# Spectral properties of locally eventually positive operator semigroups

Jonathan Mui  Jonathan Mui, School of Mathematics and Statistics, University
of Sydney, NSW 2006, Australia<EMAIL_ADDRESS>
###### Abstract.

This paper considers strongly continuous semigroups of operators on Banach
lattices which are _locally eventually positive_ , a property that was first
investigated in the context of concrete fourth-order evolution equations. We
construct a simple example to show that the typical assumptions on the
spectrum of the semigroup generator considered currently in the literature are
far from necessary in the more general setting of local eventual positivity.
Under minimal additional assumptions, we obtain results on the asymptotic
behaviour of orbits, as well as necessary conditions on the peripheral point
spectrum of locally eventually positive semigroups.

###### Key words and phrases:

Eventually positive semigroups, local eventual positivity, peripheral point
spectrum, spectral bound

###### 2010 Mathematics Subject Classification:

Primary: 47D06, 47A10. Secondary: 35B40

##  1\. Introduction

### Statement of main results

Semigroups of linear operators have long been used to study linear evolution
equations in an abstract functional analytical setting. When the underlying
space is an ordered function space, then the notion of a positive semigroup is
relevant. Today, much is known about stability and asymptotic behaviour of
positive semigroups, see for example the classic monograph [3] or the more
recent text [7]. By contrast, the study of _eventually positive semigroups_ —
in which, roughly speaking, positivity occurs only for sufficiently large
times — is still relatively young. Eventual positivity for matrix semigroups
was studied in [22], and since then has developed into a sub-field of its own
right, as a natural extension of Perron-Frobenius theory. Motivated by the
case study [8], a systematic theory of eventually positive semigroups and
resolvents on infinite-dimensional Banach lattices was developed by Daners,
Glück and Kennedy in [10, 9]. Within these papers, the reader will already
find various applications to partial differential equations. For more recent
developments, we point out in particular [11, 6] for theoretical results, and
[13, 19] for applications to fourth-order differential operators.

However, it was discovered in [17] that a still weaker positivity property was
required to describe the asymptotic behaviour of solutions to the _biharmonic
heat equation_ $u_{t}+(-\Delta)^{2}u=0$ on $\mathbb{R}^{N}$. This phenomenon,
now known as _local eventual positivity_ , has been investigated in many other
concrete PDE problems, and is the main subject of this article. The intuitive
idea is that the solution becomes eventually positive on every compact subset
of $\mathbb{R}^{N}$, but not globally. Based on this idea, we formulate an
abstract notion of a locally eventually positive semigroup on a Banach
lattice, and derive some spectral properties under quite general conditions.
We give a taste of some key results below, without yet stating the technical
definitions.

######  Theorem 1.1.

Let $(e^{tA})_{t\geq 0}$ be a real $C_{0}$-semigroup on a complex Banach
lattice $E$. Assume that the spectral bound $s(A)$ of the generator satisfies
$s(A)>-\infty$.

  1. (i)

_(Asymptotic positivity)_ If the semigroup is locally eventually positive and
the orbits are relatively compact, then for each $f\in E$ it holds that
$\operatorname{dist}(e^{tA}f,E_{+})\to 0$, where $E_{+}$ is the positive cone
of $E$.

  2. (ii)

_(Triviality of peripheral point spectrum)_ If the semigroup is strongly
locally eventually positive and the orbits of the rescaled semigroup
$(e^{t(A-s(A))})_{t\geq 0}$ are relatively weakly compact, then the
intersection of the point spectrum of the generator $\sigma_{p}(A)$ with the
line $s(A)+i\mathbb{R}$ is contained in $\\{s(A)\\}$.

The precise notion of (strong) local eventual positivity will be introduced in
Definition 3.1. Statement (i) is proved in Theorem 3.4, and Statement (ii) is
the content of Theorem 3.10. It should be noted that the above results do not
require _a priori_ assumptions on the spectrum of the semigroup generator, in
contrast to existing results on local eventual positivity in the abstract
Banach lattice setting. We will discuss this aspect further below.

### Organisation of the article

At the end of this introductory section, the reader will find a brief overview
of known results on local eventual positivity, which gives some context to the
results of this paper. Section 2 concerns the analysis of a particular locally
eventually positive semigroup which stands apart from the examples studied in
the current literature, as it shows that the usual sufficient conditions on
the spectrum to obtain local eventual positivity are far from necessary.

In Section 3, we formulate an abstract notion of local eventual positivity on
Banach lattices (Definition 3.1) and study the spectrum of generators of
locally eventually positive semigroups under mild regularity assumptions on
the semigroup. We prove the results announced in Theorem 1.1, and also give
sufficient conditions for the generator of a locally eventually positive
semigroup to have cyclic peripheral point spectrum (Corollary 3.6). Finally,
in Section 4 we investigate the question of whether the spectral bound $s(A)$
of the generator of a locally eventually positive semigroup is a spectral
value. We obtain an affirmative answer in Theorem 4.3 under quite general
conditions.

### Notations and terminology

We assume the reader is familiar with $C_{0}$-semigroups and basic notions in
the theory of vector lattices (also known as Riesz spaces). Let us fix some
conventions that will be used throughout the article.

We use the standard notations $\vee,\wedge$ to denote respectively the
supremum and infimum in a vector lattice. A normed vector lattice is a vector
lattice equipped with a _lattice norm_ , which means that if $|x|\leq|y|$,
then $\|x\|\leq\|y\|$. A _Banach lattice_ is, of course, a complete normed
vector lattice. By definition, a _complex Banach lattice_ is the
complexification of a Banach lattice over $\mathbb{R}$ — see for example [23,
II Section 11]. If $E$ is a complex Banach lattice, then the underlying Banach
space over $\mathbb{R}$ is called the _real part_ of $E$ and denoted by
$E_{\mathbb{R}}$.

For Banach lattices $E,F$, a linear operator $T:E\to F$ is called _positive_
if $Tf\geq 0$ in $F$ for all $f\geq 0$ in $E$. We write $E_{+}$ and $F_{+}$
for the positive cones in $E$ and $F$ respectively; thus an operator $T:E\to
F$ is positive if and only if $T(E_{+})\subseteq F_{+}$. If $E,F$ are complex
Banach lattices, then a bounded linear operator $T:E\to F$ is called _real_ if
$T(E_{\mathbb{R}})\subseteq F_{\mathbb{R}}$. Note that all positive operators
are real.

If $A:D(A)\subseteq E\to E$ is a closed, densely defined linear operator on a
Banach space $E$, we write $(e^{tA})_{t\geq 0}$ for the semigroup generated by
$A$. The _spectral bound_ of the generator is the quantity

$s(A):=\sup\\{\operatorname{Re}\lambda:\lambda\in\sigma(A)\\}\in[-\infty,\infty],$

with the convention that $s(A)=-\infty$ if $\sigma(A)=\emptyset$.

### Recent developments in local eventual positivity

In [17], the following phenomenon was observed in solutions to the biharmonic
heat equation $u_{t}+(-\Delta)^{2}u=0$ on $\mathbb{R}^{N}$: for every non-
zero, continuous and compactly supported initial datum $u_{0}$ with $u_{0}\geq
0$, for every compact subset $K\subset\mathbb{R}^{N}$, there exists a time
$\tau>0$ (depending on $K$ and $u_{0}$) such that the corresponding solution
$u=u(t,x)$ satisfies

$u(t,x)>0\quad\text{for all }x\in K\text{ and }t\geq\tau.$ (1.1)

In addition, the authors showed that there does _not_ exist any
$\tau\in(0,\infty)$ for which $u(t,x)>0$ for all $x\in\mathbb{R}^{N}$ and all
$t\geq\tau$. More informally, eventual positivity of the solution does not
hold ‘globally’. These features justify the term ‘local eventual positivity’.

For the biharmonic operator $(-\Delta)^{2}$ on $\mathbb{R}^{N}$, local
eventual positivity was studied further in [16], and later in [15], it was
shown that solutions to polyharmonic heat equations
$u_{t}+(-\Delta)^{\alpha}u=0$, for all $\alpha>1$, satisfy (1.1) as well. In
the recent preprint [12], the biharmonic heat equation is studied on infinite
cylinders $\mathbb{R}\times\Omega$ (with $\Omega$ a bounded domain in
$\mathbb{R}^{N}$ with smooth boundary), where it is shown that local eventual
positivity for the evolution equation depends on positivity properties for the
corresponding elliptic problem on the cross-section domain $\Omega$. On the
other hand, by building on the ideas of [9, 11], an operator-theoretic
approach to studying local eventual positivity was initiated by Arora in [5].
The results on (locally) eventually positive semigroups within the abstract
framework have thus far required rather strong assumptions on the spectrum of
the generator. In particular, the existence of isolated spectral values is
essential, since spectral projections are used extensively. From the point of
view of differential equations, this theory is therefore very effective for
problems on bounded domains, or on unbounded domains with probability measures
— see [4, Section 3.2] for a recent application in the latter setting.

On the other hand, many examples on unbounded domains do not fall within the
scope of the current operator-theoretic setting, such as polyharmonic heat
equations on Euclidean space (with the Lebesgue measure). In light of the
discussion above, for the further development of the general theory of (local)
eventual positivity, it is therefore natural to investigate what can be done
if the ‘usual’ spectral assumptions are dropped. The present article
represents a step in this direction.

##  2\. An explicit example

As we recalled in the introduction, the study of (local and global) eventual
positivity of semigroups thus far has relied on rather strong spectral
assumptions, such as requiring the spectral bound $s(A)$ of the generator to
be a pole of the resolvent operator. By contrast, in this section we show that
the generator of a locally eventually positive semigroup may have quite
pathological spectrum. The main feature of our example is a distinct ‘lack of
compactness’.

######  Example 2.1.

Consider the Banach lattice $E=L^{1}(\mathbb{R})\cap C_{0}(\mathbb{R})$ with
norm $\|f\|_{E}=\max\\{\|f\|_{1},\|f\|_{\infty}\\}$. There exists a real
$C_{0}$-semigroup $(T_{t})_{t\geq 0}$ on $E$ such that, for every $f\in
E_{+}\setminus\\{0\\}$ and every compact interval $J\subset\mathbb{R}$, there
exists $\tau\geq 0$ such that

$(T_{t}f)(x)>0\quad\text{for all }x\in J\text{ and }t\geq\tau.$

In short, $(T_{t})_{t\geq 0}$ is locally eventually positive in the sense of
property (1.1). Moreover, the generator $A$ of the semigroup satisfies
$\sigma(A)=i\mathbb{R}$.

We give the explicit construction now and postpone the discussion of the
spectrum of the generator to Proposition 2.2. Consider the functional

$\varphi(f):=\int_{\mathbb{R}}f(x)\,dx\qquad\text{for all }f\in E.$

Clearly $|\varphi(f)|\leq\|f\|_{1}\leq\|f\|_{E}$, so $\varphi\in E^{\prime}$.
Hence $G:=\ker\varphi$ is a closed subspace of $E$, and it is obviously
translation invariant. Now fix $f_{0}\in E$ such that $0<f_{0}(x)\leq 1$ for
all $x\in\mathbb{R}$ and $\int_{\mathbb{R}}f_{0}(x)\,dx=1$ (for example, one
may choose the standard Gaussian $f_{0}(x)=(2\pi)^{-1/2}e^{-x^{2}/2}$).
Consider the rank-one projection $Pf:=(\varphi\otimes
f_{0})(f)=\varphi(f)f_{0}$, and observe that $\ker P=\ker\varphi$. Then every
$f\in E$ can be expressed as

$f=Pf-(I-P)f=\varphi(f)f_{0}+[f-\varphi(f)f_{0}],$

and we obtain the direct sum decomposition

$E=F\oplus G,\qquad\text{where }F=\operatorname{rg}P\text{ and }G=\ker P.$

Let $(S_{t})_{t\geq 0}$ denote the left translation semigroup on $E$, i.e.
$(S_{t}f)(x):=f(x+t)$ for all $f\in E,t\geq 0$. We define a semigroup on $E$
by $T_{t}:=I_{F}\oplus\tau_{t}|_{G}$, which has the explicit formula

$(T_{t}f)(x)=\varphi(f)f_{0}(x)+[f(x+t)-\varphi(f)f_{0}(x+t)]\qquad t\geq
0,x\in\mathbb{R}$ (2.1)

for all $f\in E$. To see that (1.1) holds, let $f\in E_{+}\setminus\\{0\\}$ be
arbitrary, and fix a compact interval $[-R,R]$. Since $f_{0}(x)>0$ for all
$x\in\mathbb{R}$ and $f_{0}$ is continuous on the compact set $[-R,R]$, there
exists $\delta=\delta(R)>0$ such that $f_{0}(x)\geq\delta$ for all
$x\in[-R,R]$. On the other hand, $f_{0}\in C_{0}(\mathbb{R})$ implies that
there exists $M>0$ such that $0<f_{0}(x)<\delta$ whenever $|x|\geq M$. For all
$t>0$ sufficiently large such that $|x+t|\geq M$, it follows that
$0<f_{0}(x+t)<\delta$, and therefore there exists $\tau>0$ such that
$f_{0}(x)-f_{0}(x+t)>0$ for all $x\in[-R,R]$ and all $t\geq\tau$. We find

$(T_{t}f)(x)=\varphi(f)[f_{0}(x)-f_{0}(x+t)]+f(x+t)>0$ (2.2)

for all $x\in[-R,R]$ and all $t\geq\tau$.

As an additional observation, formula (2.2) shows that the time to positivity
can be chosen to depend only on the difference $f_{0}(x)-f_{0}(x+t)$, since
$\varphi(f)>0$ and $f(x+t)\geq 0$ for all $x\in\mathbb{R},t\geq 0$ and $f\in
E_{+}\setminus\\{0\\}$. Thus the semigroup can in fact be called _uniformly_
locally eventually positive.

We now show that the generator of the semigroup constructed above has spectrum
equal to the imaginary axis.

######  Proposition 2.2.

The generator $A$ of the semigroup defined by (2.1) is given by

$A=0_{|F}\oplus B_{|G},\qquad D(A)=F\oplus D(B_{|G})$

where $B_{|G}$ generates the left translation semigroup on $G$ and is thus
given by $Bf=\frac{df}{dx}$ with

$D(B_{|G})=\left\\{f\in G:\frac{df}{dx}\in G\right\\}.$

The spectrum of $A$ is $\sigma(A)=i\mathbb{R}$, and 0 is an eigenvalue with a
one-dimensional eigenspace spanned by the positive function $f_{0}$.

###### Proof.

The description of the generator follows from standard facts about subspace
semigroups and generators of translation semigroups on $C_{0}(\mathbb{R})$,
see for example [14, Chapter II, 2.2.3 & 2.2.10]. Since
$|\varphi(f)|\leq\|f\|_{1}$, and $\|f_{0}\|_{\infty}\leq 1$ and
$\|f_{0}\|_{1}=1$, from formula (2.1) we can derive the simple estimates

$\displaystyle|T_{t}f(x)|$ $\displaystyle\leq
2\|f\|_{1}+\|f\|_{\infty}\quad\text{for all }x\in\mathbb{R},\text{and}$
$\displaystyle\|T_{t}f\|_{1}$ $\displaystyle\leq 3\|f\|_{1}\quad\text{for all
}t\geq 0.$

Hence

$\|T_{t}f\|_{E}\leq 3\|f\|_{E}$

which shows that $(T_{t})_{t\geq 0}$ is a uniformly bounded semigroup. It is
also clear from (2.1) that $(T_{t})_{t\geq 0}$ extends to a bounded
$C_{0}$-group on $E$, since this is true of the translation semigroups. Hence
we deduce that $\sigma(A)\subseteq i\mathbb{R}$.

Now we show that every $\lambda\in i\mathbb{R}\setminus\\{0\\}$ is an
approximate eigenvalue of $A$. Consider the case $\lambda=i\alpha$ with
$\alpha>0$. We claim that

$w_{n}(x):=\frac{1}{\sqrt{n}}\left(1+\frac{ix}{n}\right)e^{-\frac{\alpha
x^{2}}{2n}+i\alpha x},\qquad x\in\mathbb{R},n\in\mathbb{N}$ (2.3)

forms a sequence of approximate eigenvectors for the spectral value $i\alpha$.
Firstly we observe that

$|w_{n}(x)|=\frac{1}{\sqrt{n}}\sqrt{1+\tfrac{x^{2}}{n^{2}}}e^{-\frac{\alpha
x^{2}}{2n}},$

and thus

$\|w_{n}\|_{1}\geq\frac{1}{\sqrt{n}}\int_{\mathbb{R}}e^{-\frac{\alpha
x^{2}}{2n}}\,dx=\sqrt{\frac{2\pi}{\alpha}}.$

On the other hand, it is easily seen that $\|w_{n}\|_{\infty}\to 0$ as
$n\to\infty$, and using the elementary inequality $(a+b)^{1/2}\leq
a^{1/2}+b^{1/2}$ for all $a,b\geq 0$, we estimate

$\displaystyle\int_{\mathbb{R}}|w_{n}(x)|\,dx$
$\displaystyle\leq\frac{1}{\sqrt{n}}\int_{\mathbb{R}}e^{-\frac{\alpha
x^{2}}{2n}}\,dx+\frac{1}{n\sqrt{n}}\int_{\mathbb{R}}|x|e^{-\frac{\alpha
x^{2}}{2n}}\,dx$
$\displaystyle=\sqrt{\frac{2\pi}{\alpha}}+\frac{1}{\sqrt{n}}\int_{\mathbb{R}}|y|e^{-\frac{\alpha
y^{2}}{2}}\,dy$
$\displaystyle\leq\sqrt{\frac{2\pi}{\alpha}}+\int_{\mathbb{R}}|y|e^{-\frac{\alpha
y^{2}}{2}}\,dy,$

which shows that $\sup_{n\in\mathbb{N}}\|w_{n}\|_{1}<\infty$. Hence there
exists a constant $C>0$ such that

$\sqrt{\frac{2\pi}{\alpha}}\leq\|w_{n}\|_{1}+\|w_{n}\|_{\infty}=\|w_{n}\|_{E}\leq
C$

for all $n\in\mathbb{N}$. Thus $(w_{n})_{n\in\mathbb{N}}$ is a bounded
sequence in $E$ with norms uniformly bounded away from 0. Clearly each $w_{n}$
is a smooth function. To show that $w_{n}\in D(A)$, it remains to check that
$\int_{\mathbb{R}}w_{n}\,dx=0$ for all $n\in\mathbb{N}$. Observe that

$w_{n}=(u_{n}+iv_{n})e^{i\alpha(\cdot)},\quad\text{where
}u_{n}(x)=\frac{1}{\sqrt{n}}e^{-\frac{\alpha x^{2}}{2n}},\quad
v_{n}(x)=\frac{x}{n\sqrt{n}}e^{-\frac{\alpha x^{2}}{2n}}.$

Then $v_{n}=-(1/\alpha)u_{n}^{\prime}$, and integration by parts reveals that

$\int_{\mathbb{R}}u_{n}(x)\cos(\alpha
x)\,dx=-\frac{1}{\alpha}\int_{\mathbb{R}}u_{n}^{\prime}(x)\sin(\alpha
x)\,dx=\int_{\mathbb{R}}v_{n}(x)\sin(\alpha x)\,dx.$ (2.4)

By considering odd and even symmetries together with (2.4), we find

$\displaystyle\int_{\mathbb{R}}w_{n}(x)\,dx$
$\displaystyle=\int_{\mathbb{R}}(u_{n}(x)+iv_{n}(x))(\cos(\alpha
x)+i\sin(\alpha x))\,dx$ $\displaystyle=\int_{\mathbb{R}}[u_{n}(x)\cos(\alpha
x)-v_{n}(x)\sin(\alpha x)]+i[v_{n}(x)\cos(\alpha x)+u_{n}(x)\sin(\alpha
x)]\,dx$ $\displaystyle=0$

and hence $w_{n}\in D(A)$ for all $n\in\mathbb{N}$. Finally, an easy
computation yields

$Aw_{n}-i\alpha w_{n}=w_{n}^{\prime}-i\alpha
w_{n}=\frac{1}{n\sqrt{n}}\left(1-\frac{\alpha x}{n}\right)e^{-\frac{\alpha
x^{2}}{2n}+i\alpha x}.$

Clearly $\|Aw_{n}-i\alpha w_{n}\|_{\infty}\leq C^{\prime}n^{-3/2}\to 0$ as
$n\to\infty$, for an appropriate constant $C^{\prime}>0$. Then

$\displaystyle\|Aw_{n}-i\alpha w_{n}\|_{1}$
$\displaystyle\leq\frac{1}{n\sqrt{n}}\int_{\mathbb{R}}e^{-\frac{\alpha
x^{2}}{2n}}\,dx+\frac{\alpha}{n^{2}\sqrt{n}}\int_{\mathbb{R}}|x|e^{-\frac{\alpha
x^{2}}{2n}}\,dx$ $\displaystyle=\frac{1}{n}\sqrt{\frac{2\pi}{\alpha
n}}+\frac{2\alpha}{n^{2}\sqrt{n}}\int_{0}^{\infty}xe^{-\frac{\alpha
x^{2}}{2n}}\,dx$ $\displaystyle=\frac{1}{n}\sqrt{\frac{2\pi}{\alpha
n}}+\frac{2}{n\sqrt{n}}\longrightarrow 0$

as $n\to\infty$. We conclude that $\|Aw_{n}-i\alpha w_{n}\|_{E}\to 0$ as
$n\to\infty$, and hence $(w_{n})_{n\in\mathbb{N}}$ is a sequence of
approximate eigenvectors corresponding to $\lambda=i\alpha$. If $\alpha<0$, we
simply write $0<\beta:=-\alpha$, and apply the above construction with $\beta$
in place of $\alpha$. Finally, it is obvious that $Af_{0}=0$, so
$F=\mathrm{span}(f_{0})$ is the one-dimensional eigenspace corresponding to
the eigenvalue 0. ∎

######  Remark 2.3.

Note that the spectral bound $s(A)$ of the semigroup in Example 2.1 is $0$,
and is an eigenvalue of the generator $A$. Proposition 2.2 shows that $0$ is
not a pole of the resolvent, since it is not an isolated spectral value.
Nevertheless, the associated eigenspace is one-dimensional and spanned by the
strictly positive function $f_{0}$. One may be tempted to apply [5, Theorem
3.3] to deduce at least individual local eventual positivity, but this is not
possible since the semigroup (2.1) does not converge strongly on $E$ as
$t\to\infty$.

##  3\. General results on the spectrum of locally eventually positive
semigroups

###  3.1. Lattice homomorphisms and band projections

In this section, we present some consequences of local eventual positivity
under minimal assumptions on the spectrum of the generator. Recall that in a
vector lattice $E$, two elements $x,y$ are _disjoint_ if $|x|\wedge|y|=0$, and
we write $x\perp y$. Given a non-empty subset $A\subset E$, the _disjoint
complement_ $A^{\mathrm{d}}$ of $A$ consists of all $x\in E$ for which $x\perp
y$ for all $y\in A$. A _band_ $B$ is an ideal that is _order closed_ , that
is, if $D\subset B$ is a subset such that $\sup D$ exists in $E$, then $\sup
D\in B$.

We will also require some special types of positive operators. If there is an
ideal $A$ of $E$ such that $E=A\oplus B$, then $B$ is called a _projection
band_ , and in this case, it holds that $A=B^{\mathrm{d}}$. We may then define
a positive projection $P:E\to E$ such that $\operatorname{rg}P=B$. We call $P$
the _band projection_ (or _order projection_ in some texts) onto $B$, and the
complementary projection $I-P$ satisfies
$\operatorname{rg}(I-P)=B^{\mathrm{d}}$. It is known that a projection $P$ on
a vector lattice $E$ is a band projection if and only if $0\leq P\leq I$,
where $I$ is the identity operator on $E$. Moreover, if $A,B$ are bands in $E$
with corresponding band projections $P_{A},P_{B}$, then

$A\subseteq B\iff P_{A}\leq P_{B}\iff P_{A}P_{B}=P_{B}P_{A}=P_{A}.$ (3.1)

If $P$ is the band projection onto a band $B$, then note that $I-P$ is the
band projection onto the disjoint complement $B^{\mathrm{d}}$. Proofs for
these standard facts may be found, for example, in [1, Chapter 1, Section 3].

Next, let us recall that for vector lattices $E,F$ over $\mathbb{R}$, a
_lattice homomorphism_ is a linear operator $T:E\to F$ such that $T$ preserves
the lattice operations: $T(x\vee_{E}y)=Tx\vee_{F}Ty$. It is simple to check
that $T$ is positive, and that the defining property is equivalent to the
condition $|Tx|_{F}=T|x|_{E}$. Another useful characterisation is that $T$ is
a lattice homomorphism if and only if $T$ is positive and _disjointness
preserving_ , that is,

$x\wedge_{E}y=0\Rightarrow Tx\wedge_{F}Ty=0,$ (3.2)

cf. [1, Theorem 2.14]. It follows that every band projection is a lattice
homomorphism, since if $x\wedge y=0$ in $E$, then $0\leq Px\leq x$ and $0\leq
Py\leq y$ imply $0\leq Px\wedge Py\leq x\wedge y=0$, so $Px\wedge Py=0$.

###  3.2. Local eventual and asymptotic positivity

By way of motivation, let us recall a situation described in [5, Section 7].
If $(K_{n})_{n\in\mathbb{N}}$ is an increasing sequence (i.e. $K_{n}\subseteq
K_{n+1}$ for all $n\in\mathbb{N}$) of bands in $E$ such that
$\bigcup_{n\in\mathbb{N}}K_{n}=E$, and if $P_{n}$ denotes the band projection
onto $K_{n}$, then by property (3.1), we have $0\leq P_{n}\leq P_{n+1}\leq I$
for all $n\in\mathbb{N}$. Thus we speak of an _increasing sequence_ of band
projections $(P_{n})_{n\in\mathbb{N}}$. Each projection $P_{n}$ localises a
function $f$ onto the compact subset $K_{n}$. Given a semigroup
$(e^{tA})_{t\geq 0}$, we may then ask if each localised orbit $t\mapsto
P_{n}e^{tA}f$ becomes eventually positive.

With the concepts introduced in the previous subsection, we are ready to
formulate an abstract definition of local eventual positivity for
$C_{0}$-semigroups on Banach lattices. Recall also that in a normed vector
lattice $E$, a positive element $u\in E$ is called _quasi-interior_ if the
principal ideal $E_{u}$ (i.e. the ideal generated by $u$) is dense in $E$.

######  Definition 3.1.

Let $E$ be a complex Banach lattice, and let there be given an increasing
sequence of positive operators $(P_{n})_{n\in\mathbb{N}}$ on $E$ converging
strongly to the identity. A $C_{0}$-semigroup $(e^{tA})_{t\geq 0}$ on $E$ with
generator $A$ is said to have the _LEP property_ with respect to the sequence
$(P_{n})_{n\in\mathbb{N}}$ if for every $n\in\mathbb{N}$ and $f\in
E_{+}\setminus\\{0\\}$, there exists $\tau>0$ (depending on $n$ and $f$) such
that

$P_{n}e^{tA}f\geq 0\quad\text{for all }t\geq\tau.$ (LEP)

If in addition, for all $t\geq\tau$, $P_{n}e^{tA}f$ is even a quasi-interior
point of the closed sublattice generated by $P_{n}E$ in $E$, we say that the
semigroup has the _strong LEP property_ with respect to the sequence
$(P_{n})_{n\in\mathbb{N}}$.

######  Remark 3.2.

The abbreviation LEP stands for _local eventual positivity_. Example 3.3 below
shows that Definition 3.1 is a natural abstraction of the property (1.1).

In Definition 3.1, we have admitted the more general notion of an increasing
sequence of positive operators. Nevertheless, in many applications, the
sequence $(P_{n})_{n\in\mathbb{N}}$ consists of band projections — we refer
the reader [5, Section 5] for some examples. On the other hand, note that some
commonly occurring Banach lattices do not have non-trivial band projections.
This is the case for $E=C(K)$ where $K$ is a compact, connected Hausdorff
space (see [23, Example 5, p. 63]).

######  Example 3.3.

We give two concrete examples in which the assumptions of Definition 3.1 hold.

  1. (i)

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, and let
$(K_{n})_{n\in\mathbb{N}}$ be an increasing sequence of measurable subsets of
$\Omega$ such that $\bigcup_{n\in\mathbb{N}}K_{n}=\Omega$. We consider the
Banach lattice $E=L^{p}(\Omega,\mu)$ for $1\leq p<\infty$. It is easy to see
that the set of $f\in E$ with support in $K_{n}$ is a band. For each
$n\in\mathbb{N}$, let $\chi_{n}$ denote the indicator function of the set
$K_{n}$, and define the band projection $P_{n}f:=f\chi_{n}$. Then by the
dominated convergence theorem, $P_{n}$ converges strongly to the identity. If
a $C_{0}$-semigroup $(e^{tA})_{t\geq 0}$ on $E$ has the (LEP) property with
respect to $(P_{n})_{n\in\mathbb{N}}$, this means that for every
$n\in\mathbb{N}$ and $f\in E_{+}\setminus\\{0\\}$, there exists $\tau>0$ such
that $(e^{tA}f)(x)\geq 0$ for a.e. $x\in K_{n}$, for all $t\geq\tau$.

  2. (ii)

Let $\Omega$ be a locally compact Hausdorff space, and assume there exists an
increasing sequence $(K_{n})_{n\geq 0}$ of compact subsets such that
$\bigcup_{n\geq 0}K_{n}=\Omega$. For each $n\geq 1$, let $\varphi_{n}$ be a
Urysohn function for the pair of closed sets $K_{n-1}$ and
$\overline{\Omega\setminus K_{n}}$: $\varphi_{n}$ is continuous on $\Omega$
with $0\leq\varphi_{n}\leq 1$, $\varphi_{n}\equiv 1$ on $K_{n-1}$ and
$\varphi_{n}\equiv 0$ on $\overline{\Omega\setminus K_{n}}$. On the Banach
lattice $C_{0}(\Omega)$, one verifies easily that the operators
$Q_{n}f:=\varphi_{n}f$ are lattice homomorphisms, and the sequence $Q_{n}$
increases strongly to the identity operator.

Our first result shows that if a semigroup has the (LEP) property, then under
mild regularity conditions, it is already ‘asymptotically positive’ in the
sense of (3.3) below. We remark that the notion of _individual asymptotic
positivity_ for $C_{0}$-semigroups was defined in [9, Definition 8.1] under
the additional assumptions $s(A)>-\infty$ and boundedness of the rescaled
semigroup $(e^{t(A-s(A))})_{t\geq 0}$. Given $x\in E$, the _orbit_ of $x$
under the semigroup $(e^{tA})_{t\geq 0}$ is the set $\\{e^{tA}x:t\geq 0\\}$.

######  Theorem 3.4.

Let $E$ be a complex Banach lattice, and let there be given a sequence
$(P_{n})_{n\in\mathbb{N}}$ of positive operators on $E$ increasing strongly to
the identity. Let $(e^{tA})_{t\geq 0}$ be a $C_{0}$-semigroup on $E$. If (LEP)
is satisfied with respect to the sequence $(P_{n})_{n\in\mathbb{N}}$, and the
semigroup has relatively compact orbits, then for every $x\in E_{+}$, it holds
that

$\operatorname{dist}(e^{tA}x,E_{+})\longrightarrow 0\quad\text{as
}t\to\infty.$ (3.3)

###### Proof.

Fix $x\in E_{+}\setminus\\{0\\}$. It is an elementary fact that if a sequence
of bounded linear operators on a Banach space converges strongly, then the
convergence is uniform on relatively compact subsets (see e.g. [20, III Lemma
3.7]). Since $P_{n}$ converges strongly to the identity $I$, and the set
$\\{e^{tA}x:t\geq 0\\}$ is relatively compact in $E$, we obtain

$\lim_{n\to\infty}\sup_{t\geq 0}\|(I-P_{n})e^{tA}x\|=0.$ (3.4)

By the (LEP) property, for each $n$ there exists $t_{n}\geq 0$ such that
$P_{n}e^{tA}x\geq 0$ for all $t\geq t_{n}$, and the sequence
$(t_{n})\subset[0,\infty)$ may be chosen to be strictly increasing. For each
$n\in\mathbb{N}$, we have

$\operatorname{dist}(e^{tA}x,E_{+})\leq\operatorname{dist}(P_{n}e^{tA}x,E_{+})+\operatorname{dist}((I-P_{n})e^{tA}x,E_{+})\leq\|(I-P_{n})e^{tA}x\|$

for all $t\geq t_{n}$. Now (3.4) shows that

$\sup_{t\geq t_{n}}\operatorname{dist}(e^{tA}x,E_{+})\leq\sup_{t\geq
0}\|(I-P_{n})e^{tA}x\|\longrightarrow 0$

as $n\to\infty$, which yields
$\limsup_{t\to\infty}\operatorname{dist}(e^{tA}x,E_{+})=0$. This proves the
convergence (3.3). ∎

We give a simple case in which the assumptions of the above theorem hold.

######  Corollary 3.5.

Let $(e^{tA})_{t\geq 0}$ be a $C_{0}$-semigroup on a complex Banach lattice
$E$, satisfying the (LEP) property with respect to a sequence
$(P_{n})_{n\in\mathbb{N}}$ of positive operators on $E$ increasing strongly to
the identity. If the semigroup is uniformly bounded, i.e. $\sup_{t\geq
0}\|e^{tA}\|<\infty$, and its generator $A$ has compact resolvent, then (3.3)
holds for every $x\in E_{+}$.

###### Proof.

If the generator $A$ has compact resolvent, then the orbits of the semigroup
are relatively compact, see [14, Corollary V.2.15(i)]. The conclusion then
follows from Theorem 3.4. ∎

We have shown above that relative compactness of orbits is a sufficient
condition for a semigroup satisfying (LEP) to be individually asymptotically
positive. In that case, the generator of the semigroup automatically has
interesting spectral properties. Recall that the _peripheral spectrum_ of $A$
is the part of $\sigma(A)$ lying on the line
$\\{\lambda\in\mathbb{C}:\operatorname{Re}\lambda=s(A)\\}$, and the
_peripheral point spectrum_ is the intersection of the peripheral spectrum
with the point spectrum $\sigma_{p}(A)$. A subset $S\subset\mathbb{C}$ is
called _cyclic_ if whenever $a+ib\in S$ (for some $a,b\in\mathbb{R}$), then
$a+inb\in S$ for all $n\in\mathbb{Z}$.

######  Corollary 3.6.

Let $(e^{tA})_{t\geq 0}$ be a $C_{0}$-semigroup on a complex Banach lattice
$E$ with $s(A)>-\infty$. If (LEP) is satisfied with respect to a sequence
$(P_{n})_{n\in\mathbb{N}}$ of positive operators on $E$ increasing strongly to
the identity, and the rescaled semigroup $(e^{t(A-s(A))})_{t\geq 0}$ has
relatively compact orbits, then the peripheral point spectrum of the generator
$A$ is cyclic.

###### Proof.

Since the rescaled semigroup $(e^{t(A-s(A))})_{t\geq 0}$ has relatively
compact orbits, in particular it is bounded. From this observation together
with Theorem 3.4, we find that the semigroup $(e^{tA})_{t\geq 0}$ is
individually asymptotically positive as defined in [9, Definition 8.1]. In
this case, the cyclicity of the peripheral point spectrum of $A$ follows from
[18, Theorem 6.3.2]. ∎

Let us remark that the proof in reference [18] of cyclicity of the peripheral
point spectrum for individually asymptotically positive semigroups uses the
Jacobs-deLeeuw-Glicksberg decomposition, which is also an important technical
tool in our analysis in Subsection 3.3.

We close the current subsection with some technical remarks concerning
Definition 3.1. At first glance, the definition of the strong (LEP) property
appears to be rather complicated. A notion of strong positivity that exists
currently in the literature is as follows: if $u\in E_{+}$ is a quasi-interior
point, we say that a vector $v\in E$ is _strongly positive with respect to_
$u$, denoted by $v\gg_{u}0$, if there exists a constant $c>0$ such that $v\geq
cu$. Following [5, Definition 2.4], we could say that a $C_{0}$-semigroup
$(e^{tA})_{t\geq 0}$ has the strong (LEP) property with respect to the
sequence $(P_{n})_{n\in\mathbb{N}}$ and the quasi-interior point $u$ if, for
every $n\in\mathbb{N}$ and $f\in E_{+}\setminus\\{0\\}$, there exists $\tau>0$
such that

$P_{n}e^{tA}f\gg_{P_{n}u}0\quad\text{for all }t\geq\tau.$ (3.5)

While the implied constant $c>0$ may depend on $n$, it is assumed to be
independent of $t\geq\tau$. Concrete examples in [5, Section 5] show that this
type of strong positivity does occur. However there are also natural examples
where (3.5) cannot be satisfied.

######  Example 3.7.

Consider the biharmonic heat equation $u_{t}+(-\Delta)^{2}u=0$ on
$\mathbb{R}^{N}$ with initial data $u_{0}\geq 0$, $u_{0}\not\equiv 0$, and for
simplicity assume that $u_{0}$ is continuous and compactly supported. In this
case, it was shown in [17] that for every compact subset
$K\subset\mathbb{R}^{N}$, there exists $\tau>0$ such that the corresponding
solution $u(t,\cdot)$ satisfies $u(t,x)>0$ on $K$ for all $t\geq\tau$. The
solution may be explicitly represented via an inverse Fourier transform

$u(t,x)=\frac{1}{(2\pi)^{N/2}}\int_{\mathbb{R}^{N}}e^{-t|\xi|^{4}}\hat{u}_{0}(\xi)e^{ix\cdot\xi}\,d\xi,\qquad
t>0,x\in\mathbb{R}^{N},$

where $\hat{u}_{0}$ is the Fourier transform of $u_{0}$. A simple computation
then yields the estimate

$\|u(t,\cdot)\|_{\infty}\leq
Ct^{-N/4}\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}\qquad\text{for all }t>0,$

for some constant $C>0$. Thus $\|u(t,\cdot)\|_{\infty}\to 0$ as $t\to\infty$,
which shows that the solution $u(t,\cdot)$ cannot dominate a strictly positive
multiple of a positive function on any compact set as $t\to\infty$. Hence
(3.5) does not hold. Intuitively, this is because $0$, the spectral bound of
the operator $(-\Delta)^{2}$ on $\mathbb{R}^{N}$, is not an eigenvalue.

######  Remark 3.8.

In the setting of (3.5), we have $P_{n}u>0$, i.e. $P_{n}u$ is a positive _non-
zero_ vector, for all $n\in\mathbb{N}$. Indeed, if $u\in E_{+}$ is a quasi-
interior point that belongs to the kernel of a positive operator $S$ on $E$,
then $S=0$. This is a straightforward consequence of the following useful
characterisation [23, II Theorem 6.3]: $u$ is a quasi-interior point of a
Banach lattice $E$ if and only if for each $x\in E_{+}$, the sequence
$x_{n}:=x\wedge nu$ converges to $x$ in norm.

Let $S$ be a positive operator on the Banach lattice $E$, and suppose $u$ is a
quasi-interior point. If $F$ is the closed sublattice of $E$ generated by the
image $S(E)$ and we view $S$ as an operator from $E$ to $F$, then [23, II
Proposition 6.4] shows that $Su$ is quasi-interior in $F$. If $v\in E_{+}$ is
such that $Sv\geq cSu$ for some constant $c>0$, then $E_{Su}\subset E_{Sv}$
follows from the definition of principal ideals, and we deduce that $Sv$ is
quasi-interior in $F$ as well. We have thus proved the following relationship
between the different notions of local eventual positivity.

######  Proposition 3.9.

Let $\mathcal{T}=(e^{tA})_{t\geq 0}$ be a $C_{0}$-semigroup on the complex
Banach lattice $E$. Let there be given a sequence of positive operators
$(P_{n})_{n\in\mathbb{N}}$ on $E$ increasing strongly to the identity, and a
quasi-interior point $u\in E_{+}$, such that $\mathcal{T}$ has the strong LEP
property with respect to $(P_{n})_{n\in\mathbb{N}}$ and $u$ as defined by
(3.5). Then $\mathcal{T}$ has the strong (LEP) property with respect to
$(P_{n})_{n\in\mathbb{N}}$ in the sense of Definition 3.1.

###  3.3. Local eventual strong positivity

In this section, we present a result on the triviality of the peripheral point
spectrum in the context semigroups that are strongly locally eventually
positive in the sense of Definition 3.1.

######  Theorem 3.10.

Let $E$ be a complex Banach lattice such that there exists an increasing
sequence $(Q_{n})_{n\in\mathbb{N}}$ of lattice homomorphisms converging
strongly to the identity. Let $(e^{tA})_{t\geq 0}$ be a real $C_{0}$-semigroup
on $E$ such that $s(A)>-\infty$ and the rescaled semigroup
$(e^{t(A-s(A))})_{t\geq 0}$ has relatively weakly compact orbits. If the
semigroup $(e^{tA})_{t\geq 0}$ satisfies the _strong_ (LEP) property with
respect to $(Q_{n})_{n\in\mathbb{N}}$, then the peripheral point spectrum of
the generator $A$ is contained in $\\{s(A)\\}$.

The overall strategy of the proof is similar to [18, Theorem 6.3.2]. In our
setting, however, the positivity properties of the semigroup are not ‘global’,
and thus some additional technical problems arise.

### Jacobs-deLeeuw-Glicksberg decomposition

To prove Theorem 3.10, we use the Jacobs-deLeeuw-Glicksberg decomposition,
which we review here briefly for the convenience of the reader. Firstly, if a
semigroup $\mathcal{T}\subset\mathcal{L}(E)$ has relatively weakly compact
orbits, then the semigroup operators form a relatively compact subset of
$\mathcal{L}(E)$ in the weak operator topology (WOT); see [21, Section 2.4,
Lemma 4.2]. Let $\mathcal{S}$ denote the WOT-closure of $\mathcal{T}$. Then
$\mathcal{S}$ is a compact, semitopological semigroup, and it is also abelian
if $\mathcal{T}$ is abelian. Now we introduce the _Sushkevich kernel_

$\mathcal{K}:=\bigcap_{T\in\mathcal{S}}T\mathcal{S}.$ (3.6)

It is not difficult to show that this is a non-empty subset of $\mathcal{S}$,
and is the intersection of all ideals in $\mathcal{S}$ (here, of course, we
use the algebraic definition of an ideal). Consequently $\mathcal{K}$ is the
unique minimal ideal of $\mathcal{S}$. Moreover, $\mathcal{K}$ is even a
compact group, and its unit $P$ is a projection onto the closed linear span of
the eigenvectors corresponding to the unimodular eigenvalues.

In the proofs below, we will assume without loss of generality that $s(A)=0$,
and thus speak of $e^{tA}$ in place of $e^{t(A-s(A))}$. The semigroup
$\mathcal{T}=\\{e^{tA}\\}_{t\geq 0}$ is of course abelian, and one checks that

$\mathcal{K}=\bigcap_{T\in\mathcal{S}}T\mathcal{S}\subseteq\bigcap_{t\geq
0}\overline{\\{e^{(t+s)A}:s\geq 0\\}}^{\text{WOT}}.$ (3.7)

The unit element $P\in\mathcal{K}$ is the projection onto the closed linear
span of the eigenvectors corresponding to the peripheral point spectrum
$\sigma_{p}(A)\cap i\mathbb{R}$. The range $PE$ is invariant under the
semigroup $\mathcal{T}$ since $P$ commutes with each semigroup operator, and
the restriction of the semigroup to $PE$ extends to a $C_{0}$-group, where for
each $t>0$, the inverse $(e^{tA}|_{PE})^{-1}$ is given by ${R_{t}}|_{PE}$ for
some $R_{t}\in\mathcal{K}$. All the results just quoted can be found in [21,
Section 2.4], where the general abstract theory is presented, or in [14,
Section V.2] where the presentation is adapted to operator semigroups.

### Proof of Theorem 3.10

We are now ready to give the proof of the main result. As a first step, we
show that the Sushkevich kernel $\mathcal{K}$ comprises of positive operators.
This is already known in the case of individually asymptotically positive
semigroups, c.f. the argument in [18, p. 94]. Note that the following lemma
does not require the strong (LEP) property.

######  Lemma 3.11.

Let $E$ be a complex Banach lattice such that there exists an increasing
sequence $(Q_{n})_{n\in\mathbb{N}}$ of positive operators converging strongly
to the identity. Let $(e^{tA})_{t\geq 0}$ be a real $C_{0}$-semigroup on $E$
with relatively weakly compact orbits, and assume that the semigroup has the
(LEP) property. Then the Sushkevich kernel, as defined in (3.7), consists of
positive operators.

###### Proof.

Let $R\in\mathcal{K}$ be arbitrary. The inclusion in (3.7) shows that there
exists a subnet of operators $(e^{t_{j}A})_{j\in J}$ (where $J$ is the
$R$-dependent indexing set) such that
$e^{t_{j}A}\overset{j}{\longrightarrow}R$ in the weak operator topology. Since
the semigroup is real, so is the limit operator $R$. Let $f\in
E_{+}\setminus\\{0\\}$ and $\varphi\in(E^{\prime})_{+}\setminus\\{0\\}$ be
arbitrary. Since $Q_{n}\to I$ strongly, for every $\varepsilon>0$ there is
$n_{0}\in\mathbb{N}$ such that

$|\braket{\varphi,Q_{n}Rf}-\braket{\varphi,Rf}|<\varepsilon\quad\text{for all
}n\geq n_{0}.$

Therefore

$\displaystyle\braket{\varphi,Rf}+\varepsilon$
$\displaystyle>\braket{Q_{n}^{\prime}\varphi,Rf}$
$\displaystyle=\lim_{j}\braket{Q_{n}^{\prime}\varphi,e^{t_{j}A}f}=\lim_{j}\braket{\varphi,Q_{n}e^{t_{j}A}f}\geq
0$

for all $n\geq n_{0}$. As $\varepsilon>0$ was arbitrary, this shows that
$\braket{\varphi,Rf}\geq 0$ for all $f\in E_{+}\setminus\\{0\\}$ and
$\varphi\in(E^{\prime})_{+}\setminus\\{0\\}$, hence $R$ is a positive
operator. ∎

To complete the proof of Theorem 3.10, we will require more subtle facts about
positive operators.

###### Completion of Proof of Theorem 3.10.

As before, let $P$ denote the unit of the group $\mathcal{K}$. If $P=0$ (and
hence $\mathcal{K}$ is the trivial group), then the peripheral point spectrum
is empty and the conclusion follows immediately. Hence assume $P\neq 0$. By
Lemma 3.11, $P$ is a positive projection. We have $e^{tA}P\in\mathcal{K}$ for
every $t\geq 0$ since $\mathcal{K}$ is an ideal in $\mathcal{S}$. As $P$
commutes with each operator $e^{tA}$, it follows that the semigroup
$\mathcal{T}$ restricted to the range $F:=PE$ of $P$ is a _positive_
semigroup. As remarked in Subsection 3, the restricted semigroup
$\mathcal{T}|_{F}$ extends to a $C_{0}$-group, where
$(e^{tA}|_{F})^{-1}=R_{t}|_{F}$ for some $R_{t}\in\mathcal{K}$. Now we recall
the fact that if $T\in\mathcal{L}(E)$ is an invertible operator such that both
$T$ and $T^{-1}$ are positive, then $T$ is a _lattice isomorphism_ (and the
converse is also true) — see [1, Theorem 2.15]. Therefore $\mathcal{T}|_{F}$
consists of lattice isomorphisms, which in particular are disjointness
preserving operators (recall (3.2)).

Since $P$ is a positive projection, the range $F$ is a Banach lattice with the
order inherited from $E$ and under a norm equivalent to that of $E$. For
convenience, we write $\vee$ and $\wedge$ for the supremum and infimum in $E$
respectively. Then the supremum in $F$ is given by $x\vee_{F}y=P(x\vee y)$,
and similarly for the infimum; see [23, Chapter III, Proposition 11.5] for
more details. We now show that $\dim F=1$. Assume for contradiction that $\dim
F\geq 2$. Then there exist $0\leq x,y\in F$ such that $x\perp_{F}y=0$, i.e.
$x,y$ are positive elements that are disjoint in $F$. In particular this means
$P(x\wedge y)=0$. Since $\mathcal{T}|_{F}$ is a semigroup of positive and
disjointness preserving operators, we have $e^{tA}x,e^{tA}y\geq 0$ and
$P(e^{tA}x\wedge e^{tA}y)=0$ for all $t\geq 0$. Recall that we have a sequence
$(Q_{n})_{n\in\mathbb{N}}$ of lattice homomorphisms increasing to the
identity. For every $n\in\mathbb{N}$, it follows that

$0=P(e^{tA}x\wedge e^{tA}y)\geq PQ_{n}(e^{tA}x\wedge
e^{tA}y)=P(Q_{n}e^{tA}x\wedge Q_{n}e^{tA}y)\geq 0,$

hence $P(Q_{n}e^{tA}x\wedge Q_{n}e^{tA}y)=0$ for all $t\geq 0$. Since the
semigroup $\mathcal{T}$ was assumed to have the strong (LEP) property,
$Q_{n}e^{tA}x$ and $Q_{n}e^{tA}y$ are quasi-interior points in the closed
sublattice generated by $Q_{n}E$, and so is $Q_{n}e^{tA}x\wedge Q_{n}e^{tA}y$
(see [23, Chapter II, Proposition 6.2]). However, recalling Remark 3.8, this
implies $P=0$ on $Q_{n}E$. Since this is true for each $n\in\mathbb{N}$, and
$Q_{n}$ increases to the identity on $E$, we find that $P=0$ on $E$, a
contradiction. Thus $\dim F=1$ as claimed.

The assertion of Theorem 3.10 is now obtained as follows: the restricted
semigroup $\mathcal{T}|_{F}$ is bounded on the Banach lattice $F=PE$, and
consists of lattice homomorphisms. Thus, by a classical result in the theory
of positive operator semigroups [3, C-III, Theorem 2.10], the entire spectrum
(not merely the _point_ spectrum) of the generator $A|_{F}$ is cyclic.
However, $F$ is the range of the spectral projection associated to the
peripheral point spectrum of $A$, and is one-dimensional. This implies
$\sigma_{p}(A)\cap i\mathbb{R}=\\{0\\}$, and the theorem is proved. ∎

As a straightforward consequence of Theorem 3.10, we obtain a result on strong
convergence of semigroup orbits similar to [6, Corollary 2.2] but under
slightly different positivity assumptions.

######  Corollary 3.12.

Let $E$ be a complex Banach lattice such that there exists an increasing
sequence $(Q_{n})_{n\in\mathbb{N}}$ of lattice homomorphisms converging
strongly to the identity. Let $(e^{tA})_{t\geq 0}$ be a real $C_{0}$-semigroup
on $E$ such that $s(A)>-\infty$. If the semigroup $(e^{tA})_{t\geq 0}$
satisfies the strong (LEP) property with respect to
$(Q_{n})_{n\in\mathbb{N}}$, then the following assertions are equivalent:

  1. (i)

The limit $\lim_{t\to\infty}e^{tA}f$ exists for every $f\in E$.

  2. (ii)

For each $f\in E$, the orbit $\\{e^{tA}f:t\geq 0\\}$ is relatively (strongly)
compact in $E$.

###### Proof.

Observe firstly that condition (i) implies (ii). If (ii) holds, then the
semigroup orbits are _a fortiori_ relatively weakly compact. Under both the
stated conditions, the semigroup is bounded, and thus $s(A)\leq 0$. The
Jacobs-deLeeuw-Glicksberg decomposition as described in [14, Theorem V.2.14]
yields $E=E_{s}\oplus E_{r}$, where $E_{s}=\\{f\in
E:\lim_{t\to\infty}e^{tA}f=0\\}$ and $E_{r}$ is the closed linear span of
eigenvectors corresponding to $\sigma_{p}(A)\cap i\mathbb{R}$. The conclusion
(ii)$\Rightarrow$(i) is therefore immediate if $s(A)<0$ (and we even have
$e^{tA}f\to 0$ as $t\to\infty$ for all $f\in E$). If $s(A)=0$, then Theorem
3.10 shows that the only eigenvalue of $A$ on the imaginary axis is 0, and (i)
follows as well. ∎

##  4\. Stability of locally eventually positive semigroups

The final section of this paper is concerned with the spectral bound of the
generator of semigroups satisfying the (LEP) property. In the study of
stability of semigroups, it is a natural question to ask if the spectral bound
$s(A)$ of the generator belongs to the spectrum (provided that
$\sigma(A)\neq\emptyset$ of course). It is well-known that this is true in the
case that the operator $A$ generates a _positive_ semigroup on a Banach
lattice (actually, this remains true on more general ordered Banach spaces,
see [2, Theorem 5.3.1]). Let us briefly recall the proof of this classic
result, as it is relevant to the forthcoming Theorem 4.3. It is a non-trivial
fact that if $A$ generates a positive $C_{0}$-semigroup on a Banach lattice,
then the Laplace transform representation of the resolvent

$R(\lambda,A)f=\int_{0}^{\infty}e^{-\lambda t}e^{tA}f\,dt\qquad\text{for all
}f\in E$

is valid as an improper Riemann integral for all
$\operatorname{Re}\lambda>s(A)$, which is a larger domain of convergence than
can be expected for a general $C_{0}$-semigroup. We refer to [7, Theorem 12.7]
for a direct proof. Using the positivity of the semigroup, a simple
calculation then yields

$|R(\lambda,A)f|\leq R(\operatorname{Re}\lambda,A)|f|\qquad\text{for all }f\in
E\text{ and }\operatorname{Re}\lambda>s(A),$ (4.1)

and from the above inequality, it is straightforward to obtain the conclusion
$s(A)\in\sigma(A)$—see for example [7, Corollary 12.9] for the details.

The Laplace transform representation of the resolvent was shown to hold for
individually eventually positive $C_{0}$-semigroups on Banach lattices in [10,
Proposition 7.1]. In place of inequality (4.1), we have instead the weaker
result

$|R(\lambda,A)f|\leq
R(\operatorname{Re}\lambda,A)|f|+r_{f}(\operatorname{Re}\lambda)\qquad\text{for
all }\operatorname{Re}\lambda>s(A),$ (4.2)

where $r_{f}:(s(A),\infty)\to E$ is a ‘remainder term’ that is norm-bounded as
$\operatorname{Re}\lambda\downarrow s(A)$ — see [10, Lemma 7.4] for the
details. Nevertheless, inequality (4.2) is sufficient to show that
$s(A)\in\sigma(A)$ remains true for individually eventually positive
$C_{0}$-semigroups, see [10, Theorem 7.6]. In Theorem 4.3 below, we will adopt
the same strategy to show that $s(A)\in\sigma(A)$ holds for semigroups
satisfying the (LEP) condition and an additional assumption on the essential
spectrum of the generator.

As a first step, we show that [2, Theorem 1.5.3] remains valid for eventually
positive functions on $[0,\infty)$ with values in an ordered Banach space with
_normal positive cone_ (see [2, Appendix C] for the definitions). Here it
suffices to note that all Banach lattices have normal positive cone. For
locally Bochner-integrable functions, we use the notation
$\operatorname{abs}(u)$ for the _abscissa of convergence_ and
$\operatorname{hol}(\hat{u})$ for the _abscissa of holomorphy_ of the Laplace
transform $\hat{u}$, as defined respectively on p. 27 and p. 33 of [2].

######  Lemma 4.1.

Let $X$ be an ordered Banach space with normal positive cone. Let $u\in
L^{1}_{\mathrm{loc}}([0,\infty);X)$ satisfy $\operatorname{abs}(u)<\infty$,
and suppose that there exists $\tau\geq 0$ such that $u(t)\geq 0$ for a.e.
$t\geq\tau$. Then $\operatorname{hol}(\hat{u})=\operatorname{abs}(u)$, and if
$\operatorname{abs}(u)>-\infty$, then $\hat{u}$ has a singularity at
$\operatorname{abs}(u)$.

###### Proof.

Define $v(t):=u(t+\tau)$ for all $t\geq 0$, so that $v(t)\geq 0$ for a.e.
$t\geq 0$. By [2, Theorem 1.5.3], it holds that
$\operatorname{hol}(\hat{v})=\operatorname{abs}(v)$. On the other hand, a
simple computation shows that

$\hat{u}(\lambda)=e^{-\lambda\tau}\hat{v}(\lambda)+\int_{0}^{\tau}e^{-\lambda
t}u(t)\,dt.$ (4.3)

The above formula shows that $\hat{u}$ converges if and only if $\hat{v}$
converges, and likewise $\hat{u}$ and $\hat{v}$ have the same domain of
holomorphy. Therefore
$\operatorname{hol}(\hat{u})=\operatorname{hol}(\hat{v})=\operatorname{abs}(v)=\operatorname{abs}(u)$.
∎

An essential ingredient for the proof of Theorem 4.3 is the following Laplace
transform result. It is derived similarly to [5, Proposition 6.1]; however, we
include a proof since our statement differs from the mentioned reference.

######  Proposition 4.2.

Let $E$ be a complex Banach lattice, and $T:E\to E$ a positive operator.
Suppose that $(e^{tA})_{t\geq 0}$ is a $C_{0}$-semigroup on $E$ such that
$[0,\infty)\ni t\mapsto u_{f}(t):=Te^{tA}f\in E$ is an eventually positive
function for every $f\in E_{+}$. Then, for every $f\in E$, the formula

$TR(\lambda,A)f=\int_{0}^{\infty}e^{-\lambda t}Te^{tA}f\,dt$ (4.4)

holds for all $\operatorname{Re}\lambda>\operatorname{abs}(u_{f})$, where the
integral converges as an improper Riemann integral. Moreover,
$\operatorname{abs}(u_{f})\in\sigma(A)$ and in particular,
$\operatorname{abs}(u_{f})\leq s(A)$.

###### Proof.

It is well-known that a positive operator $T:E\to E$ on a Banach lattice $E$
is automatically bounded (see for instance [1, Theorem 4.3]). Thus the formula
(4.4) holds for all $\operatorname{Re}\lambda>\omega_{0}$, where
$\omega_{0}\in\mathbb{R}$ is the growth bound of the semigroup. As
$E_{\mathbb{R}}$ spans $E$, and the positive cone of a Banach lattice is
generating (i.e. $E=E_{+}-E_{+}$), therefore it suffices to prove the
proposition for $f\in E_{+}$.

Given a positive vector $f$, the function $u_{f}(t)=Te^{tA}f$ is eventually
positive by assumption. Since we have
$\operatorname{hol}(u_{f})=\operatorname{abs}(u_{f})$ by Lemma 4.1, it follows
from the identity theorem for analytic functions that the formula (4.4) holds
for all $\operatorname{Re}\lambda>\operatorname{abs}(u_{f})$. Moreover, by
Lemma 4.1 again, $\operatorname{abs}(u_{f})$ is a singularity of the Laplace
transform $\widehat{u_{f}}$. Formula (4.4) yields that
$\operatorname{abs}(u_{f})$ is a singularity of the function $\lambda\mapsto
TR(\lambda,A)f$, and since $T$ is a bounded operator, this shows that
$R(\cdot,A)f$ has a singularity at $\operatorname{abs}(u_{f})$ as well.
Therefore $\operatorname{abs}(u_{n})\in\sigma(A)$ and
$\operatorname{abs}(u_{f})\leq s(A)$ follows immediately. ∎

Before stating the main theorem, we recall some further definitions. Let $A$
be a densely-defined closed linear operator on a Banach space $E$. The
_nullity_ of $A$ is $\mathrm{nul}(A):=\dim\ker(A)$, and the _deficiency_ of
$A$ is
$\mathrm{def}(A):=\operatorname{codim}\operatorname{rg}(A)=\dim(E/\operatorname{rg}A)$.
Recall that $A$ is called _Fredholm_ if its nullity and deficiency are both
finite. Then we can define the _Fredholm domain_ $\Delta_{F}(A)$ and the
_essential spectrum_ $\sigma_{\mathrm{ess}}(A)$ of $A$ respectively by

$\Delta_{F}(A):=\\{\lambda\in\mathbb{C}:A-\lambda I\text{ is
Fredholm}\\},\qquad\sigma_{\mathrm{ess}}(A):=\mathbb{C}\setminus\Delta_{F}(A).$
(4.5)

This leads to the natural definition of the _essential spectral bound_ :

$s_{\mathrm{ess}}(A):=\sup\\{\operatorname{Re}\lambda:\lambda\in\sigma_{\mathrm{ess}}(A)\\}\in[-\infty,\infty].$
(4.6)

Note that $\sigma_{\mathrm{ess}}(A)\subseteq\sigma(A)$, and hence
$s_{\mathrm{ess}}(A)\leq s(A)$. By [20, Theorem IV.5.28], every isolated
spectral value of $A$ with finite algebraic multiplicity belongs to
$\Delta_{F}(A)$.

We can now prove the main result of this section.

######  Theorem 4.3.

Let $(e^{tA})_{t\geq 0}$ be a $C_{0}$-semigroup on a complex Banach lattice
$E$, satisfying the (LEP) property with respect to an increasing sequence
$(P_{n})_{n\in\mathbb{N}}$ of positive operators. If
$s_{\mathrm{ess}}(A)<s(A)$, then $s(A)\in\sigma(A)$.

###### Proof.

Assume for contradiction that $s(A)\not\in\sigma(A)$. We fix $n\in\mathbb{N}$
and let $f\in E_{\mathbb{R}}$ be arbitrary. By the (LEP) property, we can
choose $\tau_{n}\geq 0$ sufficiently large so that $P_{n}e^{tA}f^{+}\geq 0$
and $P_{n}e^{tA}f^{-}\geq 0$ for all $t\geq\tau_{n}$. Then

$|P_{n}e^{tA}f|=|P_{n}e^{tA}(f^{+}-f^{-})|\leq|P_{n}e^{tA}f^{+}|+|P_{n}e^{tA}f^{-}|=P_{n}e^{tA}|f|$

for all $t\geq\tau_{n}$. From the above inequality, we deduce

$\displaystyle\left|\int_{0}^{T}e^{-\lambda t}P_{n}e^{tA}f\,dt\right|$
$\displaystyle\leq\int_{0}^{T}e^{-(\operatorname{Re}\lambda)t}|P_{n}e^{tA}f|\,dt$
$\displaystyle\leq\int_{0}^{\tau_{n}}e^{-(\operatorname{Re}\lambda)t}(|P_{n}e^{tA}f|-P_{n}e^{tA}|f|)\,dt+\int_{0}^{T}e^{-(\operatorname{Re}\lambda)t}P_{n}e^{tA}|f|\,dt$
$\displaystyle\leq
P_{n}\int_{0}^{\tau_{n}}e^{-(\operatorname{Re}\lambda)t}(|e^{tA}f|-e^{tA}|f|)\,dt+\int_{0}^{T}e^{-(\operatorname{Re}\lambda)t}P_{n}e^{tA}|f|\,dt.$

for all $T>\tau_{n}$ and $\operatorname{Re}\lambda>\operatorname{abs}(u_{n})$,
where $u_{n}(t):=P_{n}e^{tA}f$. Note that the positivity of $P_{n}$ was used
in the last inequality. If we let $T\to\infty$, then by formula (4.4) we
obtain

$|P_{n}R(\lambda,A)f|\leq
P_{n}R(\operatorname{Re}\lambda,A)|f|+P_{n}r_{n,f}(\operatorname{Re}\lambda)\qquad\text{for
all }\operatorname{Re}\lambda>\operatorname{abs}(u_{n}),$ (4.7)

where $r_{n,f}$ is defined by

$r_{n,f}(\sigma):=\int_{0}^{\tau_{n}}e^{-\sigma
t}(|e^{tA}f|-e^{tA}|f|)\,dt\qquad\text{for all }\sigma\in\mathbb{R}.$ (4.8)

Proposition 4.2 and the assumption that $s(A)\not\in\sigma(A)$ together imply
the strict inequality $\operatorname{abs}(u_{n})<s(A)$, and thus
$\max\\{\operatorname{abs}(u_{n}),s_{\mathrm{ess}}(A)\\}=:\omega<s(A)$. As
$\sigma(A)$ is a closed subset of $\mathbb{C}$, we deduce that there exists
$\lambda_{0}\in\sigma(A)$ of the form $\lambda_{0}=\delta+i\beta$ where
$\delta\in(\omega,s(A))$, $[\delta,\infty)$ belongs to the resolvent set
$\varrho(A)$, and $\beta\in\mathbb{R}\setminus\\{0\\}$. From the discussion
preceding the proof, $\lambda_{0}$ is an isolated spectral value, and hence it
is a pole of the resolvent of some order $k\in\mathbb{N}$. Let $Q_{-k}$ denote
the coefficient of the Laurent expansion of the resolvent about $\lambda_{0}$.

Now consider $\lambda=\lambda_{0}+\varepsilon$ for $\varepsilon>0$. We obtain

$|(\lambda-\lambda_{0})^{k}P_{n}R(\lambda,A)f|=\varepsilon^{k}|P_{n}R(\lambda,A)f|\leq\varepsilon^{k}P_{n}[R(\delta+\varepsilon,A)|f|+r_{n,f}(\delta+\varepsilon)]$
(4.9)

from inequality (4.7). As $\varepsilon\downarrow 0$, we have

$\varepsilon^{k}P_{n}R(\lambda,A)f\longrightarrow
P_{n}Q_{-k}f\quad\text{and}\quad\varepsilon^{k}P_{n}R(\delta+\varepsilon,A)|f|\longrightarrow
0$

in norm, since $\lambda_{0}$ is a pole of order $k$, while
$\delta=\operatorname{Re}\lambda_{0}\in\varrho(A)$. Furthermore,
$r_{n,f}(\delta+\varepsilon)$ remains norm-bounded (with $n$ fixed) as
$\varepsilon\to 0$. Thus the right-hand side of (4.9) vanishes as
$\varepsilon\downarrow 0$, implying that $P_{n}Q_{-k}=0$ on $E$, since $f\in
E_{\mathbb{R}}$ was arbitrary. The above arguments hold for each
$n\in\mathbb{N}$, and consequently we find that $P_{n}Q_{-k}=0$ for all
$n\in\mathbb{N}$. Recalling that $P_{n}$ converges strongly to the identity,
we conclude $Q_{-k}=0$, a contradiction. ∎

######  Remark 4.4.

It would be interesting to see if Theorem 4.3 remains true without the
assumption $s_{\mathrm{ess}}(A)<s(A)$, i.e. we impose only the obvious
condition $s(A)>-\infty$. We leave this as an open problem.

Under the assumptions of Theorem 4.3, we obtain the following analogue of [10,
Theorem 7.7] for LEP semigroups.

######  Theorem 4.5.

Let $(e^{tA})_{t\geq 0}$ be a $C_{0}$-semigroup on a complex Banach lattice
$E$, satisfying the (LEP) property with respect to an increasing sequence
$(P_{n})_{n\in\mathbb{N}}$ of positive operators. If
$s_{\mathrm{ess}}(A)<s(A)$, then the following assertions hold:

  1. (i)

$s(A)$ is a pole of the resolvent $R(\cdot,A)$ of some order $m\in\mathbb{N}$,
and an eigenvalue of $A$ with a positive eigenvector.

  2. (ii)

Every pole of the resolvent $R(\cdot,A)$ in the peripheral spectrum
$\sigma_{\mathrm{per}}(A)$ has order at most $m$.

###### Proof.

(i): We know that $s(A)\in\sigma(A)$ by Theorem 4.3. As shown in the
discussion preceding the proof of that Theorem, the assumption
$s_{\mathrm{ess}}(A)<s(A)$ implies that $s(A)$ is a pole of the resolvent.
Moreover, as remarked in [5, Corollary 6.2], for every $n\in\mathbb{N}$ and
$f\in E_{+}$ it holds that

$\lim_{\lambda\downarrow
s(A)}(\lambda-s(A))\operatorname{dist}(P_{n}R(\lambda,A)f,E_{+})=0;$ (4.10)

this follows in a straightforward manner from Proposition 4.2 (c.f. the
calculation in [10, Corollary 7.3]).

Let $m\in\mathbb{N}$ be the pole order of $s(A)$, and denote by $Q_{-m}$ the
coefficient of $(\lambda-s(A))^{-m}$ in the Laurent expansion of the resolvent
about $s(A)$. For every $n\in\mathbb{N}$, we have
$(\lambda-s(A))^{m}P_{n}R(\lambda,A)\to P_{n}Q_{-m}$ strongly as
$\lambda\downarrow s(A)$. Formula (4.10) then shows that

$0=\lim_{\lambda\downarrow
s(A)}(\lambda-s(A))^{m}\operatorname{dist}(P_{n}R(\lambda,A)f,E_{+})=\operatorname{dist}(P_{n}Q_{-m}f,E_{+})$

for every $n\in\mathbb{N}$ and $f\in E_{+}$. Therefore $P_{n}Q_{-m}$ is a
positive operator and non-zero for sufficiently large $n$, since $Q_{-m}\neq
0$ and $P_{n}$ converges strongly to the identity. Letting $n\to\infty$, we
find that $Q_{-m}$ is a positive, non-zero operator. It is then known that
$s(A)$ is an eigenvalue and $\operatorname{rg}Q_{-m}\neq\\{0\\}$ consists of
eigenvectors to $s(A)$. In addition, by the positivity of $Q_{-m}$, there
exists a positive eigenvector to $s(A)$. (Essential facts about the Laurent
expansion of the resolvent are collected in [10, Remark 2.1]. We note that
$Q_{-m}$ here corresponds to the operator $U^{m-1}$ in that reference).

(ii): Suppose that $\lambda_{0}:=s(A)+i\beta\in\sigma_{\mathrm{per}}(A)$, with
$0\neq\beta\in\mathbb{R}$, is a pole of the resolvent. Let $n\in\mathbb{N}$
and $f\in E_{\mathbb{R}}$ be arbitrary. For $\lambda=\lambda_{0}+\varepsilon$
with $\varepsilon>0$, we obtain

$|(\lambda-\lambda_{0})^{k}P_{n}R(\lambda,A)f|\leq\varepsilon^{k}P_{n}[R(s(A)+\varepsilon,A)|f|+r_{n,f}(s(A)+\varepsilon)]$

for every $k\in\mathbb{N}$, as in Formula (4.9). If $m$ is the pole order of
$s(A)$ and $k>m$, then the right-hand side of the above inequality converges
in norm to $0$ for each $n\in\mathbb{N}$ and $f\in E_{\mathbb{R}}$ as
$\varepsilon\downarrow 0$. However, we may choose some $f\in E_{\mathbb{R}}$
and sufficiently large $n$ such that the left-hand side has a non-zero limit
as $\varepsilon\downarrow 0$. Thus if $k$ is the pole order of $\lambda_{0}$,
then $k\leq m$. ∎

## Acknowledgements

Part of this work was initiated during a pleasant visit to the Bergische
Universität Wuppertal and the Technische Universität Dresden in June 2022.
Together with the University of Sydney, I am grateful for the financial
support of these three institutions. I thank Jochen Glück in particular for
sharing a number of useful ideas and insights, and I also thank Sahiba Arora
and Daniel Daners for valuable discussions.

## References

  * [1] C.D. Aliprantis, and O. Burkinshaw, _Positive operators_. Springer, Dordrecht, 2006. Reprint of the 1985 original.
  * [2] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, _Vector-valued Laplace transforms and Cauchy problems_ , 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. DOI: 10.1007/978-3-0348-0087-7
  * [3] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, _One-parameter semigroups of positive operators_ , Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. DOI: 10.1007/BFb0074922
  * [4] D. Addona, F. Gregorio, A. Rhandi and C. Tacelli, _Bi-Kolmogorov type operators and weighted Rellich’s inequalities_ , Nonlinear Diff. Eq. Appl. 29 (2022), No. 2, Paper No. 13, 37. DOI: 10.1007/s00030-021-00747-y
  * [5] S. Arora, _Locally eventually positive operator semigroups_ , J. Operator Theory 88 (2022), No. 1, 203–242. DOI: 10.7900/jot.2021jan26.2316
  * [6] S. Arora and J. Glück, _Spectrum and convergence of eventually positive operator semigroups_ , Semigroup Forum 103 (2021), No. 3, 791–811. DOI: 10.1007/s00233-021-10204-y
  * [7] A. Bátkai, M. Kramar Fijavž and A. Rhandi, _Positive Operator Semigroups_ , Operator Theory: Advances and Applications, vol. 257, Birkhäuser/Springer, Cham, 2017. DOI:10.1007/978-3-319-42813-0
  * [8] D. Daners, _Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator_ , Positivity 18 (2014), No. 2, 235-256. DOI: 10.1007/s11117-013-0243-7
  * [9] D. Daners, J. Glück, and J. B. Kennedy, _Eventually and asymptotically positive semigroups on Banach lattices_ , J. Differential Equations 261 (2016), 2607–2649. DOI: 10.1016/j.jde.2016.05.007
  * [10] D. Daners, J. Glück, and J. B. Kennedy, _Eventually positive semigroups of linear operators_ , J. Math. Anal. Appl. 433 (2016), 1561–1593. DOI: 10.1016/j.jmaa.2015.08.050
  * [11] D. Daners and J. Glück, _A criterion for the uniform eventual positivity of operator semigroups_ , Integral Equations and Operator Theory, 90 (2018), No. 4, Paper No. 46, 19. DOI: 10.1007/s00020-018-2478-y
  * [12] D. Daners, J. Glück, and J. Mui, _Local uniform convergence and eventual positivity of solutions to biharmonic heat equations_ , Preprint available at https://arxiv.org/abs/2111.02753, 2021.
  * [13] R. Denk, M. Kunze, and D. Ploß, _The bi-Laplacian with Wentzell boundary conditions on Lipschitz domains_ , Integral Equations and Operator Theory 93 (2021), Paper No. 13, 26. DOI: 10.1007/s00020-021-02624-w
  * [14] K-J. Engel, and R. Nagel, _One-parameter semigroups for linear evolution equations_ , vol. 194. Springer, Berlin, 2000.
  * [15] L. C. F. Ferreira and V. A. Ferreira, Jr., _On the eventual local positivity for polyharmonic heat equations_ , Proc. Amer. Math. Soc. 147 (2019), 4329–4341. DOI: 10.1090/proc/14565
  * [16] A. Ferrero, F. Gazzola, and H.-C. Grunau, _Decay and eventual local positivity for biharmonic parabolic equations_ , Discrete Contin. Dyn. Syst. 21 (2008), 1129–1157. DOI: 10.3934/dcds.2008.21.1129
  * [17] F. Gazzola and H.-C. Grunau, _Eventual local positivity for a biharmonic heat equation in $\mathbb{R}^{n}$_, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), 83–87. DOI: 10.3934/dcdss.2008.1.83
  * [18] J. Glück, _Invariant sets and long time behaviour of operator semigroups_ , PhD Thesis, Universität Ulm, 2016. DOI: 10.18725/OPARU-4238
  * [19] F. Gregorio and D. Mugnolo, _Bi-Laplacians on graphs and networks_ , J. Evol. Equ. 20 (2020), 191–232. DOI: 10.1007/s00028-019-00523-7
  * [20] T. Kato, _Perturbation theory for linear operators_ , Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.
  * [21] U. Krengel, _Ergodic theorems_ , De Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter & Co., Berlin, 1985.
  * [22] D. Noutsos and M.J. Tsatsomeros, _Reachability and holdability of nonnegative states_ , SIAM J. Matrix Anal. Appl., 30 (2008), No. 2, 700–712. DOI: 10.1137/070693850
  * [23] H.H. Schaefer, _Banach lattices and positive operators_ , Die Grundlagen der Mathematischen Wissenschaft, Band 215. Springer-Verlag, New York-Heidelberg, 1974.


# A reproduction of the Milky Way’s Faraday rotation measure map in galaxy
simulations from global to local scales

Stefan Reissl1, Ralf S. Klessen1,2, Eric W. Pellegrini3,
Daniel Rahner1, Rüdiger Pakmor3, Robert Grand4,
Facundo Gómez5,6, Federico Marinacci7, and Volker Springel3 E-mail:
<EMAIL_ADDRESS>
1Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische
Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany
2Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches
Rechnen,Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
3Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748
Garching, Germany
4Astrophysics Research Institute, Liverpool John Moores University, 146
Brownlow Hill, Liverpool, L3 5RF, UK
5Instituto de Investigación Multidisciplinar en Ciencia y Tecnología,
Universidad de La Serena, Raúl Bitrán 1305, La Serena, Chile
6Departamento de Física y Astronomía, Universidad de La Serena, Av. Juan
Cisternas 1200 Norte, La Serena, Chile
7Institute for Theory and Computation, Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA

Magnetic fields are of critical importance for our understanding of the origin
and long-term evolution of the Milky Way. This is due to their decisive role
in the dynamical evolution of the interstellar medium (ISM) and their
influence on the star-formation process [22, 11, 13, 4]. Faraday rotation
measures (RM) along many different sightlines across the Galaxy are a primary
means to infer the magnetic field topology and strength from observations [23,
24, 40, 15]. However, the interpretation of the data has been hampered by the
failure of previous attempts to explain the observations in theoretical models
and to synthesize a realistic multi-scale all-sky RM map [3, 7, 27]. We here
utilize a cosmological magnetohydrodynamic (MHD) simulation of the formation
of the Milky Way, augment it with a novel star cluster population synthesis
model for a more realistic structure of the local interstellar medium [31,
29], and perform detailed polarized radiative transfer calculations on the
resulting model [5, 33]. This yields a faithful first principles prediction of
the Faraday sky as observed on Earth. The results reproduce the observations
of the Galaxy not only on global scales, but also on local scales of
individual star-forming clouds. They also imply that the Local Bubble [1, 42]
containing our Sun dominates the RM signal over large regions of the sky.
Modern cosmological MHD simulations of the Milky Way’s formation, combined
with a simple and plausible model for the fraction of free electrons in the
ISM, explain the RM observations remarkably well, thus indicating the
emergence of a firm theoretical understanding of the genesis of magnetic
fields in our Universe across cosmic time.

Magnetic fields significantly influence the kinematical and morphlogical
properties of the ISM and contribute to regulating the birth of new
generations of stars [18]. To better understand this connection, several
observational techniques have been developed and perfected over the last
century. For example, dust polarization measurements of aligned dust grains
[2, 30] and synchrotron emission [4, 20, 35] allow us to infer the projected
line-of-sight (LOS) field orientation, while estimates of the LOS field
strength can be obtained from the Zeeman effect [9]. However, these methods
are only applicable to a limited set of parameters [5], and additional
uncertainties arise from our incomplete understanding of the microphysics
involved such as grain alignment mechanisms [2]. A complementary approach is
to determine the characteristic Faraday rotation measure (RM). It is based on
the fact that polarized radiation can change its polarization angle as it
passes though a magnetized and ionized medium. The observed signal depends on
the magnetic field strength and direction as well as on the density of free
electrons, and on the radiation frequency [37].

Since the early 1960’s, numerous attempts were made to reconstruct an all-sky
RM map of the Milky Way [23, 24, 40] from observations of pulsars and extra-
galactic background sources. Complementary synthetic data has proven its worth
for systematically interpreting and analyzing this plethora of observations
[3, 7, 27]. However, all existing approaches are hampered by the fact that the
distribution of thermal electrons as well as the detailed structure of the
magnetic field are not well known. While the large-scale properties are
usually well constrained [25, 14], crucial information about individual star-
forming regions and clouds is missing, which is needed to reproduce the
observed small-scale features.

Our approach goes beyond the current state-of-the-art and employs data from a
high-resolution cosmological simulation to reconstruct the large-scale
properties of the galaxy combined with a novel star cluster population
synthesis model, which introduces the missing small-scale physics.
Specifically, we take the Au-6 galaxy from the Auriga project [12], which is
able to reproduce the global star-formation rate and structure of the Milky
Way very well, while at the same time predicting the amplification of minute
primordial magnetic seed fields to micro-Gauss strength over secular
timescales. We keep the overall gas density and magnetic field structure, but
we discard the original stellar population and distribution of free electrons.
We then synthesize a new population of star clusters and calculate the
corresponding radiative and mechanical feedback based on the WARPFIELD cloud-
cluster evolution method [31, 32]. Next, we obtain the corresponding emission
from each cluster across the electromagnetic spectrum [28], use the polarized
radiative transfer code POLARIS [34] to build up the spatially varying
interstellar radiation field in the galaxy, and from that reconstruct the
distribution of free electrons in the ISM, as discussed by Pellegrini and
colleagues [29]. Finally, we perform a second POLARIS sweep to calculate
synthetic Faraday RM all-sky maps [35] from varying positions within the
galaxy.

The total integrated angle of linear polarization for an observer follows as

$\chi_{\rm obs}=\chi_{\rm source}+RM\times\left(\frac{c}{\nu}\right)^{2}\,,$
(1)

where $\chi_{\rm source}$ is the polarization angle of the source and $\nu$ is
the observed frequency. The integral

$RM=\frac{1}{2\pi}\frac{e^{2}}{m_{\rm
e}^{2}c^{4}}\int_{0}^{s_{\mathrm{obs}}}n_{\rm th}(s)B_{||}(s)\mathrm{d}s$ (2)

defines the rotation measure [6] for the non-relativistic limit along the LOS
$s$ towards the observer, $c$ is the speed of light, $e$ and $m_{\rm e}$ are
the electron charge and mass, respectively, and $B_{||}$ is the LOS magnetic
field component. Altogether we compute high-resolution Faraday RM all-sky maps
with a total number of $N=786,432$ pixels for ten distinct observer positions
within the Au-6 galaxy (which we denote P01 - P10). They are placed at roughly
the same distance to the galactic center as the Sun (that is at galactocentric
radii $8\,\mathrm{kpc}\leq R\leq 10\,\mathrm{kpc}$) and are located in a gas
density cavity similar to the Local Bubble that defines our own Galactic
environment [1, 42], as illustrated in Figure 6 of Pellegrini and colleagues
[29]..

Figure 1: _All-sky Faraday RM map of the Milky Way and of a model galaxy from
the Auriga suite of cosmological simulations._ In the top row we show the
observed maps of O12 (left) and the data from H22 (right). In the bottom row
(left) we provide the synthetic RM map constructed from the original Au-6 data
[27]. The lack of signal at large galactic latitudes and the failure to
reproduce small-scale features is evident. This problem is remedied by
combining the galaxy with a realistic small-scale model of star-cluster
formation and evolution and a self-consistent reconstruction of the density of
free electrons [29], while keeping the original magnetic field structure of
the cosmological simulation [26]. The resulting resulting RM map is shown at
the bottom right and very similar to the Milky Way observations.

Figure 2:  _Average magnitude of the Faraday RM along the Galactic longitude
$l$ (left panel) and Galactic latitude $b$ (right panel)._ The magnitude of
the Faraday RM has been averaged over latitude within $|b|<1.2^{\circ}$ when
shown as a function of longitude for all of our all-sky RM maps. The amount of
RM differs for each of the observer positions (synthetic and our own within
the Milky Way) as a result of the local electron distributions and magnetic
field directions, respectively. However, all RM profiles show the same trend
from positive to negative values at $l=0^{\circ}$ as the large scale toriodal
field component reverses its direction with respect to the observer. Also, the
general synthetic and observed trends agree well with a peak at $b=0^{\circ}$
and a minimum towards the poles.

Figure 3:  _Multipole spectrum of RM maps for ten different observer positions
compared to observational data for the Milky Way._ Left panel: Direct
multipole spectrum as a function of multipole moment $\ell$ for ten different
observer positions (P01-P10) within the origin Au-6 galaxy and our modified
galaxy version and compares with the reconstructed RM observational data of
O12 and H22. A multipole moment of $\ell=100$ corresponds roughly to a
resolution of $14.4^{\circ}$. At higher $\ell$ the synthetic map of the
modified galaxy shows more structure all over the sky in comparison with the
reconstructed O12 and H22 data. All spectra are normalized at $\ell=10$ for
better comparison. Right panel: Histogram of the RM healpix maps shown in Fig.
1. All distributions are normalized by their peak values.

The existing RM all-sky maps of the Milky Way are built from an ensemble of
extragalactic polarized radio sources. Certain patches of the sky with missing
data or with reduced coverage are reconstructed applying Bayesian statistics,
and consequently, these regions appear to be smoother than the average. We
compare our results with the observed all-sky RM maps presented by Oppermann
and collaborators [24] and by Hutschenreuter and colleagues [15], who included
a larger source number and employed a more sophisticated reconstruction
algorithm (O12 and H22 hereafter). We note that it has recently been pointed
out [38] that high RM sightlines may have gone undetected for decades, either
from instrumental limitations or from biases in the source selection, and that
this introduces a systematic shift of these maps towards low RM values.

Figure 1 shows the O12 and H22 data (top row) as well as two synthetic RM maps
from the Auriga galaxy (bottom row) from one exemplary observers position
(P01) at the solar circle. The synthetic map at the left takes the galaxy as
is [27], based on the data of the cosmological simulation only, and the right
one combines the simulation data with the detailed model for star-cluster
formation and evolution [29] described above. It is immediately obvious that
the Auriga-only maps misses small-scale features, whereas the full model
agrees remarkably well the observations. It has a comparable level of
fluctuations on small angular scales and gives the right RM magnitude in all
regions of the sky. We also note that all maps exhibit a reversal of the
magnetic field direction at the Galactic center, as indicated by a transition
from positive to negative RM values. This transition is a distinct feature of
the magnetic field morphology of the Milky Way, revealing a globally toroidal
field structure. It is also well visible in Figure 2, where we plot the
average RM along the Galactic longitude $l$ (left) and latitude $b$ (right)
coordinates for all observer positions in the model galaxy. The field reversal
in the center is a global property of the disk and therefore independent of
the location of the measurement. Similarly, all models exhibit the highest
amount of RM near the disk midplane ($b=0^{\circ}$) with decreasing values
towards the poles ($b=\pm 90^{\circ}$), again in agreement with O12 and H22,
indicating that the magnetic field quickly decreases further up and down into
the Galactic halo. We also mention that our model slightly overestimates the
amount of RM at large $b$ with an offset of $0-30\ \mathrm{rad\ m}^{-2}$.
Since the signal at large latitudes $b$ is mostly produced by material that is
nearby [27], this deviation emphasizes again the importance of properly
accounting for the immediate surrounding of the Sun, and it is an indication
of the limitations of current galaxy formation simulations when it comes to
reproducing the Local Bubble [1, 42].

For a more quantitative comparison, we compute the multipole expansion of the
RM maps and display the result in Figure 3. Our synthetic RM spectra are
consistent with the O12 and H22 data for all multipole moments $\ell\lesssim
30$ at all observer positions, whereas we are predicting more small-scale
structures at larger $\ell$. We speculate that this may be a result of the
bias in the Bayesian statistics of O12 and H22 towards lower RM values [38].
In contrast, the expansion of the original AU-6 galaxy [27] deviates from the
observations already above $\ell\approx 10$ indicating again importance of
small-scale physics for the interpretation of the Milky Way data. We stress
that including a realistic star-cluster population synthesis model is
indispensable for reproducing the observed small-scale (large $\ell$)
features. We also note that similar conclusions have been reached by Beck and
collaborators [3], however, they did so by introducing a small-scale random
magnetic field component still lacking the contribution of a proper multi-
scale thermal electron model.

Altogether, we find that the general trends in the RM maps presented here
agree well for different positions within the galaxy, and they all reproduce
the observed data. Consequently, we conclude that current high-resolution MHD
simulations of formation and evolution of the Milky Way in a cosmological
context, combined with adequate models of star formation and stellar feedback,
can well explain the properties of magnetic fields in spiral galaxies. This
marks an important step forward in our theoretical understanding of magnetic
field amplification by various forms of the dynamo process acting in these
systems [27, 36]. We note that the methods presented here can also be applied
to RM observations in the high-redshift Universe [21] and can thus help to
monitor the genesis of magnetic fields over cosmic timescales. However, we
also caution that most of the RM signal above and below the Galactic plane
might be dominated by the local environment [27]. Subsequently, maps of the
Faraday rotation of the Milky Way cannot be adequately interpreted without
knowledge of the conditions in our Local Bubble [1, 42]. This has been
proposed before [40, 27], and it is also implied by the dust polarization
measurements of the Planck satellite [1]. Distinct observers in different
parts of the Galaxy would see different local magnetic field configurations
and electron densities. Our results suggest that current measurements of the
Milky Way RM carry a level of uncertainty that was previously not fully
appreciated and that can only be accounted for on a statistical basis by
detailed modeling efforts as presented here.

## Methods

To construct the synthetic all-sky Faraday RM map of the Milky Way, we take
one of the simulations from the Auriga project [12]. The galaxy Au-6 is the
result of a very high-resolution cosmological MHD zoom-in simulation from
initial conditions that are specifically selected to reproduce key features of
the Local Group. The calculation includes line cooling, stellar evolution,
galactic winds, and the growth of black holes and their associated AGN
feedback [12]. All simulations include self-consistently evolving magnetic
fields on a Voronoi grid [25] with the moving mesh code AREPO [39]. The galaxy
Au-6 is selected as Milky Way analogue based on its size, total mass, and star
formation rate.

In the next step we follow the procedure as outlined by Pellegrini and
colleagues [29] (hereafter P21) and synthesize a new cluster population and
electron fractions that are more faithful to star-forming physics and small
scale density structures know from observations. To do so, we replace the
original star particles and electron fractions from the simulation and instead
reconstruct this information from first principles employing the WARPFIELD
cloud/cluster evolution model [31, 32]. It follows the time evolution and
structure of the stellar wind bubble, HII region, and protodissociation region
(PDR) surrounding a cluster of massive stars in spherical symmetry. WARPFIELD
accounts self-consistently for the physics of stellar winds, supernovae,
radiation pressure, ionization, and gravity. It solves explicitly for the
density structure adopted by the gas in response to the action of these
various feedback processes, and therefore allows one to account for the
evolution of the luminosity and emerging spectrum.

We determine the local gas mass within equidistant annuli of the Au-6 galaxy,
using linearly spaced radial bins originating at the galactic center-of-mass
and sample the cluster mass function by randomly depositing mass with a rate
obtained from the Kennicutt-Schmidt [17] relation. Per annulus and cluster a
random location is selected. If the total gas mass within a distance less than
$50\ \mathrm{pc}$ from that location is larger than the cluster mass we have
drawn, then the cluster is placed there, and the corresponding gas mass is
subtracted from the original Voronoi grid of the Au-6 galaxy. As a result of
this procedure, the star clusters are typically inserted near density peaks.
This is consistent with observations, where young clusters are seen in the
vicinity of dense molecular gas, but no longer are deeply embedded into their
parental clouds due to efficient stellar feedback [19]. This leads to a
realistic stellar population that is characterized by the number of clusters
for a given mass and age in each annulus. An illustratino of this approach is
presented in Figure 6 of P21.

The physical properties of each cluster including all emission properties are
obtained from the WARPFIELD database [31, 32] combined with the spectral
synthesis code CLOUDY111http://www.nublado.org/ $\rm v17.00$ [10]. This
information [28] is then used to compute the impact of ionizing radiation from
each individual cluster on the ambient Galactic ISM in order to (re)populate
the entire Au-6 Voronoi grid with thermal electrons. We assume solar
metallicity, and cosmic ray ionization rates and an interstellar radiation
field (ISRF) that is representative of the Milky Way [18]. Note that we
neglect the population of old field stars in our analysis, because they do not
contribute to the ISRF at ionizing frequencies. The resulting distribution of
free electrons agrees very well with the observationally-inspired models for
the Milky Way by Cordes & Lazio [8] and by Yao et al. [41]. This is
illustrated in Figure 8 of P21. For a detailed description of the magnetic
field structure of the Auriga-6 galaxy, we refer to Pakmor et al. [26]. With
this approach, we have all necessary information to compute the integral (2)
through arbitrary sightlines through the galaxy.

Finally, we employ the radiative transfer (RT) code
POLARIS222http://www1.astrophysik.uni-kiel.de/${\sim}$polaris/ [34] capable of
dust polarization calculations [33] as well as Zeeman splitting line RT [5,
33] and solve the RT problem on the native Voronoi grid of AREPO simulations.
We calculate the resulting synthetic Faraday RM all-sky maps from ten selected
positions in the Au-6 galaxy integrating equation (2) for many different
sightlines across the entire galaxy and covering the entire sky with high
angular resolution [35]. For our calculations of the synthetic RM maps we
apply a HEALPIX333http://healpix.jpl.nasa.gov resolution of
$N_{\mathrm{side}}=256$ leading to a total amount of $786\ 432$ pixel. This
resolution is identical to the observed H22 map [16] but larger than O12 map
by a factor of four. The location of the fictitious observers in the Au-6
galaxy are selected such that the distance to the galactic center is between
$8\,$kpc and $10\,$kpc, with the solar values being 8.5 kpc, and that they are
placed in a Local Bubble like region, where previous supernovae have created a
low-density cavity, as illustrated in Figure 6 of P21.

## Code Availability

Cluster properties and ionizaiton are calulated with the WARPFIELD code [31,
32] and the spectral synthesis code CLOUDY $\rm v17.00$
(http://www.nublado.org/), respectively. Cosmological simulations are
performed by the moving mesh code AREPO [39] (https://arepo-code.org/wp-
content/userguide/index.html) and the RT post-processing we utilize the RT
code POLARIS [34] (https://portia.astrophysik.uni-kiel.de/polaris/). We used
Python and its associated libraries including astropy, numpy, and matplotlib
for data analysis and presentation.

## Corresponding Author

The corresponding author is Stefan Reissl. Please send any requests for
further information or data to<EMAIL_ADDRESS>
## Acknowledgements

S.R., R.S.K., E.W.P., and D.R. acknowledge support from the Deutsche
Forschungsgemeinschaft in the Collaborative Research Center (SFB 881, ID
138713538) “The Milky Way System” (subprojects A1, B1, B2, and B8) and from
the Heidelberg Cluster of Excellence (EXC 2181, ID 390900948) “STRUCTURES: A
unifying approach to emergent phenomena in the physical world, mathematics,
and complex data”, funded by the German Excellence Strategy. R.S.K. also
thanks for funding form the European Research Council in the ERC Synergy Grant
“ECOGAL – Understanding our Galactic ecosystem: From the disk of the Milky Way
to the formation sites of stars and planets” (ID 855130). RG acknowledges
support from an STFC Ernest Rutherford Fellowship (ST/W003643/1).” F.A.G.
acknowledges financial support from CONICYT through the project FONDECYT
Regular Nr. 1181264, and funding from the Max Planck Society through a Partner
Group grant. The project benefited from computing resources provided by the
State of Baden-Württemberg through bwHPC and DFG through grant INST 35/1134-1
FUGG, and from the data storage facility SDS@hd supported through grant INST
35/1314-1 FUGG. The Heidelberg team also thank for computing time provided by
the Leibniz Computing Center (LRZ) for project pr74nu.

## Author Contributions

S.R. has run all polarized radiative transfer calculations and has performed
most of the analysis. The text was jointly written by S.R. and R.S.K. The
WARPFIELD cloud-cluster evolution model was mostly contributed by E.W.P. and
D.R. The Augiga-6 data and support with the data handling have been provided
by R.P., R.G., F.G., F.M., and V.S.

## References

  * [1] M. I. R. Alves, F. Boulanger, K. Ferrière, and L. Montier.  The Local Bubble: a magnetic veil to our Galaxy.  A&A, 611:L5, April 2018.
  * [2] B.-G. Andersson, A. Lazarian, and J. E. Vaillancourt.  Interstellar Dust Grain Alignment.  ARA&A, 53:501–539, August 2015.
  * [3] M. C. Beck, A. M. Beck, R. Beck, K. Dolag, A. W. Strong, and P. Nielaba.  New constraints on modelling the random magnetic field of the MW.  JCAP, 5:056, May 2016.
  * [4] R. Beck.  Magnetic fields in spiral galaxies.  A&ARv, 24:4, December 2015.
  * [5] R. Brauer, S. Wolf, S. Reissl, and F. Ober.  Magnetic fields in molecular clouds: Limitations of the analysis of Zeeman observations.  A&A, 601:A90, May 2017.
  * [6] B. J. Burn.  On the depolarization of discrete radio sources by Faraday dispersion.  MNRAS, 133:67, 1966.
  * [7] I. Butsky, J. Zrake, J.-h. Kim, H.-I. Yang, and T. Abel.  Ab Initio Simulations of a Supernova-driven Galactic Dynamo in an Isolated Disk Galaxy.  ApJ, 843:113, July 2017.
  * [8] J. M. Cordes and T. J. W. Lazio.  NE2001.I. A New Model for the Galactic Distribution of Free Electrons and its Fluctuations.  ArXiv Astrophysics e-prints, July 2002.
  * [9] R. M. Crutcher.  Magnetic Fields in Molecular Clouds: Observations Confront Theory.  ApJ, 520:706, August 1999.
  * [10] G. J. Ferland, M. Chatzikos, F. Guzmán, M. L. Lykins, P. A. M. van Hoof, R. J. R. Williams, N. P. Abel, N. R. Badnell, F. P. Keenan, R. L. Porter, and P. C. Stancil.  The 2017 Release Cloudy.  RMXAA, 53:385, October 2017.
  * [11] K. M. Ferrière.  The interstellar environment of our galaxy.  RvMP, 73:1031, October 2001.
  * [12] R. J. J. Grand, F. A. Gómez, F. Marinacci, R. Pakmor, V. Springel, D. J. R. Campbell, C. S. Frenk, A. Jenkins, and S. D. M. White.  The Auriga Project: the properties and formation mechanisms of disc galaxies across cosmic time.  MNRAS, 467:179, May 2017.
  * [13] C. Heiles and M. Haverkorn.  Magnetic Fields in the Multiphase Interstellar Medium.  SSRv, 166:293, May 2012.
  * [14] P. Hennebelle.  The FRIGG project: From intermediate galactic scales to self-gravitating cores.  A&A, 611:A24, March 2018.
  * [15] S. Hutschenreuter, C. S. Anderson, S. Betti, G. C. Bower, J. A. Brown, M. Brüggen, E. Carretti, T. Clarke, A. Clegg, A. Costa, S. Croft, C. V. Eck, B. M. Gaensler, F. de Gasperin, M. Haverkorn, G. Heald, C. L. H. Hull, M. Inoue, M. Johnston-Hollitt, J. Kaczmarek, C. Law, Y. K. Ma, D. MacMahon, S. A. Mao, C. Riseley, S. Roy, R. Shanahan, T. Shimwell, J. Stil, C. Sobey, S. P. O’Sullivan, C. Tasse, V. Vacca, T. Vernstrom, P. K. G. Williams, M. Wright, and T. A. Enßlin.  The Galactic Faraday rotation sky 2020.  A&A, 657:A43, January 2022.
  * [16] Sebastian Hutschenreuter and Torsten A. Enßlin.  The Galactic Faraday depth sky revisited.  A&A, 633:A150, January 2020.
  * [17] R. C. Kennicutt.  Star Formation in Galaxies Along the Hubble Sequence.  ARA&A, 36:189, 1998.
  * [18] R. S. Klessen and S. C. O. Glover.  Physical Processes in the Interstellar Medium.  Star Formation in Galaxy Evolution: Connecting Numerical Models to Reality, Saas-Fee Advanced Course, Volume 43. ISBN 978-3-662-47889-9. Springer-Verlag Berlin Heidelberg, 2016, p. 85, 43:85, 2016\.
  * [19] Mark R. Krumholz, Christopher F. McKee, and Joss Bland-Hawthorn.  Star Clusters Across Cosmic Time.  ARA&A, 57:227–303, August 2019.
  * [20] A. Lazarian, K. H. Yuen, H. Lee, and J. Cho.  Synchrotron Intensity Gradients as Tracers of Interstellar Magnetic Fields.  ApJ, 842:30, June 2017.
  * [21] S. A. Mao, C. Carilli, B. M. Gaensler, O. Wucknitz, C. Keeton, A. Basu, R. Beck, P. P. Kronberg, and E. Zweibel.  Detection of microgauss coherent magnetic fields in a galaxy five billion years ago.  Nature Astronomy, 1:621, August 2017.
  * [22] Christopher F. McKee and Eve C. Ostriker.  Theory of Star Formation.  ARA&A, 45(1):565–687, September 2007.
  * [23] D. Morris and G. L. Berge.  Direction of the Galactic Magnetic Field in the Vicinity of the Sun.  ApJ, 139:1388, May 1964.
  * [24] N. Oppermann, H. Junklewitz, G. Robbers, M. R. Bell, T. A. Enßlin, A. Bonafede, R. Braun, J. C. Brown, T. E. Clarke, I. J. Feain, B. M. Gaensler, A. Hammond, L. Harvey-Smith, G. Heald, M. Johnston-Hollitt, U. Klein, P. P. Kronberg, S. A. Mao, N. M. McClure-Griffiths, S. P. O’Sullivan, L. Pratley, T. Robishaw, S. Roy, D. H. F. M. Schnitzeler, C. Sotomayor-Beltran, J. Stevens, J. M. Stil, C. Sunstrum, A. Tanna, A. R. Taylor, and C. L. Van Eck.  An improved map of the Galactic Faraday sky.  A&A, 542:A93, June 2012.
  * [25] R. Pakmor, F. Marinacci, and V. Springel.  Magnetic Fields in Cosmological Simulations of Disk Galaxies.  ApJ, 783:L20, March 2014.
  * [26] Rüdiger Pakmor, Facundo A. Gómez, Robert J. J. Grand, Federico Marinacci, Christine M. Simpson, Volker Springel, David J. R. Campbell, Carlos S. Frenk, Thomas Guillet, Christoph Pfrommer, and Simon D. M. White.  Magnetic field formation in the Milky Way like disc galaxies of the Auriga project.  MNRAS, 469(3):3185–3199, August 2017.
  * [27] Rüdiger Pakmor, Thomas Guillet, Christoph Pfrommer, Facundo A. Gómez, Robert J. J. Grand, Federico Marinacci, Christine M. Simpson, and Volker Springel.  Faraday rotation maps of disc galaxies.  MNRAS, 481(4):4410–4418, December 2018.
  * [28] E. W. Pellegrini, D. Rahner, S. Reissl, S. C. O. Glover, R. S. Klessen, L. Rousseau-Nepton, and R. Herrera-Camus.  WARPFIELD-EMP: The self-consistent prediction of emission lines from evolving H II regions in dense molecular clouds.  MNRAS, 496(1):339–363, July 2020.
  * [29] Eric W. Pellegrini, Stefan Reissl, Daniel Rahner, Ralf S. Klessen, Simon C. O. Glover, Rüdiger Pakmor, Rodrigo Herrera-Camus, and Robert J. J. Grand.  WARPFIELD population synthesis: the physics of (extra-)Galactic star formation and feedback-driven cloud structure and emission from sub-to-kpc scales.  MNRAS, 498(3):3193–3214, November 2020.
  * [30] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. I. R. Alves, M. Arnaud, D. Arzoumanian, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, N. Bartolo, E. Battaner, K. Benabed, A. Benoît, A. Benoit-Lévy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, A. Bracco, C. Burigana, E. Calabrese, J.-F. Cardoso, A. Catalano, H. C. Chiang, P. R. Christensen, L. P. L. Colombo, C. Combet, F. Couchot, B. P. Crill, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, C. Dickinson, J. M. Diego, H. Dole, S. Donzelli, O. Doré, M. Douspis, A. Ducout, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, D. Falceta-Gonçalves, E. Falgarone, K. Ferrière, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frejsel, S. Galeotta, S. Galli, K. Ganga, T. Ghosh, M. Giard, E. Gjerløw, J. González-Nuevo, K. M. Górski, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, V. Guillet, D. L. Harrison, G. Helou, P. Hennebelle, S. Henrot-Versillé, C. Hernández-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, W. A. Holmes, A. Hornstrup, K. M. Huffenberger, G. Hurier, A. H. Jaffe, T. R. Jaffe, W. C. Jones, M. Juvela, E. Keihänen, R. Keskitalo, T. S. Kisner, J. Knoche, M. Kunz, H. Kurki-Suonio, G. Lagache, J.-M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, R. Leonardi, F. Levrier, M. Liguori, P. B. Lilje, M. Linden-Vørnle, M. López-Caniego, P. M. Lubin, J. F. Macías-Pérez, D. Maino, N. Mandolesi, A. Mangilli, M. Maris, P. G. Martin, E. Martínez-González, S. Masi, S. Matarrese, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, M.-A. Miville-Deschênes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, C. B. Netterfield, F. Noviello, D. Novikov, I. Novikov, N. Oppermann, C. A. Oxborrow, L. Pagano, F. Pajot, R. Paladini, D. Paoletti, F. Pasian, L. Perotto, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, N. Ponthieu, G. W. Pratt, S. Prunet, J.-L. Puget, J. P. Rachen, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, I. Ristorcelli, G. Rocha, M. Rossetti, G. Roudier, J. A. Rubiño-Martín, B. Rusholme, M. Sandri, D. Santos, M. Savelainen, G. Savini, D. Scott, J. D. Soler, V. Stolyarov, R. Sudiwala, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, M. Tucci, G. Umana, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, L. A. Wade, B. D. Wandelt, I. K. Wehus, N. Ysard, D. Yvon, and A. Zonca.  Planck intermediate results. XXXV. Probing the role of the magnetic field in the formation of structure in molecular clouds.  A&A, 586:A138, February 2016.
  * [31] Daniel Rahner, Eric W. Pellegrini, Simon C. O. Glover, and Ralf S. Klessen.  Winds and radiation in unison: a new semi-analytic feedback model for cloud dissolution.  MNRAS, 470:4453, 2017.
  * [32] Daniel Rahner, Eric W. Pellegrini, Simon C. O. Glover, and Ralf S. Klessen.  WARPFIELD 2.0: feedback-regulated minimum star formation efficiencies of giant molecular clouds.  MNRAS, 483(2):2547–2560, February 2019.
  * [33] S. Reissl, A. M. Stutz, R. Brauer, E. W. Pellegrini, D. R. G. Schleicher, and R. S. Klessen.  Magnetic fields in star-forming systems (I): idealized synthetic signatures of dust polarization and Zeeman splitting in filaments.  MNRAS, 481:2507, December 2018.
  * [34] S. Reissl, S. Wolf, and R. Brauer.  Radiative transfer with POLARIS. I.: Analysis of magnetic fields through synthetic dust continuum polarization measurements.  A&A, 593:A87, September 2016.
  * [35] Stefan Reissl, Robert Brauer, Ralf S. Klessen, and Eric W. Pellegrini.  Radiative Transfer with POLARIS. II.: Modeling of Synthetic Galactic Synchrotron Observations.  ApJ, 885(1):15, November 2019.
  * [36] M. Rieder and R. Teyssier.  A small-scale dynamo in feedback-dominated galaxies as the origin of cosmic magnetic fields - I. The kinematic phase.  MNRAS, 457:1722, April 2016.
  * [37] G. B. Rybicki and A. P. Lightman.  Radiative processes in astrophysics.  1979\.
  * [38] R. Shanahan, S. J. Lemmer, J. M. Stil, H. Beuther, Y. Wang, J. Soler, L. D. Anderson, F. Bigiel, S. C. O. Glover, P. Goldsmith, R. S. Klessen, N. M. McClure-Griffiths, S. Reissl, M. Rugel, and R. J. Smith.  Strong Excess Faraday Rotation on the Inside of the Sagittarius Spiral Arm.  APJL, 887(1):L7, December 2019.
  * [39] V. Springel.  E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh.  MNRAS, 401:791, January 2010.
  * [40] X. H. Sun, T. L. Landecker, B. M. Gaensler, E. Carretti, W. Reich, J. P. Leahy, N. M. McClure-Griffiths, R. M. Crocker, M. Wolleben, M. Haverkorn, K. A. Douglas, and A. D. Gray.  Faraday Tomography of the North Polar Spur: Constraints on the Distance to the Spur and on the Magnetic Field of the Galaxy.  ApJ, 811:40, September 2015.
  * [41] J. M. Yao, R. N. Manchester, and N. Wang.  A New Electron-density Model for Estimation of Pulsar and FRB Distances.  ApJ, 835(1):29, January 2017.
  * [42] Catherine Zucker, Alyssa A. Goodman, João Alves, Shmuel Bialy, Michael Foley, Joshua S. Speagle, Josefa Groschedl, Douglas P. Finkbeiner, Andreas Burkert, Diana Khimey, and Cameren Swiggum.  Star formation near the Sun is driven by expansion of the Local Bubble.  Nature, 601(7893):334–337, January 2022.