(t){\mathbf{x}}_{0}\|^{2}=-2\langle{\sqrt{B_{H}}P(t){\mathbf{x}}_{0}},{\sqrt{B_{H}}P(t){\mathbf{x}}_{0}}\rangle.$
(2.15)

Using the assumption on ${\mathbf{x}}_{0}$, we have that
$\sqrt{B_{H}}P(t){\mathbf{x}}_{0}=\sum_{k=m}^{\infty}\frac{t^{k}}{k!}\sqrt{B_{H}}(-B)^{k}{\mathbf{x}}_{0}$.
Substituting this in (2.15) gives

${q}^{\prime}(t)=-2\frac{t^{2m}}{m!\cdot
m!}\langle{\sqrt{B_{H}}B^{m}{\mathbf{x}}_{0}},{\sqrt{B_{H}}B^{m}{\mathbf{x}}_{0}}\rangle+\sum_{k=2m+1}^{\infty}u_{k}t^{k}$
(2.16)

for some scalars $u_{k}$. Since this sequence converges absolutely (The
expression in (2.15) is real analytic), we can integrate this expression term-
wise to obtain $q(t)$. Using that $q(0)=\|{\mathbf{x}}_{0}\|^{2}=1$, we find

$q(t)=1-\frac{2}{2m+1}\frac{t^{2m+1}}{m!\cdot
m!}\langle{\sqrt{B_{H}}B^{m}{\mathbf{x}}_{0}},{\sqrt{B_{H}}B^{m}{\mathbf{x}}_{0}}\rangle+\sum_{k=2m+1}^{\infty}\frac{u_{k}}{k+1}t^{k+1},$
(2.17)

which shows (2.14).

Now, since the positive conservative-dissipative matrix $B$ is hypocoercive
with HC-index ${m_{HC}}\in{\mathord{\mathbb{N}}}_{0}$, there exists a
normalized vector ${\mathbf{x}}_{0}$ such that

$\|\sqrt{B_{H}}B^{j}{\mathbf{x}}_{0}\|_{2}=0\quad\text{for }0\leq
j\leq{m_{HC}}-1,\quad\text{and }\sqrt{B_{H}}B^{{m_{HC}}}{\mathbf{x}}_{0}\neq
0\,,$ (2.18)

but none that would satisfy instead also
$\sqrt{B_{H}}B^{{m_{HC}}}{\mathbf{x}}_{0}=0$. Hence, for such
${\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}\big{)}$
the identity (2.14) holds with $m={m_{HC}}$. Finally, taking the supremum over
all initial conditions
${\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}_{0}\|_{2}=1$ (which is a closed set) yields the estimates
(2.11a) and (2.11b) with $a=2{m_{HC}}+1$ and

$\begin{split}c_{1}:=&\frac{1}{2}\min_{{\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\widetilde{T}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}_{0}\|_{2}=1}\tfrac{2}{(2{m_{HC}}+1)!}\binom{2{m_{HC}}}{{m_{HC}}}\langle{{\mathbf{x}}_{0}},{(B^{*})^{{m_{HC}}}B_{H}B^{m_{HC}}{\mathbf{x}}_{0}}\rangle\\\
=&\frac{1}{(2{m_{HC}}+1)!}\binom{2{m_{HC}}}{{m_{HC}}}\min_{{\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\widetilde{T}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}_{0}\|_{2}=1}\langle{{\mathbf{x}}_{0}},{(B_{A}^{*})^{{m_{HC}}}B_{H}B_{A}^{m_{HC}}{\mathbf{x}}_{0}}\rangle\
.\end{split}$ (2.19)

In the last step we used

$B^{{m_{HC}}}{\mathbf{x}}_{0}=B_{A}B^{{m_{HC}}-1}{\mathbf{x}}_{0}=\ldots=B_{A}^{{m_{HC}}}{\mathbf{x}}_{0}\,,$
(2.20)

since all other terms vanish due to
${\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}\big{)}$.
Due to (2.18), $c_{1}>0$.

In the proof of Part (b) we shall actually improve this constant by the factor
$\binom{2{m_{HC}}}{{m_{HC}}}^{-2}$, see Lemma A.3.

Upper bound: Second, we prove that there exists $c_{2}>0$ such that

$\|P(t)\|_{2}\leq 1-c_{2}t^{a}+{\mathcal{O}}(t^{a+1})\quad\text{as }t\to
0^{+}\ $ (2.21a) or, equivalently, $\|P(t)\|_{2}^{2}\leq
1-2c_{2}t^{a}+{\mathcal{O}}(t^{a+1})\quad\text{as }t\to 0^{+}\ $ (2.21b)

holds with $a=2{m_{HC}}+1$. 666In inequalities, such as (2.21), the Landau
symbol ${\mathcal{O}}$ is used in the sense that there exist constants
$\epsilon,M>0$ such that $\|P(t)\|_{2}\leq 1-c_{2}t^{a}+M\ t^{a+1}$ for
$t\in[0,\epsilon)$. Here, we consider the case ${m_{HC}}=1$ whereas the
general case ${m_{HC}}\in{\mathord{\mathbb{N}}}$ is treated in Appendix A.

If matrix $B$ has hypocoercivity index ${m_{HC}}=1$, there exists $\kappa>0$
such that

$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{m_{HC}}=\sum_{j=0}^{{m_{HC}}}(B^{*})^{j}B_{H}B^{j}=B_{H}+B^{*}B_{H}B\geq\kappa
I>0\,.$ (2.22)

Since ${m_{HC}}=1$, $B^{*}B_{H}B$ is positive definite on $\ker(B_{H})$. For
${\mathbf{x}}\in{\mathord{\mathbb{C}}}^{n}$ with $\|{\mathbf{x}}\|=1$, we
define

$\lambda_{\mathbf{x}}:=\langle{{\mathbf{x}}},{B_{H}{\mathbf{x}}}\rangle\geq 0\
,\quad\mu_{\mathbf{x}}:=\langle{{\mathbf{x}}},{B^{*}B_{H}B{\mathbf{x}}}\rangle\geq
0\ ,\quad\text{such that }\lambda_{\mathbf{x}}+\mu_{\mathbf{x}}\geq\kappa>0\
.$ (2.23)

Note that
$\|P(t){\mathbf{x}}_{0}\|_{2}^{2}=\langle{{\mathbf{x}}_{0}},{Q(t){\mathbf{x}}_{0}}\rangle$
where

$Q(t):=P^{*}(t)\,P(t)=e^{-B^{*}t}\,e^{-Bt}=\sum_{j=0}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j}\binom{j}{k}(-B^{*})^{k}(-B)^{j-k}\
.$ (2.24)

Since
${\big{\|}\sum_{k=0}^{j}\binom{j}{k}(-B^{*})^{k}(-B)^{j-k}\big{\|}_{2}}\leq(2\|B\|_{2})^{j}$,
the Taylor series for the matrix family $Q(t)$ converges uniformly on bounded
$t$-intervals. Hence we have

$\|P(t)\|_{2}^{2}=\|Q(t)\|_{2}=\|Q_{j}(t)\|_{2}+{\mathcal{O}}(t^{j+1})\,,$
(2.25)

where $Q_{j}(t)$ denotes the Taylor expansion (2.24), but truncated after the
$t^{j}$-term. Consider

$Q_{3}(t)=\sum_{j=0}^{3}\frac{t^{j}}{j!}\
U_{j}=I+tU_{1}+\frac{t^{2}}{2!}U_{2}+\frac{t^{3}}{3!}U_{3}.$ (2.26)

Then, its spectral norm satisfies

$\|Q_{3}(t)\|_{2}:=\sup_{\|{\mathbf{x}}_{0}\|=1}\|Q_{3}(t){\mathbf{x}}_{0}\|_{2}=\sup_{\|{\mathbf{x}}_{0}\|=1}\langle{{\mathbf{x}}_{0}},{Q_{3}(t){\mathbf{x}}_{0}}\rangle,$
(2.27)

for sufficiently small $t\geq 0$. The latter identity holds, since the self-
adjoint matrix $Q_{3}(t)$ satisfies $Q_{3}(0)=I$.

Let ${\mathbf{x}}$ be a unit vector, we estimate each expression
$\langle{{\mathbf{x}}},{U_{j}{\mathbf{x}}}\rangle$, $j=1,2,3$ separately as

$\langle{{\mathbf{x}}},{U_{1}{\mathbf{x}}}\rangle=-2\langle{{\mathbf{x}}},{B_{H}{\mathbf{x}}}\rangle=-2\lambda_{\mathbf{x}}\quad\text{using}\quad\lambda_{\mathbf{x}}=\langle{{\mathbf{x}}},{B_{H}{\mathbf{x}}}\rangle\geq
0\ .$

Next, since $U_{2}=2(B^{*}B_{H}+B_{H}B)$, so that by the Cauchy–Schwarz
inequality,

$\langle{{\mathbf{x}}},{U_{2}{\mathbf{x}}}\rangle=2\langle{{\mathbf{x}}},{(B^{*}B_{H}+B_{H}B){\mathbf{x}}}\rangle\leq
4\|\sqrt{B_{H}}{\mathbf{x}}\|\|\sqrt{B_{H}}B{\mathbf{x}}\|=4\sqrt{\lambda_{\mathbf{x}}}\sqrt{\mu_{\mathbf{x}}}$

using (2.23). In the same way, we derive for
$U_{3}=-2(B_{H}B^{2}+2B^{*}B_{H}B+(B^{*})^{2}B_{H})$ that

$\langle{{\mathbf{x}}},{U_{3}{\mathbf{x}}}\rangle\leq-4\langle{{\mathbf{x}}},{B^{*}B_{H}B{\mathbf{x}}}\rangle-2\langle{{\mathbf{x}}},{(B_{H}B^{2}+(B^{*})^{2}B_{H}){\mathbf{x}}}\rangle\\\
\leq-4\mu_{\mathbf{x}}+4\|{\sqrt{B_{H}}{\mathbf{x}}}\|\|{\sqrt{B_{H}}B^{2}{\mathbf{x}}}\|=-4\mu_{\mathbf{x}}+4\sqrt{\lambda_{\mathbf{x}}}\|{\sqrt{B_{H}}B^{2}{\mathbf{x}}}\|.$

Altogether then,

$\begin{split}\langle{\mathbf{x}},Q_{3}(t){\mathbf{x}}\rangle&\leq
1-2\lambda_{\mathbf{x}}\
t+2\sqrt{\lambda_{\mathbf{x}}}\sqrt{\mu_{\mathbf{x}}}\
t^{2}-\frac{2}{3}\mu_{\mathbf{x}}\
t^{3}+\frac{2}{3}\sqrt{\lambda_{\mathbf{x}}}\|{\sqrt{B_{H}}B^{2}}\|\ t^{3}\\\
&=1-2t\Big{(}\sqrt{\lambda_{\mathbf{x}}}-\frac{\sqrt{\mu_{\mathbf{x}}}}{2}t\Big{)}^{2}-\frac{1}{6}\mu_{\mathbf{x}}\
t^{3}+\frac{2}{3}\sqrt{\lambda_{\mathbf{x}}}\|{\sqrt{B_{H}}B^{2}}\|\
t^{3}.\end{split}$ (2.28)

To estimate (2.28), we will distinguish two cases for
$\sqrt{\lambda_{\mathbf{x}}}\in[0,\|\sqrt{B_{H}}\|]$: If
$\sqrt{\lambda_{\mathbf{x}}}$ is sufficiently small we will use the third term
in (2.28) to compensate the non-negative fourth term in (2.28), while in the
other case we will use the second term to do so.

Case a: If

$\sqrt{\lambda_{\mathbf{x}}}\leq\alpha\quad\text{with }\alpha:=\min\Bigg{(}1,\
\frac{\kappa}{2\big{(}1+4\|{\sqrt{B_{H}}B^{2}}\|\big{)}}\Bigg{)}>0,$ (2.29)

then the following estimates hold: Due to (2.23) and
$\sqrt{\lambda_{\mathbf{x}}}\leq 1$ we deduce that
$\mu_{\mathbf{x}}\geq\kappa-\lambda_{\mathbf{x}}\geq\kappa-\sqrt{\lambda_{\mathbf{x}}}$
which implies that

$\langle{\mathbf{x}},Q_{3}(t){\mathbf{x}}\rangle\leq
1-\frac{1}{6}\mu_{\mathbf{x}}\
t^{3}+\frac{2}{3}\sqrt{\lambda_{\mathbf{x}}}\|{\sqrt{B_{H}}B^{2}}\|\ t^{3}\leq
1-\frac{\kappa}{12}\ t^{3},\qquad\text{for }t\geq 0\ .$ (2.30)

Case b: If $\alpha\leq\sqrt{\lambda_{\mathbf{x}}}\leq\|{\sqrt{B_{H}}}\|$ then
we restrict the time interval to obtain a similar estimate. Using $\alpha$
from (2.29), define

$t_{1}:=\frac{\alpha}{\sqrt{\|{B^{*}B_{H}B}\|}}.$ (2.31)

For $t\in[0,t_{1}]$, we use $0\leq\mu_{\mathbf{x}}\leq\|{B^{*}B_{H}B}\|$ to
derive

$\sqrt{\lambda_{\mathbf{x}}}-\frac{\sqrt{\mu_{\mathbf{x}}}}{2}t\geq\alpha-\frac{\sqrt{\mu_{\mathbf{x}}}}{2}t_{1}\geq\frac{\alpha}{2}>0.$

Consequently, the expression in (2.28) can be estimated as

$\begin{split}\langle{\mathbf{x}},Q_{3}(t){\mathbf{x}}\rangle&\leq
1-\frac{\alpha^{2}}{2}t+\frac{2}{3}\|{\sqrt{B_{H}}}\|\
\|{\sqrt{B_{H}}B^{2}}\|\ t^{3}\\\ &\leq
1-t^{3}\Big{(}\frac{\alpha^{2}}{2}\frac{1}{t_{1}^{2}}-\frac{2}{3}\|{\sqrt{B_{H}}}\|\
\|{\sqrt{B_{H}}B^{2}}\|\Big{)}.\end{split}$

Then, there exists $t_{*}\in(0,t_{1}]$ such that

$\langle{\mathbf{x}},Q_{3}(t){\mathbf{x}}\rangle\leq 1-\frac{\kappa}{12}\
t^{3},\qquad\text{for }t\in[0,t_{*}]\ .$ (2.32)

Together the estimates (2.30) and (2.32) as well as (2.25), (2.27) prove the
upper bound (2.21b) with $a=3$ for ${m_{HC}}=1$, e.g. with $c_{2}:=\kappa/24$.
For ${m_{HC}}\geq 2$, see Lemma A.1 in Appendix A.

To sum up, due to (2.11a) and Lemma A.1, there exist constants $c_{1},c_{2}>0$
such that $a=2{m_{HC}}+1$ and

$1-c_{1}t^{a}+{\mathcal{O}}(t^{a+1})\leq\|P(t)\|_{2}\leq
1-c_{2}t^{a}+{\mathcal{O}}(t^{a+1})\quad\text{as }t\to 0^{+}\ ,$ (2.33)

e.g. choosing $c_{1}$ as in (2.19) and $c_{2}$ as in Lemma A.1, respectively.
Moreover, due to (2.10), $\|P(t)\|_{2}$ is analytic on $[0,t_{0}]$. This
implies that the propagator norm satisfies (2.8) for some
$c\in{\mathord{\mathbb{R}}}$, which satisfies $0<c_{2}\leq c\leq c_{1}$.

For the reverse direction suppose that (2.8) is satisfied for some constant
$c>0$ and an exponent $a>0$. Then, as we shall show, for some least finite
value of $m\in{\mathord{\mathbb{N}}}_{0}$,
$T_{m}=\sum_{j=0}^{m}B_{A}^{j}B_{H}(B_{A}^{*})^{j}>0$ has to hold, or
equivalently,

$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{m}=\sum_{j=0}^{m}(B^{*})^{j}B_{H}B^{j}>0\,.$

Otherwise, for arbitrarily large values of
$\widetilde{m}\in{\mathord{\mathbb{N}}}$, we could find unit vectors
${\mathbf{x}}_{0}$ such that
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{\widetilde{m}-1}{\mathbf{x}}_{0}=0$,
and then by the first part of this proof, we would have for such a vector
$\|P(t){\mathbf{x}}_{0}\|_{2}^{2}\geq
1-c_{\widetilde{m}}t^{2\widetilde{m}+1}+\mathcal{O}(t^{2\widetilde{m}+2})$
with some $c_{\widetilde{m}}\geq 0$, because of (2.17). But for sufficiently
large $\widetilde{m}$, this is incompatible with (2.8). Thus we conclude that,
whenever (2.8) is valid for any $a>0$, $B$ has a finite hypocoercivity index
${m_{HC}}\in{\mathord{\mathbb{N}}}_{0}$, and then necessarily, $a\in
2{\mathord{\mathbb{N}}}_{0}+1$.

Part (b): So far we have proved that the propagator norm satisfies (2.33),
e.g. choosing $c_{1}$ as in (2.19) and $c_{2}$ as the lower bound of $c$ in
(A.2), respectively. Using the improved upper bound $c_{1}$ in (A.34) and
lower bound $c_{2}$ in (A.2) for the multiplicative constant $c$ we realize
that $c_{2}=c=c_{1}$ such that (2.9) holds. In the final identity of (2.9) we
used again (2.20) to reveal that $c$ is proportional to $B_{A}^{2m_{HC}}$.
This finishes the proof. ∎

######  Remark 2.8.

The statement of Theorem 2.7(b) was inspired by a detailed perturbation
analysis, which will be presented in [2].

For $\epsilon$-dependent ODE systems of the form (1.5), Theorem 2.7(b) implies
the following result:

######  Corollary 2.9.

Consider the $\epsilon$-dependent ODE (1.5) with system matrix $B=\epsilon
A+C$ where $\epsilon\in{\mathord{\mathbb{R}}}$. If $B=\epsilon A+C$ is
hypocoercive for $\epsilon\neq 0$, then the coefficient $c=c_{\epsilon}$ in
the Taylor expansion of the propagator norm (2.8) satisfies

$c=c_{\epsilon}=\epsilon^{2m_{HC}}\
\frac{1}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}\min_{{\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\widetilde{T}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}_{0}\|_{2}=1}\langle{\mathbf{x}}_{0},(A^{*})^{{m_{HC}}}CA^{m_{HC}}{\mathbf{x}}_{0}\rangle\
.$ (2.34)

######  Remark 2.10.

As already briefly mentioned in Example 2 of [6], the hypocoercivity index
$m_{HC}$ is upper semicontinuous w.r.t. the matrix $B$: An arbitrarily small
perturbation of $B$ can lower but not increase the index. This is consistent
with the result in (2.8): All Taylor coefficients are there of course
continuous in $B$. But a small perturbation of $B$ may lower, but not increase
the number $a$ of the first non-vanishing monomial in this Taylor series
(beyond the constant 1).

######  Remark 2.11.

Note that the leading exponent in (2.8) can only be odd. This is related to
the local behavior of trajectories that decay the worst (in the vicinity of a
stationary point $t_{0}$ of $\|{\mathbf{x}}(t)\|$), see the yellow curve in
Fig. 1, left. Since the system (1.1) is assumed to be conservative-
dissipative, such a trajectory, of course, cannot behave locally like
$1-c(t-t_{0})^{a}$ with $a$ even.

######  Remark 2.12.

Special cases of the above theorem were pointed out to us by Laurent Miclo: In
§1 of [19] the short time decay behavior of the Goldstein-Taylor model (a
linear transport equation with relaxation term) was determined as
$1-\frac{t^{3}}{3}+o(t^{3})$. Actually, this model is a PDE. But since it is
considered on a torus in $x$, each of its spatial Fourier modes (except of the
0-mode) satisfies a conservative-dissipative ODE system with hypocoercivity
index 1 (see [3] for details of this modal decomposition). Hence, mode by
mode, the result from [19] is an example for Theorem 2.7. For closely related
BGK-models with hypocoercivity index 2 and 3 we refer to [4].

In [13] the short time decay behavior of a kinetic Fokker–Planck equation on
the torus in $x$ was computed as $1-\frac{t^{3}}{12}+o(t^{3})$. Again, in
Fourier space and by using a Hermite function basis in velocity, this model
can be written as an (infinite dimensional) conservative-dissipative system
with hypocoercivity index 1 (see §2.1 of [13]). In that paper it was also
mentioned that the decay exponent in (2.8) can be seen as some “order of
hypocoercivity” of the generator.

For degenerate Fokker–Planck equations, the hypocoercivity index can also be
related to the regularization rate for short times: In [20, Theorem A.12] the
regularization of initial data from a weighted $L^{2}$ space into a weighted
$H^{1}$ space is derived, and in [20, Theorem A.15], [10, Theorem 4.8] it is
generalized to entropy functionals and their corresponding Fisher
informations. In all these cases the regularization rate is $t^{-a}$ with
$a=2{m_{HC}}+1$ (somewhat related to Theorem 2.7 above).

By definition, the propagator norm of an ODE (1.1) is given as the envelope of
the norm of a family of solutions, see e.g. [8] and Figure 1. But, maybe
surprisingly, even its precise short time behavior is _not_ given by the norm
of any specific solution. This is illustrated in the following example.

######  Example 2.13.

We consider ODE (1.1) with matrix

$B=\begin{pmatrix}1&-3/10\\\ 3/10&0\end{pmatrix}.$ (2.35)

The eigenvalues of $B$ are $\lambda_{1}=1/10$ and $\lambda_{2}=9/10$, and the
eigenvalues of $B_{H}$ are $0$ and $1$. Thus, matrix $B$ is hypocoercive with
hypocoercivity index ${m_{HC}}=1$. Following (the first part of) the proof of
Theorem 2.7, solutions starting in ${\mathbf{x}}_{0}$ satisfying (2.18) are
used to establish the desired lower bound of the propagator norm. The kernel
of $B_{H}$ is one-dimensional and it is spanned by
${\mathbf{x}}_{0}=\binom{0}{1}$. The solution of (1.1) with initial condition
${\mathbf{x}}(0)={\mathbf{x}}_{0}$ is given by

${\mathbf{x}}(t)=\tfrac{1}{8}\begin{pmatrix}3&-3\\\
9&-1\end{pmatrix}\binom{e^{-t/10}}{e^{-9t/10}}$

and its squared norm satisfies

$\|{\mathbf{x}}(t)\|^{2}_{2}=\tfrac{45}{32}e^{-t/5}-\tfrac{9}{16}e^{-t}+\tfrac{5}{32}e^{-9t/5}\sim
1-0.06\ t^{3}+{\mathcal{O}}(t^{4})\quad\text{for }t\to 0^{+}.$

By contrast, due to [8, Proposition 4.2], the squared propagator norm
satisfies

$\begin{split}\|e^{-Bt}\|^{2}_{2}\leq&e^{-t}\tfrac{1}{16}\Big{(}\sqrt{(25\cosh(8t/10)-9)^{2}-16^{2}}+25\cosh(8t/10)-9\Big{)}\\\
&\sim 1-0.015\ t^{3}+{\mathcal{O}}(t^{4})\quad\text{for }t\to
0^{+}.\end{split}$ (2.36)

Thus the propagator norm decays slower than the solution starting at the
vector ${\mathbf{x}}_{0}$ which satisfies (2.18) with ${m_{HC}}=1$, see also
Figure 1.

In fact, the sharp constant $c=0.015$ in (2.36) cannot be obtained by any
single trajectory, but rather by a family of trajectories starting at the one-
parameter family of normalized initial conditions,
${\mathbf{x}}_{\tau},\,\tau\geq 0$ emerging from ${\mathbf{x}}_{0}$: More
concretely, using
${\mathbf{x}}_{\tau}:=\big{(}-\frac{3\tau}{20},1\big{)}^{\top}\big{/}\sqrt{1+\frac{9\tau^{2}}{400}}$
(as constructed in Lemma A.4 below) yields

$\|e^{-Bt}\,{\mathbf{x}}_{t}\|^{2}_{2}\sim 1-0.015\
t^{3}+{\mathcal{O}}(t^{4})\quad\text{for }t\to 0^{+}.$

Note that we used here the initial conditions ${\mathbf{x}}_{t}$ (i.e.
$\tau=t$), and that we have
$\frac{\,\operatorname{d}\\!missing}{\,\operatorname{d}\\!\tau}{\mathbf{x}}_{\tau}(\tau=0)=\big{(}-\frac{3}{20},0\big{)}^{\top}=\frac{1}{2}B{\mathbf{x}}_{0}$.

Figure 1: Evolution of (1.1) with matrix $B$ from (2.35): Comparison between
the propagator norm (red line), its upper exponential envelope
$1.25\exp(-t/10)$ (green line), and the norm of the solution with initial
condition ${\mathbf{x}}(0)=\binom{0}{1}$ (blue line) and that with initial
condition ${\mathbf{x}}(0)=\binom{-0.1}{\sqrt{0.99}}$ (yellow line), all
plotted on two time scales.

###  2.3 Numerical illustration of the short time decay and the waiting time
$t_{0}$

Next we shall illustrate the decay behavior on two examples of dimension
$n=4$. In particular we shall consider the $\epsilon$-dependence of the three
phases, the asymptotic phase close to $t=0$ (as characterized in Theorem 2.7),
the intermediate phase (characterized by the waiting time $t_{0}$ and the
exponential decay for large time (see (1.3)).

######  Example 2.14.

Consider the matrix family $B_{\epsilon}:=\epsilon A+C$, $\epsilon\neq 0$ with

$A=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&-1&0&0\\\
-1&0&0&0\end{pmatrix}\,,\qquad C=\operatorname{diag}(0,0,1,1)\ ,$ (2.37)

which satisfies $m_{HC}(B_{\epsilon})=1$.

Figure 2 shows the spectral norm of the semigroup
$P_{\epsilon}(t):=e^{-B_{\epsilon}t}$ as a function of time and for several
values of $\epsilon$ (differing from each other by the factor $\sqrt{2}$). The
numerically observed waiting times (to decrease the solution norm by the
factor $1/e$) are very close to $t_{0}\sim 1/\epsilon^{2}$. They increase by a
factor of about 2 when passing from $\epsilon$ to $\sqrt{2}\epsilon$, at least
in the asymptotic regime $\epsilon\to 0$. A close-up of the same figure around
$t=0$ shows that the asymptotic behavior of the semigroup norm is like
$\|P_{\epsilon}(t)\|_{2}\sim 1-c_{\epsilon}\,t^{3}$ with
$c_{\epsilon}=\tilde{c}\,\epsilon^{2}$, and by recalling that
$m_{HC}(B_{\epsilon})=1$ implies $a=3$ in (2.8). Following (2.34), the
multiplicative factor is $\tilde{c}=\frac{1}{12}$ (compare with e.g. [8]).

Figure 2: The decay of $\|P_{\epsilon}(t)\|_{2}$ is given for six values of
$\epsilon$. Left: For $t$ away from 0, this semigroup decays almost
exponentially. With the logarithmic scale used here, the horizontal black line
corresponds to $1/e$. The waiting times (defined as intersection with the line
$1/e$) behave like ${\mathcal{O}}(\epsilon^{-2})$. We remark that the kink in
the blue curve is _not_ a numerical artifact.
Right: This double logarithmic plot shows $1-\|P_{\epsilon}(t)\|_{2}\sim
c_{\epsilon}\,t^{3}$ for small time, more precisely for $t\in[e^{-9},e^{5}]$.
The curves have slope 3, and $c_{\epsilon}=\tilde{c}\,\epsilon^{2}$. The plot
also shows the quite sharp transition from the initial algebraic behavior
$1-\tilde{c}\,\epsilon^{2}\,t^{3}$ to the exponential behavior
$c_{\epsilon}^{*}\,e^{-\tilde{\mu}\,\epsilon^{2}\,t}$.

######  Example 2.15.

This example is analogous to Example 2.14, but for the matrix family
$B_{\epsilon}:=\epsilon A+C$, $\epsilon\neq 0$ with

$A=\begin{pmatrix}0&1&0&0\\\ -1&0&1&0\\\ 0&-1&0&1\\\
0&0&-1&0\end{pmatrix}\,,\qquad C=\operatorname{diag}(0,0,0,1)\ ,$ (2.38)

which satisfies $m_{HC}(B_{\epsilon})=3$.

Figure 3 shows the spectral norm of the semigroup
$P_{\epsilon}(t):=e^{-B_{\epsilon}t}$. The numerically observed waiting times
(to decrease the solution norm by the factor $1/e$) are very close to
$t_{0}\sim 4/\epsilon^{2}$. A close-up around $t=0$ shows that the asymptotic
behavior of the semigroup norm is like $\|P_{\epsilon}(t)\|_{2}\sim
1-c_{\epsilon}\,t^{7}$ with $c_{\epsilon}=\tilde{c}\,\epsilon^{6}$, and by
recalling that $m_{HC}(B_{\epsilon})=3$ implies $a=7$ in (2.8). Following
(2.34), the multiplicative factor is $\tilde{c}=\frac{1}{100800}$ (see e.g.
[2]).

Figure 3: The decay of $\|P_{\epsilon}(t)\|_{2}$ is given for six values of
$\epsilon$. Left: For $t$ away from 0, this semigroup decays almost
exponentially. With the logarithmic scale used here, the horizontal black line
corresponds to $1/e$. The waiting times (defined as intersection with the line
$1/e$) behave like ${\mathcal{O}}(\epsilon^{-2})$.
Right: This double logarithmic plot shows $1-\|P_{\epsilon}(t)\|_{2}\sim
c_{\epsilon}\,t^{7}$ for small time, more precisely for
$t\in[e^{-1.5},e^{4}]$. The curves have slope 7, and $c_{\epsilon}=\tilde{c}\
\epsilon^{6}$. The plot also shows the quite sharp transition from the initial
algebraic behavior $1-\tilde{c}\ \epsilon^{6}\,t^{7}$ to the exponential
behavior $c_{\epsilon}^{*}\ e^{-\tilde{\mu}\ \epsilon^{2}\ t}$.

##  Appendix A Auxiliary results to prove short time decay of propagator norm

To finish the proof of Theorem 2.7, we shall prove the following upper bound
for the propagator norm:

######  Lemma A.1.

Let the ODE system (1.1) be conservative-dissipative, and let the system
matrix $B$ be (hypo)coercive with hypocoercivity index
${m_{HC}}\in{\mathord{\mathbb{N}}}$. Then, there exist constants
$c_{2},M,t_{2}>0$ such that

$\|e^{-Bt}\|_{2}\leq 1-c_{2}t^{a}+Mt^{a+1}\qquad\text{for }t\in[0,t_{2}]\ ,$
(A.1)

where $a=2{m_{HC}}+1$. Moreover, the multiplicative factor $c$ in (2.8)
satisfies

$c\geq c_{2}\qquad\text{with
}c_{2}:=\frac{1}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}\min_{{\mathbf{x}}\in\ker\big{(}\widetilde{\widetilde{T}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}\|=1}\langle{{\mathbf{x}}},{(B^{*})^{m_{HC}}B_{H}B^{m_{HC}}{\mathbf{x}}}\rangle.$
(A.2)

###### Proof of Lemma A.1.

First we note that the hypocoercivity of $B$ implies $B_{H}\neq 0$ due to
(1.8). Following the proof of Theorem 2.7, we consider
$\|e^{-Bt}\|_{2}^{2}=\lambda_{\max}\big{(}Q(t)\big{)}$ for small $t>0$, where

$Q(t)=e^{-B^{*}t}e^{-Bt}=\sum_{j=0}^{\infty}\frac{t^{j}}{j!}\ U_{j}$ (A.3)

with

$U_{j}=(-1)^{j}\sum_{k=0}^{j}\binom{j}{k}(B^{*})^{k}B^{j-k}\,,\quad\text{satisfying
}\|U_{j}\|_{2}\leq(2\|B\|_{2})^{j}\,,\quad j\in{\mathord{\mathbb{N}}}_{0}\,.$
(A.4)

To compute
$\lambda_{\max}\big{(}Q(t)\big{)}=\max_{\|{\mathbf{x}}\|_{2}=1}{\mathbf{x}}^{*}Q(t){\mathbf{x}}$,
we consider the $t$-dependent function
$g({\mathbf{x}};t):={\mathbf{x}}^{*}Q(t){\mathbf{x}}-1$ with ${\mathbf{x}}$ in
the sphere $\mathcal{S}:=\\{{\mathbf{x}}\in{\mathord{\mathbb{C}}}^{n}\ |\
\|{\mathbf{x}}\|_{2}=1\\}$. For $a=2{m_{HC}}+1$, we denote the Taylor series
for $Q(t)$ and $g(t)$ truncated after the $t^{a}/a!$ term with $Q_{a}(t)$ and
$g_{a}({\mathbf{x}};t)$, respectively. We recall that $U_{0}=I$ and
$U_{1}=-2B_{H}$.

Let the matrices $U_{j}$, $j\in{\mathord{\mathbb{N}}}_{0}$ denote the
coefficients of $t^{j}/j!$ in the Taylor expansion (2.24) and (A.3), such that

$U_{j}:=(-1)^{j}\sum_{k=0}^{j}\binom{j}{k}(B^{*})^{k}B^{j-k}\,,\qquad
j\in{\mathord{\mathbb{N}}}_{0}\,,$ (A.5)

and note that each $U_{j}$ is self-adjoint. By Pascal’s identity
$\binom{j}{k}=\binom{j-1}{k-1}+\binom{j-1}{k}$ with the usual convention that
$\binom{j}{-1}=\binom{j}{j+1}=0$,

$\displaystyle\sum_{k=0}^{j}\binom{j}{k}(B^{*})^{k}B^{j-k}$
$\displaystyle=\sum_{k=0}^{j}\left(\binom{j-1}{k-1}(B^{*})^{k}B^{j-k}+\binom{j-1}{k}(B^{*})^{k}B^{j-k}\right)$
$\displaystyle=\sum_{k=0}^{j}\left(\binom{j-1}{k-1}(B^{*})^{k-1}B^{*}B^{j-k}+\binom{j-1}{k}(B^{*})^{k}BB^{j-1-k}\right)$
$\displaystyle=\sum_{k=0}^{j-1}\binom{j-1}{k}(B^{*})^{k}(B^{*}+B)B^{j-1-k}=2\sum_{k=0}^{j-1}\binom{j-1}{k}(B^{*})^{k}B_{H}B^{j-1-k}\,.$

Consequently,

$U_{j}=(-1)^{j}\
2\sum_{k=0}^{j-1}\binom{j-1}{k}(B^{*})^{k}B_{H}B^{j-1-k}\,,\qquad
j\in{\mathord{\mathbb{N}}}\,.$ (A.6)

First we outline the strategy of the proof, say for the case ${m_{HC}}=1$,
i.e. $a=3$: If ${\mathbf{x}}\in\ker(U_{1})=\ker(B_{H})$ with
$\|{\mathbf{x}}\|_{2}=1$, then (2.18) with ${\mathbf{x}}_{0}={\mathbf{x}}$
holds. Since ${m_{HC}}=1$, $B^{*}B_{H}B$ is positive definite on
$\ker(B_{H})$. Consequently, ${\mathbf{x}}^{*}U_{2}{\mathbf{x}}=0$,
${\mathbf{x}}^{*}U_{3}{\mathbf{x}}<0$, such that
$g_{a}({\mathbf{x}};t)=\tfrac{t^{3}}{3!}{\mathbf{x}}^{*}U_{3}{\mathbf{x}}<0$
for $t>0$. By contrast, if ${\mathbf{x}}\notin\ker U_{1}$, we have
${\mathbf{x}}^{*}U_{1}{\mathbf{x}}=-2{\mathbf{x}}^{*}B_{H}{\mathbf{x}}<0$.
Hence, for ${\mathbf{x}}\notin\ker U_{1}$,

$g_{a}({\mathbf{x}};t)=-\hat{c}t+{\mathcal{O}}(t^{2})\leq-\tilde{c}t^{3}\quad\text{for
}t\to 0^{+}$

follows for some $\hat{c},\tilde{c}>0$ that depend on ${\mathbf{x}}$. Since
$g_{a}({\mathbf{x}};t)$ depends continuously on ${\mathbf{x}}$, it is possible
to combine these two estimates with a constant $c$ that is independent of
${\mathbf{x}}\in\mathcal{S}$. Since $(\ker U_{1})^{c}\cap\mathcal{S}$ is not
compact, we do not obtain a uniform estimate “automatically”. So, the key
aspect is here to obtain a uniform decay estimate for ${\mathbf{x}}$ “close to
$\ker U_{1}$”, in the sense that
$-\epsilon\leq{\mathbf{x}}^{*}U_{1}{\mathbf{x}}\leq 0$.

Step 1. Matrices with hypocoercivity index ${m_{HC}}=1$. We suppose that
matrix $B$ has hypocoercivity index ${m_{HC}}=1$, i.e. there exists $\kappa>0$
such that

$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{m_{HC}}=\sum_{j=0}^{{m_{HC}}}(B^{*})^{j}B_{H}B^{j}=B_{H}+B^{*}B_{H}B\geq\kappa
I>0\,.$ (A.7)

Since ${m_{HC}}=1$, $B^{*}B_{H}B$ is positive definite on $\ker(B_{H})$. Our
goal is to estimate $g({\mathbf{x}};t)$ on $\mathcal{S}$. For
${\mathbf{x}}\in\mathcal{S}$, we define

$\lambda_{\mathbf{x}}:=\langle{{\mathbf{x}}},{B_{H}{\mathbf{x}}}\rangle\geq 0\
,\quad\mu_{\mathbf{x}}:=\langle{{\mathbf{x}}},{B^{*}B_{H}B{\mathbf{x}}}\rangle\geq
0\ ,\quad\text{such that }\lambda_{\mathbf{x}}+\mu_{\mathbf{x}}\geq\kappa>0\
.$ (A.8)

Step 1a. Consider ${\mathbf{x}}\in\mathcal{S}$ with
$\lambda_{\mathbf{x}}\leq\delta$ where $\delta\in(0,\kappa)$ will be chosen
later and $\kappa>0$ such that (A.7) holds. The key idea (to estimate
$g({\mathbf{x}};t)$ for $t\in[0,1]$) is to collect the terms $t^{j}$ of order
$j$ less than $a=2{m_{HC}}+1=3$ in a quadratic form which is non-positive.
Therefore, we use (A.6) and Lemma A.2 with $U=(-B)^{*}$, $V=U_{1}$, $W=-B$ and
$m={m_{HC}}-1=0$, to rewrite $g({\mathbf{x}};t)$ as

$\begin{split}g({\mathbf{x}};t)={\mathbf{x}}^{*}\Big{(}\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\
U_{j}\Big{)}{\mathbf{x}}&=\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}((-B)^{k}{\mathbf{x}})^{*}U_{1}(-B)^{j-k-1}{\mathbf{x}}\\\
&=t\Bigg{(}\sum_{k=0}^{\infty}\frac{1}{(k+1)!}t^{k}((-B)^{k}{\mathbf{x}})^{*}\Bigg{)}U_{1}\Bigg{(}\sum_{\ell=0}^{\infty}\frac{1}{(\ell+1)!}t^{\ell}(-B)^{\ell}{\mathbf{x}}\Bigg{)}\\\
&\quad+\frac{t^{3}}{3!}\tbinom{2}{1}\tfrac{1}{4}((-B){\mathbf{x}})^{*}U_{1}(-B){\mathbf{x}}+\sum_{j=4}^{\infty}\frac{t^{j}}{j!}\sum_{k=1}^{j-2}\binom{j-1}{k}\Delta^{(1)}_{j,k}((-B)^{k}{\mathbf{x}})^{*}U_{1}(-B)^{j-k-1}{\mathbf{x}}.\end{split}$

The first term is non-positive since $U_{1}=-2B_{H}\leq 0$. The second term is
retained. The third term can be estimated from above by $M_{4}\ t^{4}$ with
$M_{4}:=\sum_{j=4}^{\infty}\tfrac{1}{j!}(2\|B\|_{2})^{j}$ by using (A.6),
(A.4), and $\Delta^{(1)}_{j,k}<1$. Altogether, we derive the estimate

$g({\mathbf{x}};t)\leq\frac{1}{3!\
2}\big{(}(B{\mathbf{x}})^{*}U_{1}B{\mathbf{x}}\big{)}\ t^{3}+M_{4}\
t^{4}=-\frac{\mu_{\mathbf{x}}}{3!}\ t^{3}+M_{4}\ t^{4}.$ (A.9)

To establish a uniform negative upper bound for
$(B{\mathbf{x}})^{*}U_{1}B{\mathbf{x}}$ for ${\mathbf{x}}\in\mathcal{S}$ with
$\lambda_{\mathbf{x}}\leq\delta$, we use (A.8) to deduce
$\mu_{\mathbf{x}}\geq\kappa-\lambda_{\mathbf{x}}\geq\kappa-\delta>0$ since
$\delta\in(0,\kappa)$. For $\delta\in(0,\kappa)$, define

$\mu_{\delta}:=\min_{{\mathbf{x}}\in\mathcal{S}\text{ with
}\lambda_{\mathbf{x}}\leq\delta}\mu_{\mathbf{x}}=\min_{{\mathbf{x}}\in\mathcal{S}\text{
with
}\lambda_{\mathbf{x}}\leq\delta}\langle{{\mathbf{x}}},{B^{*}B_{H}B{\mathbf{x}}}\rangle$
(A.10)

such that $\mu_{\delta}\geq\kappa-\delta>0$ and

$\mu_{0}:=\lim_{\delta\to
0}\mu_{\delta}=\min_{{\mathbf{x}}\in\ker\big{(}\widetilde{\widetilde{T}}_{0}\big{)},\
\|{\mathbf{x}}\|=1}\langle{{\mathbf{x}}},{B^{*}B_{H}B{\mathbf{x}}}\rangle.$
(A.11)

Then, we derive from (A.9) that

$g({\mathbf{x}};t)\leq-\frac{\mu_{\delta}}{3!}t^{3}+M_{4}\ t^{4}\qquad\text{
for all ${\mathbf{x}}\in\mathcal{S}$ with $\lambda_{\mathbf{x}}\leq\delta$ and
$t\in[0,1]$}.$ (A.12)

Step 1b. Consider ${\mathbf{x}}\in\mathcal{S}$ with
$\lambda_{\mathbf{x}}>\delta$ where $\delta\in(0,\kappa)$ will be chosen later
and $\kappa>0$ such that (A.7) holds. For $t\in[0,1]$, we deduce

$g({\mathbf{x}};t):={\mathbf{x}}^{*}\Big{(}\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\
U_{j}\Big{)}{\mathbf{x}}=t{\mathbf{x}}^{*}U_{1}{\mathbf{x}}+\sum_{j=2}^{\infty}\frac{t^{j}}{j!}\
{\mathbf{x}}^{*}U_{j}{\mathbf{x}}\leq-2\lambda_{\mathbf{x}}t+M_{2}\ t^{2}\,,$
(A.13)

since $\big{|}\sum_{j=2}^{\infty}\frac{t^{j}}{j!}\
{\mathbf{x}}^{*}U_{j}{\mathbf{x}}\big{|}\leq t^{2}M_{2}$ with
$M_{2}:=\sum_{j=2}^{\infty}\frac{1}{j!}\ (2\|B\|_{2})^{j}$. Then,
$-2\lambda_{\mathbf{x}}t+M_{2}t^{2}\leq-\lambda_{\mathbf{x}}t$ for all $0\leq
t\leq{\lambda_{\mathbf{x}}}/{M_{2}}$. For any given $c>0$, the estimate
$-\lambda_{\mathbf{x}}t\leq-ct^{3}$ holds if $0\leq
t\leq\sqrt{{\lambda_{\mathbf{x}}}/{c}}$. Define
$t_{\delta}:=\min\\{{\delta}/{M_{2}},\sqrt{{\delta}/{c}},1\\}$. Then, we
derive

$g({\mathbf{x}};t)\leq-ct^{3}\qquad\text{for all ${\mathbf{x}}\in\mathcal{S}$
with $\lambda_{\mathbf{x}}>\delta$ and $t\in[0,t_{\delta}]$.}$ (A.14)

To sum up, choosing any $\delta\in(0,\kappa)$, the estimate (A.12) is derived.
Then, for $c:=\mu_{\delta}/3!$, there exists a (sufficiently small)
$t_{\delta}>0$ (as defined in Step 1b) such that the estimate (A.14) holds.
Consequently, we obtain

$g({\mathbf{x}};t)\leq-\frac{\mu_{\delta}}{3!}t^{3}+M_{4}\ t^{4}\quad\text{for
all }{\mathbf{x}}\in\mathcal{S}\text{ and }t\in[0,t_{\delta}].$ (A.15)

This shows (A.1) with $c_{2}:=c/2=\mu_{\delta}/(3!\ 2)$ and $a=2{m_{HC}}+1=3$.

Step 1c. To prove the second statement in Lemma A.1, we improve the estimate
of $c$ as follows: By definition, the time $t_{\delta}$ satisfies
$\lim_{\delta\to 0}t_{\delta}=0$. To derive (the sharp) lower estimate (A.2)
on the multiplicative factor $c$, we consider the Taylor expansion (2.8) of
the propagator norm, use estimate (A.15), and take the limit $\delta\to 0$:

$-2c=\lim_{\delta\to 0}\frac{\|e^{-B\
t_{\delta}}\|_{2}^{2}-1}{t_{\delta}^{3}}=\lim_{\delta\to
0}\frac{g({\mathbf{x}};t_{\delta})}{t_{\delta}^{3}}\leq\lim_{\delta\to
0}\Big{(}-\frac{\mu_{\delta}}{3!}+M_{4}\
t_{\delta}\Big{)}=-\frac{\mu_{0}}{3!}.$ (A.16)

Hence, we identified a lower estimate for the multiplicative factor $c$ in
(2.8) as

$c\geq\frac{\mu_{0}}{3!\ 2}=\frac{1}{3!\
2}\min_{{\mathbf{x}}\in\ker\big{(}\widetilde{\widetilde{T}}_{0}\big{)},\
\|{\mathbf{x}}\|=1}\langle{{\mathbf{x}}},{B^{*}B_{H}B{\mathbf{x}}}\rangle.$
(A.17)

This finishes the proof of the second statement in Lemma A.1 in the case
${m_{HC}}=1$.

Step 2. Matrices with hypocoercivity index ${m_{HC}}\geq 2$. For matrices $B$
with hypocoercivity index ${m_{HC}}\geq 2$, i.e. there exists $\kappa>0$ such
that

$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{m_{HC}}=\sum_{j=0}^{{m_{HC}}}(B^{*})^{j}B_{H}B^{j}\geq\kappa
I>0\ ,$ (A.18)

we generalize this procedure as follows: We define, for
${\mathbf{x}}\in\mathcal{S}$,

$\lambda_{\mathbf{x}}:=\langle{{\mathbf{x}}},{\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}{\mathbf{x}}}\rangle\geq
0\
,\quad\mu_{\mathbf{x}}:=\langle{{\mathbf{x}}},{(B^{*})^{m_{HC}}B_{H}B^{m_{HC}}{\mathbf{x}}}\rangle\geq
0\ ,\quad\text{such that }\lambda_{\mathbf{x}}+\mu_{\mathbf{x}}\geq\kappa>0\
.$ (A.19)

Step 2a. Consider ${\mathbf{x}}\in\mathcal{S}$ with
$\lambda_{\mathbf{x}}\leq\delta$ where $\delta\in(0,\kappa)$ will be chosen
later and $\kappa>0$ such that (A.18) holds. For $t\in[0,1]$, we derive as in
Step 2b (see (A.27) below):

$\begin{split}g({\mathbf{x}};t)&\leq\frac{(B^{{m_{HC}}}{\mathbf{x}})^{*}U_{1}B^{{{m_{HC}}}}{\mathbf{x}}}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}t^{2{{m_{HC}}}+1}+M_{2{{m_{HC}}}+2}\
t^{2{{m_{HC}}}+2}\\\
&=-\frac{2}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}\mu_{\mathbf{x}}t^{2{{m_{HC}}}+1}+M_{2{{m_{HC}}}+2}\
t^{2{{m_{HC}}}+2}\,,\end{split}$ (A.20)

with
$M_{2{{m_{HC}}}+2}:=\sum_{j=2{{m_{HC}}}+2}^{\infty}\tfrac{1}{j!}(2\|B\|_{2})^{j}>0$.
To establish a uniform negative upper bound for $-\mu_{\mathbf{x}}$ for
${\mathbf{x}}\in\mathcal{S}$ with $\lambda_{\mathbf{x}}\leq\delta$, we use
(A.19) to deduce
$\mu_{\mathbf{x}}\geq\kappa-\lambda_{\mathbf{x}}\geq\kappa-\delta>0$ since
$\delta\in(0,\kappa)$. For $\delta\in(0,\kappa)$, define

$\mu_{\delta}:=\min_{{\mathbf{x}}\in\mathcal{S}\text{ with
}\lambda_{\mathbf{x}}\leq\delta}\mu_{\mathbf{x}}=\min_{{\mathbf{x}}\in\mathcal{S}\text{
with
}\lambda_{\mathbf{x}}\leq\delta}\langle{{\mathbf{x}}},{(B^{*})^{m_{HC}}B_{H}B^{m_{HC}}{\mathbf{x}}}\rangle$
(A.21)

such that $\mu_{\delta}\geq\kappa-\delta>0$ and

$\mu_{0}:=\lim_{\delta\to
0}\mu_{\delta}=\min_{{\mathbf{x}}\in\ker\big{(}\widetilde{\widetilde{T}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}\|=1}\langle{{\mathbf{x}}},{(B^{*})^{m_{HC}}B_{H}B^{m_{HC}}{\mathbf{x}}}\rangle.$
(A.22)

Then, we derive from (A.20) that

$g({\mathbf{x}};t)\leq-c\ t^{2{{m_{HC}}}+1}+M_{2{{m_{HC}}}+2}\
t^{2{{m_{HC}}}+2}\qquad\text{ for all ${\mathbf{x}}\in\mathcal{S}$ with
$\lambda_{\mathbf{x}}\leq\delta$ and $t\in[0,1]$},$ (A.23a) where
$c:=\frac{2\mu_{\delta}}{({2{{m_{HC}}}+1})!\ \binom{2{{m_{HC}}}}{{m_{HC}}}}\
.$ (A.23b)

Step 2b. Next, we shall show the following statement: Consider
${\mathbf{x}}\in\mathcal{S}$ with $\lambda_{\mathbf{x}}>\delta$ where
$\kappa>0$ such that (A.18) holds. For given $c>0$, there exists
$t_{\delta}>0$ such that

$g({\mathbf{x}};t)\leq-c\ t^{2{m_{HC}}+1}\qquad\text{for all
${\mathbf{x}}\in\mathcal{S}$ with $\lambda_{\mathbf{x}}>\delta$ and
$t\in[0,t_{\delta}]$.}$ (A.24)

First, we decompose the sphere $\mathcal{S}$ into the (non-disjoint) closed
subsets

$\begin{split}\mathcal{C}_{0}&:=\\{{\mathbf{x}}\in\mathcal{S}\ |\
{\mathbf{x}}^{*}U_{1}{\mathbf{x}}\leq-\epsilon\\}\,,\\\
\mathcal{C}_{1}&:=\\{{\mathbf{x}}\in\mathcal{S}\ |\
-\epsilon\leq{\mathbf{x}}^{*}U_{1}{\mathbf{x}}\ \land\
(B{\mathbf{x}})^{*}U_{1}B{\mathbf{x}}\leq-\epsilon\\}\,,\\\
\mathcal{C}_{2}&:=\\{{\mathbf{x}}\in\mathcal{S}\ |\
-\epsilon\leq{\mathbf{x}}^{*}U_{1}{\mathbf{x}}\ \land\
-\epsilon\leq(B{\mathbf{x}})^{*}U_{1}B{\mathbf{x}}\ \land\
(B^{2}{\mathbf{x}})^{*}U_{1}B^{2}{\mathbf{x}}\leq-\epsilon\\}\,,\\\ &\vdots\\\
\mathcal{C}_{m}&:=\\{{\mathbf{x}}\in\mathcal{S}\ |\ \forall
k\in\\{0,\ldots,m-1\\}:\
-\epsilon\leq(B^{k}{\mathbf{x}})^{*}U_{1}B^{k}{\mathbf{x}}\ \land\
(B^{m}{\mathbf{x}})^{*}U_{1}B^{m}{\mathbf{x}}\leq-\epsilon\\}\,,\\\ &\vdots\\\
\mathcal{C}_{{m_{HC}}-1}&:=\\{{\mathbf{x}}\in\mathcal{S}\ |\ \forall
k\in\\{0,\ldots,{m_{HC}}-2\\}:\
-\epsilon\leq(B^{k}{\mathbf{x}})^{*}U_{1}B^{k}{\mathbf{x}}\ \\\ &\hskip
150.0pt\land\
(B^{{m_{HC}}-1}{\mathbf{x}})^{*}U_{1}B^{{m_{HC}}-1}{\mathbf{x}}\leq-\epsilon\\}\,,\end{split}$
(A.25)

as well as $\mathcal{C}_{{m_{HC}}}:=\\{{\mathbf{x}}\in\mathcal{S}\ |\ \forall
k\in\\{0,\ldots,{m_{HC}}-1\\}:\
-\epsilon\leq(B^{k}{\mathbf{x}})^{*}U_{1}B^{k}{\mathbf{x}}\ \\}$, for some
positive parameter $\epsilon$ to be determined next.

We show that there exists $\epsilon>0$ such that
$\mathcal{C}_{{m_{HC}}}\subseteq\\{{\mathbf{x}}\in\mathcal{S}\ |\
\lambda_{\mathbf{x}}\leq\delta\\}$: Consider
${\mathbf{x}}\in\mathcal{C}_{{m_{HC}}}$. Then, ${\mathbf{x}}$ satisfies

$\epsilon/2\geq\langle{{\mathbf{x}}},{(B^{*})^{k}B_{H}B^{k}{\mathbf{x}}}\rangle\quad\text{
for }k=0,\ldots,{m_{HC}}-1\ ,$

which (upon summing up) implies that

${m_{HC}}\epsilon/2\geq\langle{{\mathbf{x}}},{\sum_{k=0}^{{m_{HC}}-1}(B^{*})^{k}B_{H}B^{k}\
{\mathbf{x}}}\rangle=\langle{{\mathbf{x}}},{\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}{\mathbf{x}}}\rangle=\lambda_{\mathbf{x}}\
.$

Choosing

$\epsilon:=\frac{2\delta}{{m_{HC}}}\quad\text{ implies
}\mathcal{C}_{{m_{HC}}}\subseteq\\{{\mathbf{x}}\in\mathcal{S}\ |\
\lambda_{\mathbf{x}}\leq\delta\\}\ .$ (A.26)

Hence, the already established estimate (A.23) holds in particular for
${\mathbf{x}}\in\mathcal{C}_{{m_{HC}}}$ and $t\in[0,\,1]$. Using
$\bigcup_{j=0}^{{m_{HC}}-1}\mathcal{C}_{j}\supseteq\\{{\mathbf{x}}\in\mathcal{S}\
|\ \lambda_{\mathbf{x}}>\delta\\}$ (as the complementary inclusion of (A.26)),
we are left to prove the estimate (A.24) for all
${\mathbf{x}}\in\bigcup_{j=0}^{{m_{HC}}-1}\mathcal{C}_{j}$:

For all $\ell\in\\{0,\ldots,{m_{HC}}-1\\}$,
${\mathbf{x}}\in\mathcal{C}_{\ell}$ and $t\in[0,1]$, the key idea (to estimate
$g({\mathbf{x}};t)$) is to collect the terms $t^{j}$ of order $j$ less than
$2\ell+1$ in a quadratic form which is non-positive. Therefore, we use again
(A.6) and Lemma A.2 with $U=-B^{*}$, $V=U_{1}$, $W=-B$ and $m=\ell-1$, to
rewrite $g({\mathbf{x}};t)$ as

$\begin{split}g({\mathbf{x}};t)&={\mathbf{x}}^{*}\Big{(}\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\
U_{j}\Big{)}{\mathbf{x}}\\\
&=\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}((-B)^{k}{\mathbf{x}})^{*}U_{1}(-B)^{j-k-1}{\mathbf{x}}\\\
&=\sum_{j=0}^{\ell-1}\frac{t^{2j+1}}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2j+1)!}{(k+2j+1)!}\tbinom{k+j}{j}t^{k}((-B)^{k+j}{\mathbf{x}})^{*}\Bigg{)}U_{1}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2j+1)!}{(k+2j+1)!}\tbinom{k+j}{j}t^{k}(-B)^{k+j}{\mathbf{x}}\Bigg{)}\\\
&\quad+\frac{t^{2\ell+1}}{({2\ell+1})!}\binom{2\ell}{\ell}\Delta^{(\ell)}_{{2\ell+1},\ell}((-B)^{\ell}{\mathbf{x}})^{*}U_{1}(-B)^{\ell}{\mathbf{x}}\\\
&\quad+\sum_{j=2\ell+2}^{\infty}\frac{t^{j}}{j!}\sum_{k=\ell}^{j-\ell-1}\binom{j-1}{k}\Delta^{(\ell)}_{j,k}((-B)^{k}{\mathbf{x}})^{*}U_{1}(-B)^{j-k-1}{\mathbf{x}}.\end{split}$
(A.27a) The first term is non-positive since $U_{1}=-2B_{H}\leq 0$. The second
term is estimated using the assumption ${\mathbf{x}}\in\mathcal{C}_{\ell}$ and
the identity $\Delta^{(\ell)}_{{2\ell+1},\ell}=\binom{2\ell}{\ell}^{-2}$. The
third term can be estimated from above by $M_{2\ell+2}\ t^{2\ell+2}$ with
$M_{2\ell+2}:=\sum_{j=2\ell+2}^{\infty}\tfrac{1}{j!}(2\|B\|_{2})^{j}>0$ by
using (A.6), (A.4), and $\Delta^{(\ell)}_{j,k}\leq 1$ for $0\leq\ell\leq k\leq
j-\ell-1$. Altogether, we obtain the estimate
$g({\mathbf{x}};t)\leq-\frac{\epsilon}{({2\ell+1})!\binom{2\ell}{\ell}}t^{2\ell+1}+M_{2\ell+2}\
t^{2\ell+2}.$ (A.27b)

For given $c>0$ (e.g. as in (A.23b)), there exists $\tilde{t}_{\ell}>0$
(depending on $\epsilon$, with $\lim_{\epsilon\to 0}\tilde{t}_{\ell}=0$) such
that

$g({\mathbf{x}};t)\leq-c\ t^{2{m_{HC}}+1}\qquad\text{for all
${\mathbf{x}}\in\mathcal{C}_{\ell}$ and $t\in[0,\tilde{t}_{\ell}]$.}$ (A.28)

Choosing $\epsilon=2\delta/{m_{HC}}$ as in (A.26) (such that
$\bigcup_{j=0}^{{m_{HC}}-1}\mathcal{C}_{j}\supseteq\\{{\mathbf{x}}\in\mathcal{S}\
|\ \lambda_{\mathbf{x}}>\delta\\}$) and

$t_{\delta}:=\min\\{\tilde{t}_{\ell}\ ,\ \ell=0,\ldots,{m_{HC}}-1\\}\ ,$
(A.29)

implies the estimate (A.24).

To sum up, for any fixed $\delta\in(0,\kappa)$, the estimate (A.23) with
multiplicative constant $c$ in (A.23b) is proven in Step 2a. Then, for
$\epsilon=2\delta/{m_{HC}}$ and $c$ in (A.23b), there exists a (sufficiently
small) $t_{\delta}>0$ (as defined in (A.29)) such that the estimate (A.24)
holds. Consequently, we obtain

$g({\mathbf{x}};t)\leq-\frac{2\mu_{\delta}}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}t^{2{{m_{HC}}}+1}+M_{2{{m_{HC}}}+2}\
t^{2{{m_{HC}}}+2}\qquad\text{ for all ${\mathbf{x}}\in\mathcal{S}$ with
$t\in[0,t_{\delta}]$.}$ (A.30)

This shows (A.1) with $c_{2}:=c/2$, $c$ as defined in (A.23b) and
$a=2{m_{HC}}+1$. This finishes the proof of the first statement in Lemma A.1
for ${m_{HC}}\in{\mathord{\mathbb{N}}}$.

Step 2c. To prove the second statement in Lemma A.1, we improve the estimate
of $c$ as follows: By definition, the time $t_{\delta}$ depends on $\delta$
(since $\epsilon=2\delta/{m_{HC}}$) such that $\lim_{\delta\to
0}t_{\delta}=0$. To derive the (sharp) lower estimate (A.2) on the
multiplicative factor $c$, we consider the Taylor expansion (2.8) of the
propagator norm, use estimate (A.30), and take the limit $\delta\to 0$ such
that

$-2c=\lim_{\delta\to 0}\frac{\|e^{-B\
t_{\delta}}\|_{2}^{2}-1}{t_{\delta}^{2{m_{HC}}+1}}\leq\lim_{\delta\to
0}\Bigg{(}-\frac{2\mu_{\delta}}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}+M_{2{{m_{HC}}}+2}\
t_{\delta}\Bigg{)}=-\frac{2\mu_{0}}{({2{{m_{HC}}}+1})!\binom{2{{m_{HC}}}}{{m_{HC}}}}.$
(A.31)

This proves the lower estimate (A.2) for the multiplicative factor $c$ in
(2.8). ∎

The proof of Lemma A.1 uses the following identity:

######  Lemma A.2.

Let $U,V,W\in{\mathord{\mathbb{C}}}^{n\times n}$. For all
$m\in{\mathord{\mathbb{N}}}_{0}$, the following identity holds

$\begin{split}&\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}U^{k}VW^{j-k-1}\\\
&=\sum_{j=0}^{m}\frac{t^{2j+1}}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{(}\sum_{k=0}^{\infty}\frac{(2j+1)!}{(k+2j+1)!}\binom{k+j}{j}t^{k}U^{k+j}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\frac{(2j+1)!}{(\ell+2j+1)!}\binom{\ell+j}{j}t^{\ell}W^{\ell+j}\Bigg{)}\\\
&\quad+\sum_{j=2m+3}^{\infty}\frac{t^{j}}{j!}\sum_{k=m+1}^{j-m-2}\binom{j-1}{k}\Delta^{(m+1)}_{j,k}U^{k}VW^{j-k-1}\,,\end{split}$
(A.32)

where
$\Delta^{(m)}_{j,k}:=\frac{\binom{k}{m}\binom{j-k-1}{m}}{\binom{k+m}{m}\binom{j-k-1+m}{m}}$
for all $m\leq k$ and $m\leq j-k-1$.

###### Proof.

We will prove the identity by induction. For $m=0$, we have to prove the
identity

$\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}U^{k}VW^{j-k-1}\\\
=t\Bigg{(}\sum_{k=0}^{\infty}\tfrac{1}{(k+1)!}t^{k}U^{k}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\tfrac{1}{(\ell+1)!}t^{\ell}W^{\ell}\Bigg{)}+\sum_{j=3}^{\infty}\frac{t^{j}}{j!}\sum_{k=1}^{j-2}\tbinom{j-1}{k}\tfrac{k(j-k-1)}{(k+1)(j-k)}U^{k}VW^{j-k-1}$

since
$\Delta^{(1)}_{j,k}=\frac{\binom{k}{1}\binom{j-k-1}{1}}{\binom{k+1}{1}\binom{j-k}{1}}=\tfrac{k(j-k-1)}{(k+1)(j-k)}$.
The first term on the right hand side can be written by the Cauchy product
formula as

$\displaystyle
t\Bigg{(}\sum_{k=0}^{\infty}\tfrac{1}{(k+1)!}t^{k}U^{k}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\tfrac{1}{(\ell+1)!}t^{\ell}W^{\ell}\Bigg{)}$
$\displaystyle=t\sum_{j=0}^{\infty}t^{j}\sum_{k=0}^{j}\tfrac{1}{(k+1)!}U^{k}V\tfrac{1}{(j-k+1)!}W^{j-k}=\sum_{j=0}^{\infty}t^{j+1}\sum_{k=0}^{j}\tfrac{1}{(k+1)!}U^{k}V\tfrac{1}{(j-k+1)!}W^{j-k}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tfrac{j!}{(k+1)!(j-k)!}U^{k}VW^{j-k-1}=\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}\tfrac{j}{(k+1)(j-k)}U^{k}VW^{j-k-1}$
$\displaystyle=\sum_{j=1}^{2}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}U^{k}VW^{j-k-1}+\sum_{j=3}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}\tfrac{j}{(k+1)(j-k)}U^{k}VW^{j-k-1}\,.$

Therefore,

$\displaystyle
t\Bigg{(}\sum_{k=0}^{\infty}\tfrac{1}{(k+1)!}t^{k}U^{k}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\tfrac{1}{(\ell+1)!}t^{\ell}W^{\ell}\Bigg{)}+\sum_{j=3}^{\infty}\frac{t^{j}}{j!}\sum_{k=1}^{j-2}\tbinom{j-1}{k}\tfrac{k(j-k-1)}{(k+1)(j-k)}U^{k}VW^{j-k-1}$
$\displaystyle=\sum_{j=1}^{2}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}U^{k}VW^{j-k-1}+\sum_{j=3}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}\tfrac{j}{(k+1)(j-k)}U^{k}VW^{j-k-1}+\sum_{j=3}^{\infty}\frac{t^{j}}{j!}\sum_{k=1}^{j-2}\tbinom{j-1}{k}\tfrac{k(j-k-1)}{(k+1)(j-k)}U^{k}VW^{j-k-1}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}U^{k}VW^{j-k-1}\,.$

We assume that the formula holds for $m\in{\mathord{\mathbb{N}}}_{0}$ and
prove it for $m+1$. First, we use again the Cauchy product formula to derive

$\displaystyle\frac{t^{2m+3}}{(2m+3)!}\frac{1}{\binom{2m+2}{m+1}}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2m+3)!}{(k+2m+3)!}\tbinom{k+m+1}{m+1}t^{k}U^{k+m+1}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\tfrac{(2m+3)!}{(\ell+2m+3)!}\tbinom{\ell+m+1}{m+1}t^{\ell}W^{\ell+m+1}\Bigg{)}$
$\displaystyle=\frac{t^{2m+3}}{(2m+3)!}\frac{1}{\binom{2m+2}{m+1}}\sum_{j=0}^{\infty}t^{j}\sum_{k=0}^{j}\tfrac{(2m+3)!}{(k+2m+3)!}\tbinom{k+m+1}{m+1}U^{k+m+1}V\tfrac{(2m+3)!}{(j-k+2m+3)!}\tbinom{j-k+m+1}{m+1}W^{j-k+m+1}$
$\displaystyle=\sum_{j=0}^{\infty}\frac{t^{j+2m+3}}{(j+2m+3)!}\sum_{k=0}^{j}\tfrac{(j+2m+3)!}{(2m+3)!\binom{2m+2}{m+1}}\tfrac{(2m+3)!}{(k+2m+3)!}\tbinom{k+m+1}{m+1}U^{k+m+1}V\tfrac{(2m+3)!}{(j-k+2m+3)!}\tbinom{j-k+m+1}{m+1}W^{j-k+m+1}$
$\displaystyle=\sum_{j=2m+3}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-2m-3}\tfrac{j!}{(2m+3)!\binom{2m+2}{m+1}}\tfrac{(2m+3)!}{(k+2m+3)!}\tbinom{k+m+1}{m+1}U^{k+m+1}V\tfrac{(2m+3)!}{(j-k)!}\tbinom{j-k-m-2}{m+1}W^{j-k-m-2}$
$\displaystyle=\sum_{j=2m+3}^{\infty}\frac{t^{j}}{j!}\sum_{k=m+1}^{j-m-2}\tfrac{j!}{(2m+3)!\binom{2m+2}{m+1}}\tfrac{(2m+3)!}{(k+m+2)!}\tbinom{k}{m+1}\tfrac{(2m+3)!}{(j-k+m+1)!}\tbinom{j-k-1}{m+1}U^{k}VW^{j-k-1}\,.$

Therefore,

$\displaystyle\sum_{j=0}^{m+1}\frac{t^{2j+1}}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{(}\sum_{k=0}^{\infty}\frac{(2j+1)!}{(k+2j+1)!}\binom{k+j}{j}t^{k}U^{k+j}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\frac{(2j+1)!}{(\ell+2j+1)!}\binom{\ell+j}{j}t^{\ell}W^{\ell+j}\Bigg{)}$
$\displaystyle\quad+\sum_{j=2m+5}^{\infty}\frac{t^{j}}{j!}\sum_{k=m+2}^{j-m-3}\binom{j-1}{k}\Delta^{(m+2)}_{j,k}U^{k}VW^{j-k-1}$
$\displaystyle=\sum_{j=0}^{m}\frac{t^{2j+1}}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{(}\sum_{k=0}^{\infty}\frac{(2j+1)!}{(k+2j+1)!}\binom{k+j}{j}t^{k}U^{k+j}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\frac{(2j+1)!}{(\ell+2j+1)!}\binom{\ell+j}{j}t^{\ell}W^{\ell+j}\Bigg{)}$
$\displaystyle\quad+\sum_{j=2m+3}^{\infty}\frac{t^{j}}{j!}\sum_{k=m+1}^{j-m-2}\tfrac{j!}{(2m+3)!\binom{2m+2}{m+1}}\tfrac{(2m+3)!}{(k+m+2)!}\tbinom{k}{m+1}\tfrac{(2m+3)!}{(j-k+m+1)!}\tbinom{j-k-1}{m+1}U^{k}VW^{j-k-1}$
$\displaystyle\quad+\sum_{j=2m+5}^{\infty}\frac{t^{j}}{j!}\sum_{k=m+2}^{j-m-3}\binom{j-1}{k}\Delta^{(m+2)}_{j,k}U^{k}VW^{j-k-1}\,.$
(A.33)

Using
$\Delta^{(m+2)}_{j,k}=\Delta^{(m+1)}_{j,k}\frac{(k-m-1)(j-k-m-2)}{(k+m+2)(j-k+m+1)}$
we deduce that (A.33) equals:

$\displaystyle=\sum_{j=0}^{m}\frac{t^{2j+1}}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{(}\sum_{k=0}^{\infty}\frac{(2j+1)!}{(k+2j+1)!}\binom{k+j}{j}t^{k}U^{k+j}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\frac{(2j+1)!}{(\ell+2j+1)!}\binom{\ell+j}{j}t^{\ell}W^{\ell+j}\Bigg{)}$
$\displaystyle\quad+\sum_{j=2m+3}^{\infty}\frac{t^{j}}{j!}\sum_{k=m+1}^{j-m-2}\tbinom{j-1}{k}\Delta^{(m+1)}_{j,k}\tfrac{(2m+3)j}{(k+m+2)(j-k+m+1)}U^{k}VW^{j-k-1}$
$\displaystyle\quad+\sum_{j=2m+5}^{\infty}\frac{t^{j}}{j!}\sum_{k=m+2}^{j-m-3}\tbinom{j-1}{k}\Delta^{(m+1)}_{j,k}\frac{(k-m-1)(j-k-m-2)}{(k+m+2)(j-k+m+1)}U^{k}VW^{j-k-1}$
$\displaystyle=\sum_{j=0}^{m}\frac{t^{2j+1}}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{(}\sum_{k=0}^{\infty}\frac{(2j+1)!}{(k+2j+1)!}\binom{k+j}{j}t^{k}U^{k+j}\Bigg{)}V\Bigg{(}\sum_{\ell=0}^{\infty}\frac{(2j+1)!}{(\ell+2j+1)!}\binom{\ell+j}{j}t^{\ell}W^{\ell+j}\Bigg{)}$
$\displaystyle\quad+\sum_{j=2m+3}^{\infty}\frac{t^{j}}{j!}\sum_{k=m+1}^{j-m-2}\binom{j-1}{k}\Delta^{(m+1)}_{j,k}U^{k}VW^{j-k-1}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}U^{k}VW^{j-k-1}\,,$

where we used the induction hypothesis, i.e. (A.32), in the final equality.
This finishes the proof. ∎

To determine the optimal multiplicative factor $c$ in (2.8) (see Theorem
2.7(b)), we shall next derive an improved upper estimate for $c$, compared to
(2.19):

######  Lemma A.3.

Let the ODE system (1.1) be conservative-dissipative, and let the system
matrix $B$ be (hypo)coercive with hypocoercivity index
${m_{HC}}\in{\mathord{\mathbb{N}}}$. Then, the multiplicative factor $c$ in
(2.8) satisfies

$c\leq c_{1}\qquad\text{with
}c_{1}:=\frac{1}{(2{m_{HC}}+1)!\binom{2{m_{HC}}}{{m_{HC}}}}\min_{{\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\widetilde{T}}_{{m_{HC}}-1}\big{)},\
\|{\mathbf{x}}_{0}\|_{2}=1}\langle{B^{m_{HC}}{\mathbf{x}}_{0}},{B_{H}B^{m_{HC}}{\mathbf{x}}_{0}}\rangle.$
(A.34)

###### Proof.

As illustrated by Example 2.13, see also Figure 1, the propagator norm is in
general not determined by the norm of one specific solution. Instead we
consider a parameterized family of solutions pertaining to initial values
${\mathbf{x}}_{\tau}$, $\tau\in[0,1]$:

Due to Lemma A.4 below, for
${\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}\big{)}$
(with $\|{\mathbf{x}}_{0}\|_{2}=1$), there exist real constants $b_{\ell}$,
$\ell=1,\ldots,{m_{HC}}$ such that

${\mathbf{x}}_{\tau}:={\mathbf{x}}_{0}+\sum_{\ell=1}^{{m_{HC}}}b_{\ell}\tau^{\ell}B^{\ell}{\mathbf{x}}_{0}$

satisfies

$g({\mathbf{x}}_{\tau};\tau)=-2c_{1}({\mathbf{x}}_{0})\tau^{2{m_{HC}}+1}+{{\mathcal{O}}}(\tau^{2{m_{HC}}+2}),\quad\text{for
}\tau\in[0,1],$ (A.35)

with $c_{1}({\mathbf{x}}_{0})$ defined in (A.40) and $\lim_{\tau\to
0}{\mathbf{x}}_{\tau}={\mathbf{x}}_{0}$. To normalize the family of vectors
${\mathbf{x}}_{\tau}$, $\tau\in[0,1]$ we define

$\widetilde{{\mathbf{x}}}_{\tau}:=\frac{{\mathbf{x}}_{\tau}}{\|{\mathbf{x}}_{\tau}\|_{2}},$
(A.36)

still satisfying $\lim_{\tau\to
0}\widetilde{{\mathbf{x}}}_{\tau}={\mathbf{x}}_{0}$. For $\tau\in[0,1]$, we
estimate the propagator norm as

$\big{\|}e^{-B\tau}\big{\|}_{2}^{2}\geq\big{\|}e^{-Bt}\widetilde{{\mathbf{x}}}_{\tau}\big{\|}_{2}^{2}\Big{|}_{t=\tau}=\frac{\big{\|}e^{-B\tau}{\mathbf{x}}_{\tau}\big{\|}_{2}^{2}}{\|{\mathbf{x}}_{\tau}\|_{2}^{2}}=\frac{\|{\mathbf{x}}_{\tau}\|_{2}^{2}+g({\mathbf{x}}_{\tau};\tau)}{\|{\mathbf{x}}_{\tau}\|_{2}^{2}}=1-2\frac{c_{1}({\mathbf{x}}_{0})}{\|{\mathbf{x}}_{\tau}\|_{2}^{2}}\tau^{2{m_{HC}}+1}+\frac{{{\mathcal{O}}}(\tau^{2{m_{HC}}+2})}{\|{\mathbf{x}}_{\tau}\|_{2}^{2}},$
(A.37)

where we used definition (A.20) and (A.35). To derive a bound for the
multiplicative factor $c$ in (2.8), we consider the Taylor expansion (2.8) of
the propagator norm, use estimate (A.37) for $\tau>0$, and take the limit
$\tau\to 0$ such that

$-2c=\lim_{\tau\to
0}\frac{\big{\|}e^{-B\tau}\big{\|}_{2}^{2}-1}{\tau^{2{m_{HC}}+1}}\geq\lim_{\tau\to
0}\frac{g({\mathbf{x}}_{\tau};\tau)}{\|{\mathbf{x}}_{\tau}\|_{2}^{2}\
\tau^{2{m_{HC}}+1}}=\lim_{\tau\to
0}\bigg{(}-2\frac{c_{1}({\mathbf{x}}_{0})}{\|{\mathbf{x}}_{\tau}\|_{2}^{2}}+\frac{{{\mathcal{O}}}(\tau)}{\|{\mathbf{x}}_{\tau}\|_{2}^{2}}\bigg{)}=-2c_{1}({\mathbf{x}}_{0}).$

Hence, $c\leq c_{1}({\mathbf{x}}_{0})$ for all normalized vectors
${\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}\big{)}$.
Taking the minimum of $c_{1}({\mathbf{x}}_{0})$ over all normalized vectors
${\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}\big{)}$
yields the upper bound for the multiplicative factor $c$ as given in (A.34).
This finishes the proof. ∎

The proof of Lemma A.3 uses the following construction of a vector function:

######  Lemma A.4.

Let the ODE system (1.1) be conservative-dissipative, and let the system
matrix $B$ be hypocoercive with hypocoercivity index
${m_{HC}}\in{\mathord{\mathbb{N}}}$. Then, for each
${\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}\big{)}$,
there exists a polynomial vector function
${\mathbf{x}}_{\tau}\in{\mathord{\mathbb{C}}}^{n}$, $\tau\in[0,1]$ of the form

${\mathbf{x}}_{\tau}={\mathbf{x}}_{0}+\sum_{\ell=1}^{{m_{HC}}}b_{\ell}\tau^{\ell}B^{\ell}{\mathbf{x}}_{0},\qquad\text{for
a suitable choice of
}b_{\ell}\in{\mathord{\mathbb{R}}},\quad\ell=1,\ldots,{m_{HC}},$ (A.38)

such that

$g({\mathbf{x}}_{\tau};\tau):={\mathbf{x}}_{\tau}^{*}\Big{(}\sum_{j=1}^{\infty}\frac{\tau^{j}}{j!}\
U_{j}\Big{)}{\mathbf{x}}_{\tau}=-2c_{1}({\mathbf{x}}_{0})\tau^{a}+{\mathcal{O}}(\tau^{a+1})\qquad\text{for
}\tau\in[0,1],$ (A.39)

where $a=2{m_{HC}}+1$ and

$c_{1}({\mathbf{x}}_{0}):=\frac{\big{\|}\sqrt{B_{H}}B^{m_{HC}}{\mathbf{x}}_{0}\big{\|}_{2}^{2}}{(2{m_{HC}}+1)!\
\binom{2{m_{HC}}}{{m_{HC}}}}.$ (A.40)

###### Proof.

For ${\mathbf{x}}\in{\mathord{\mathbb{C}}}^{n}$ and $\tau\in[0,1]$, consider
$g({\mathbf{x}};\tau):={\mathbf{x}}^{*}\Big{(}\sum_{j=1}^{\infty}\frac{\tau^{j}}{j!}\
U_{j}\Big{)}{\mathbf{x}}$ using $U_{j}$ in the form (A.6). Following Lemma A.2
with $U=-B^{*}$, $V=U_{1}$, $W=-B$, and $m={m_{HC}}-1$, we rewrite
$g({\mathbf{x}};\tau)$ as

$\begin{split}g({\mathbf{x}};\tau)&=\sum_{j=1}^{\infty}\frac{\tau^{j}}{j!}\sum_{k=0}^{j-1}\tbinom{j-1}{k}((-B)^{k}{\mathbf{x}})^{*}U_{1}(-B)^{j-k-1}{\mathbf{x}}\\\
&=\sum_{j=0}^{{m_{HC}}-1}\frac{\tau^{2j+1}}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2j+1)!}{(k+2j+1)!}\tbinom{k+j}{j}\tau^{k}((-B)^{k+j}{\mathbf{x}})^{*}\Bigg{)}U_{1}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2j+1)!}{(k+2j+1)!}\tbinom{k+j}{j}\tau^{k}(-B)^{k+j}{\mathbf{x}}\Bigg{)}\\\
&\quad+\frac{\tau^{2{m_{HC}}+1}}{({2{m_{HC}}+1})!}\binom{2{m_{HC}}}{{m_{HC}}}\Delta^{({m_{HC}})}_{{2{m_{HC}}+1},{m_{HC}}}((-B)^{m_{HC}}{\mathbf{x}})^{*}U_{1}(-B)^{{m_{HC}}}{\mathbf{x}}\\\
&\quad+\sum_{j=2{m_{HC}}+2}^{\infty}\frac{\tau^{j}}{j!}\sum_{k={m_{HC}}}^{j-{m_{HC}}-1}\binom{j-1}{k}\Delta^{({m_{HC}})}_{j,k}((-B)^{k}{\mathbf{x}})^{*}U_{1}(-B)^{j-k-1}{\mathbf{x}}\\\
&=-2\sum_{j=0}^{{m_{HC}}-1}\frac{\tau}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{\|}\sqrt{B_{H}}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2j+1)!}{(k+2j+1)!}\tbinom{k+j}{j}\tau^{k+j}(-B)^{k+j}{\mathbf{x}}\Bigg{)}\Bigg{\|}_{2}^{2}\\\
&\quad-2\frac{\tau^{2{m_{HC}}+1}}{({2{m_{HC}}+1})!\
\binom{2{m_{HC}}}{{m_{HC}}}}\big{\|}\sqrt{B_{H}}B^{{m_{HC}}}{\mathbf{x}}\big{\|}_{2}^{2}+{\mathcal{O}}(\tau^{2{m_{HC}}+2})\,,\end{split}$
(A.41)

using that $U_{1}=-2B_{H}$ is a negative semi-definite Hermitian matrix, and
the identity

$\Delta^{({m_{HC}})}_{{2{m_{HC}}+1},{m_{HC}}}=\binom{2{m_{HC}}}{{m_{HC}}}^{-2},$

to rewrite the first and second term, respectively. The third term in (A.41)
can be bounded by $M_{2{m_{HC}}+2}\ \tau^{2{m_{HC}}+2}$ with
$M_{2{m_{HC}}+2}:=\sum_{j=2{m_{HC}}+2}^{\infty}\tfrac{1}{j!}(2\|B\|_{2})^{j}>0$,
using that $\Delta^{({m_{HC}})}_{j,k}\leq 1$ for $0\leq{m_{HC}}\leq k\leq
j-{m_{HC}}-1$ and $\tau\in[0,1]$.

Step 1. To estimate the second term in the last identity of (A.41) for
${\mathbf{x}}={\mathbf{x}}_{\tau}$, we use a polynomial ansatz for
${\mathbf{x}}_{\tau}$:

For $\tau\in[0,1]$, we consider the ansatz

${\mathbf{x}}_{\tau}:=\sum_{\ell=0}^{{m_{HC}}}\tau^{\ell}{\mathbf{x}}_{\ell}\qquad\text{with
the given
}{\mathbf{x}}_{0}\in\ker\big{(}\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{T}$}}_{{m_{HC}}-1}\big{)}\text{
and some ${\mathbf{x}}_{\ell}\in{\mathord{\mathbb{C}}}^{n}$,
$\ell=1,\ldots,{m_{HC}}$ to be chosen.}$ (A.42)

Then, we observe that

$\big{\|}\sqrt{B_{H}}B^{{m_{HC}}}{\mathbf{x}}_{\tau}\big{\|}_{2}^{2}=\big{\|}\sqrt{B_{H}}B^{{m_{HC}}}\sum_{\ell=0}^{{m_{HC}}}\tau^{\ell}{\mathbf{x}}_{\ell}\big{\|}_{2}^{2}=\big{\|}\sqrt{B_{H}}B^{{m_{HC}}}{\mathbf{x}}_{0}\big{\|}_{2}^{2}+{{\mathcal{O}}}(\tau),$
(A.43)

such that the second term in the last identity of (A.41) satisfies

$-2\frac{\tau^{2{m_{HC}}+1}}{({2{m_{HC}}+1})!\
\binom{2{m_{HC}}}{{m_{HC}}}}\big{\|}\sqrt{B_{H}}B^{{m_{HC}}}{\mathbf{x}}_{\tau}\big{\|}_{2}^{2}=-2\frac{\tau^{2{m_{HC}}+1}}{({2{m_{HC}}+1})!\
\binom{2{m_{HC}}}{{m_{HC}}}}\big{\|}\sqrt{B_{H}}B^{{m_{HC}}}{\mathbf{x}}_{0}\big{\|}_{2}^{2}+{{\mathcal{O}}}(\tau^{2{m_{HC}}+2}).$
(A.44)

Step 2. To estimate the first term in the last identity of (A.41) for
${\mathbf{x}}={\mathbf{x}}_{\tau}$, we refine the ansatz (A.42) for
${\mathbf{x}}_{\tau}$ as follows:

Consider (A.42) with

${\mathbf{x}}_{\ell}:=b_{\ell}B^{\ell}{\mathbf{x}}_{0},\quad\text{with some
}b_{\ell}\in{\mathord{\mathbb{R}}},\quad\ell=1,\ldots,{m_{HC}}\text{ to be
chosen.}$ (A.45)

We shall construct the coefficients $b_{\ell}\in{\mathord{\mathbb{R}}}$,
$\ell=1,\ldots,{m_{HC}}$ and set $b_{0}:=1$ such that the first term in the
last identity of (A.41) satisfies

$\sum_{j=0}^{{m_{HC}}-1}\frac{\tau}{(2j+1)!}\frac{1}{\binom{2j}{j}}\Bigg{\|}\sqrt{B_{H}}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2j+1)!}{(k+2j+1)!}\tbinom{k+j}{j}\tau^{k+j}(-B)^{k+j}{\mathbf{x}}_{\tau}\Bigg{)}\Bigg{\|}_{2}^{2}={{\mathcal{O}}}(\tau^{2{m_{HC}}+2}).$
(A.46)

Each term in the outer sum is non-negative. Therefore, for
$j=0,\ldots,{m_{HC}}-1$, we consider each

$S_{j}:=\Bigg{\|}\sqrt{B_{H}}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2j+1)!}{(k+2j+1)!}\tbinom{k+j}{j}\tau^{k+j}(-B)^{k+j}\sum_{\ell=0}^{{m_{HC}}}b_{\ell}\tau^{\ell}B^{\ell}{\mathbf{x}}_{0}\Bigg{)}\Bigg{\|}_{2}^{2}$
(A.47)

separately, and construct $b_{\ell}$, $\ell=1,\ldots,{m_{HC}}$ iteratively
such that $S_{j}={{\mathcal{O}}}(\tau^{2{m_{HC}}+1})$.

Starting with $\ell=1$, we determine $b_{\ell}=b_{1}$ by considering
$S_{{m_{HC}}-\ell}=S_{{m_{HC}}-1}$: Using

$\sqrt{B}_{H}{\mathbf{x}}_{0}=\ldots=\sqrt{B}_{H}B^{{m_{HC}}-1}{\mathbf{x}}_{0}=0,$
(A.48)

we can rewrite $S_{{m_{HC}}-1}$ as

$\begin{split}S_{{m_{HC}}-1}&=\Bigg{\|}\sqrt{B_{H}}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2{m_{HC}}-1)!}{(k+2{m_{HC}}-1)!}\tbinom{k+{m_{HC}}-1}{{m_{HC}}-1}\tau^{k+{m_{HC}}-1}(-B)^{k+{m_{HC}}-1}\sum_{\ell=0}^{{m_{HC}}}b_{\ell}\tau^{\ell}B^{\ell}{\mathbf{x}}_{0}\Bigg{)}\Bigg{\|}_{2}^{2}\\\
&=\Big{\|}\sqrt{B_{H}}\Big{(}\tau^{{m_{HC}}-1}(-B)^{{m_{HC}}-1}b_{1}\tau
B{\mathbf{x}}_{0}+\tfrac{(2{m_{HC}}-1)!}{(2{m_{HC}})!}\tbinom{{m_{HC}}}{{m_{HC}}-1}\tau^{{m_{HC}}}(-B)^{{m_{HC}}}{\mathbf{x}}_{0}+{{\mathcal{O}}}(\tau^{{m_{HC}}+1})\Big{)}\Big{\|}_{2}^{2}\\\
&=\tau^{2{m_{HC}}}\Big{\|}\sqrt{B_{H}}\Big{(}b_{1}-\tfrac{(2{m_{HC}}-1)!}{(2{m_{HC}})!}\tbinom{{m_{HC}}}{{m_{HC}}-1}\Big{)}(-B)^{m_{HC}}{\mathbf{x}}_{0}\Big{\|}_{2}^{2}+{{\mathcal{O}}}\big{(}\tau^{2{m_{HC}}+1}\big{)}.\end{split}$
(A.49)

Choosing

$b_{1}:=\tfrac{(2{m_{HC}}-1)!}{(2{m_{HC}})!}\tbinom{{m_{HC}}}{{m_{HC}}-1}=\tfrac{1}{2}$
(A.50)

yields $S_{{m_{HC}}-1}={{\mathcal{O}}}(\tau^{2{m_{HC}}+1})$.

Subsequently, for $\ell=2,\ldots,{m_{HC}}$, we determine $b_{\ell}$ by
considering $S_{{m_{HC}}-\ell}$ and using (A.48):

$\begin{split}S_{{m_{HC}}-\ell}&=\Bigg{\|}\sqrt{B_{H}}\Bigg{(}\sum_{k=0}^{\infty}\tfrac{(2{m_{HC}}-2\ell+1)!}{(k+2{m_{HC}}-2\ell+1)!}\tbinom{k+{m_{HC}}-\ell}{{m_{HC}}-\ell}\tau^{k+{m_{HC}}-\ell}(-B)^{k+{m_{HC}}-\ell}\sum_{p=0}^{{m_{HC}}}b_{p}\tau^{p}B^{p}{\mathbf{x}}_{0}\Bigg{)}\Bigg{\|}_{2}^{2}\\\
&=\Bigg{\|}\sqrt{B_{H}}\Bigg{(}\sum_{k=0}^{\ell}\tfrac{(2{m_{HC}}-2\ell+1)!}{(k+2{m_{HC}}-2\ell+1)!}\tbinom{k+{m_{HC}}-\ell}{{m_{HC}}-\ell}\tau^{k+{m_{HC}}-\ell}(-B)^{k+{m_{HC}}-\ell}b_{\ell-k}\tau^{\ell-k}B^{\ell-k}{\mathbf{x}}_{0}\Bigg{)}\Bigg{\|}_{2}^{2}+{{\mathcal{O}}}(\tau^{2{m_{HC}}+1})\\\
&=\tau^{2{m_{HC}}}\Bigg{\|}\sqrt{B_{H}}\Bigg{(}\sum_{k=0}^{\ell}\tfrac{(2{m_{HC}}-2\ell+1)!}{(k+2{m_{HC}}-2\ell+1)!}\tbinom{k+{m_{HC}}-\ell}{{m_{HC}}-\ell}b_{\ell-k}(-1)^{\ell-k}\Bigg{)}(-B)^{{m_{HC}}}{\mathbf{x}}_{0}\Bigg{\|}_{2}^{2}+{{\mathcal{O}}}(\tau^{2{m_{HC}}+1}).\end{split}$

Using $b_{p}$, $p=0,\ldots,\ell-1$ from the previous steps, and choosing

$b_{\ell}:=-(-1)^{\ell}\sum_{k=1}^{\ell}\tfrac{(2{m_{HC}}-2\ell+1)!}{(k+2{m_{HC}}-2\ell+1)!}\tbinom{k+{m_{HC}}-\ell}{{m_{HC}}-\ell}b_{\ell-k}(-1)^{\ell-k}$
(A.51)

yields $S_{{m_{HC}}-\ell}={{\mathcal{O}}}(\tau^{2{m_{HC}}+1})$. Choosing these
$b_{\ell}$, we have verified (A.46). Thus, using the ansatz (A.38) for
${\mathbf{x}}_{\tau}$ implies that $g({\mathbf{x}}_{\tau};\tau)$ from (A.41)
equals the r.h.s. of (A.44). This proves that the identity (A.39) holds for
$\tau\in[0,1]$. ∎

Acknowledgment: The first author (FA) was supported by the Austrian Science
Fund (FWF) via the FWF-funded SFB # F65. The second author (AA) was supported
by the Austrian Science Fund (FWF), partially via the FWF-doctoral school
”Dissipation and dispersion in non-linear partial differential equations” (#
W1245) and the FWF-funded SFB # F65. The third author (EC) was partially
supported by U.S. N.S.F. grant DMS 1501007.

The second author (AA) acknowledges fruitful discussions with Laurent Miclo on
the possible connection between the hypocoercivity index and the short time
decay rate, which eventually led to Theorem 2.7. The authors are grateful to
the anonymous referee, whose suggestions helped to simplify the proof of
Theorem 2.7.

## References

  * [1]
  * [2] F. Achleitner and A. Arnold.  The hypocoercivity index for ODEs: a refined short time analysis.  work in progress, 2022.
  * [3] F. Achleitner, A. Arnold, and E. A. Carlen.  On linear hypocoercive BGK models.  In From particle systems to partial differential equations. III, volume 162 of Springer Proc. Math. Stat., pages 1–37. Springer, Cham, 2016.
  * [4] F. Achleitner, A. Arnold, and E. A. Carlen.  On multi-dimensional hypocoercive BGK models.  Kinet. Relat. Models, 11(4):953–1009, 2018.
  * [5] F. Achleitner, A. Arnold, and E. A. Carlen.  The hypocoercivity index for the intermediate and large time behavior of ODEs.  work in progress, 2022.
  * [6] F. Achleitner, A. Arnold, and V. Mehrmann.  Hypocoercivity and controllability in linear semi-dissipative hamiltonian ordinary differential equations and differential-algebraic equations.  ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, page e202100171.
  * [7] F. Achleitner, A. Arnold, and V. Mehrmann.  Hypocoercivity and hypocontractivity concepts for linear dynamical systems.  preprint arXiv:2204.13033, 2022.
  * [8] F. Achleitner, A. Arnold, and B. Signorello.  On optimal decay estimates for ODEs and PDEs with modal decomposition.  In Stochastic dynamics out of equilibrium, volume 282 of Springer Proc. Math. Stat., pages 241–264. Springer, Cham, 2019.
  * [9] F. Achleitner, A. Arnold, and D. Stürzer.  Large-time behavior in non-symmetric Fokker–Planck equations.  Riv. Math. Univ. Parma (N.S.), 6(1):1–68, 2015.
  * [10] A. Arnold and J. Erb.  Sharp entropy decay for hypocoercive and non-symmetric Fokker–Planck equations with linear drift.  preprint arXiv:1409.5425, 2014.
  * [11] A. Arnold, C. Schmeiser, and B. Signorello.  Propagator norm and sharp decay estimates for Fokker–Planck equations with linear drift.  Comm. Math. Sc., 20(4):1047–1080, 2022.
  * [12] J. Dolbeault, C. Mouhot, and C. Schmeiser.  Hypocoercivity for linear kinetic equations conserving mass.  Trans. Amer. Math. Soc., 367(6):3807–3828, 2015.
  * [13] S. Gadat and L. Miclo.  Spectral decompositions and $L^{2}$-operator norms of toy hypocoercive semi-groups.  Kinet. Relat. Models, 6(2):317–372, 2013.
  * [14] D. Hinrichsen and A. J. Pritchard.  Mathematical systems theory. I, volume 48 of Texts in Applied Mathematics.  Springer-Verlag, Berlin, 2005.
  * [15] L. Hörmander.  Hypoelliptic second order differential equations.  Acta Math., 119:147–171, 1967.
  * [16] T. Kato.  Perturbation theory for linear operators.  Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Reprint of the 1980 edition.
  * [17] L. Kohaupt.  Differential calculus for some $p$-norms of the fundamental matrix with applications.  J. Comput. Appl. Math., 135(1):1–21, 2001.
  * [18] C. Mehl, V. Mehrmann, and P. Sharma.  Stability radii for linear Hamiltonian systems with dissipation under structure-preserving perturbations.  SIAM J. Matrix Anal. Appl., 37(4):1625–1654, 2016.
  * [19] L. Miclo and P. Monmarché.  Étude spectrale minutieuse de processus moins indécis que les autres.  In Séminaire de Probabilités XLV, volume 2078 of Lecture Notes in Math., pages 459–481. Springer, Cham, 2013.
  * [20] C. Villani.  Hypocoercivity.  Mem. Amer. Math. Soc., 202(950):iv+141, 2009.