Modified Green-Hyperbolic Operators

Modified Green-Hyperbolic Operators††This paper is a contribution to the
Special Issue on Global Analysis on Manifolds in honor of Christian Bär for
his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Baer.html

Christopher J. FEWSTER

C.J. Fewster

Department of Mathematics, University of York, Heslington, York YO10 5DD, UK
<EMAIL_ADDRESS>https://www.york.ac.uk/maths/people/chris-fewster/

Received April 20, 2023, in final form July 29, 2023; Published online August
08, 2023

Green-hyperbolic operators – partial differential operators on globally
hyperbolic spacetimes that (together with their formal duals) possess advanced
and retarded Green operators – play an important role in many areas of
mathematical physics. Here, we study modifications of Green-hyperbolic
operators by the addition of a possibly nonlocal operator acting within a
compact subset $K$ of spacetime, and seek corresponding ‘$K$-nonlocal’
generalised Green operators. Assuming the modification depends holomorphically
on a parameter, conditions are given under which $K$-nonlocal Green operators
exist for all parameter values, with the possible exception of a discrete set.
The exceptional points occur precisely where the modified operator admits
nontrivial smooth homogeneous solutions that have past- or future-compact
support. Fredholm theory is used to relate the dimensions of these spaces to
those corresponding to the formal dual operator, switching the roles of future
and past. The $K$-nonlocal Green operators are shown to depend holomorphically
on the parameter in the topology of bounded convergence on maps between
suitable Sobolev spaces, or between suitable spaces of smooth functions. An
application to the LU factorisation of systems of equations is described.

Green-hyperbolic operators; nonlocal equations; Fredholm theory

35R01; 35R09; 47B93

Dedicated to Christian Bär
on the occasion of his sixtieth birthday

##  1 Introduction

Linear partial differential operators are the workhorses of mathematical
physics, providing the simplest models of classical and quantum field theories
from which more complicated interacting models may be built. In general
relativity or other nonlinear theories, linear operators appear whenever the
theory is linearised, for example, to study the stability of solutions, or the
propagation of gravitational waves.

A particularly useful general class of operators acting between spaces of
smooth sections of vector bundles over globally hyperbolic Lorentzian
manifolds has been introduced by Bär [2] under the name ‘Green-hyperbolic
operators’. Green-hyperbolicity is a generalisation, rather than a
specialisation, of hyperbolicity: a Green-hyperbolic operator need not be
hyperbolic, and there are examples that are elliptic, or of indefinite type.
The defining property of a Green-hyperbolic operator is that it should possess
advanced and retarded Green operators, along with its formal dual, and from
this simple algebraic requirement many other properties flow, as described
elegantly by Bär. In particular, the Green operators are unique, continuous,
and have extensions that are continuous inverses to (extensions of) the Green-
hyperbolic operator on various spaces of smooth or distributional sections.

Examples of Green-hyperbolic operators include (a) normally hyperbolic
operators, such as the wave operator $\Box$ and its variants allowing for the
inclusion of potentials and external vector potentials, (b) first order
symmetric hyperbolic systems on globally hyperbolic spacetimes [2]; (c) any
operator whose square is Green-hyperbolic, thus incorporating Dirac type
operators; (d) operators whose solution theory may be related to one of the
above, such as the (non-hyperbolic) Proca operator $-\delta\mathrm{d}+m^{2}$,
whose Green operators are obtained from those of
${-(\delta\mathrm{d}+\mathrm{d}\delta)+m^{2}}$ on smooth $1$-form fields. See
[2] for these and other examples.

The purpose of this paper is to study modifications of Green-hyperbolic
operators, that can lead outside the class of partial differential operators.
For simplicity, we mainly study the case of scalar operators but there is no
hindrance (beyond those of notation!) to extending our results to the general
bundle case. The operators we consider are of the form $P+A$, where $P$ is
Green-hyperbolic and $A$ is a continuous linear self-map of $C^{\infty}(M)$
(not necessarily a differential operator) whose range is contained in
$C^{\infty}_{K}(M)$, the smooth functions supported in a compact subset
$K\subset M$. Without loss of generality, we may always assume that $K$ is
_topologically regular_ , that is, equal to the closure of its interior. An
operator of this type is potentially nonlocal, though the nonlocality is, as
it were, localised within $K$. By the kernel theorem, any such operator can be
represented as

$A\phi=\int_{M}T(x,y)\phi(y)\mu_{y},$

where $\mu$ is a smooth density, and $T\in{\mathscr{D}}^{\prime}(M\times M)$
has support in $K\times M$ and is semi-regular in its first slot (i.e., $T$
belongs to the nuclear tensor product
$C^{\infty}(M)\widehat{\otimes}{\mathscr{D}}^{\prime}(M)$). Operators of this
type include, but go beyond, differential and pseudodifferential operators
supported in $K$ (for which the singular support of $T$ is confined to the
diagonal).

There are several applications for operators of this type. In perturbative
algebraic quantum field theory (pAQFT) they arise from the class of regular
interactions, and the results proved here establish the existence of suitable
Green operators needed in [17], for example. Another application is to
noncommutative potential scattering [23], where equations such as

$P\phi+w\star\phi=0$

appear as toy models for the dynamics of classical and quantum fields on a
noncommutative spacetime. Here $P$ is a Green-hyperbolic operator, $w$ is a
fixed smooth function, and $\star$ is a noncommutative deformation of
multiplication, differing only from pointwise multiplication inside a compact
set $K$ [24]. The application of the results obtained here to such models will
be discussed elsewhere [13]. See also [7] for a slightly different context in
which nonlocal equations have appeared recently.

One may in fact develop an entire theory of nonlocal Green-hyperbolic
operators and this is done in the companion paper [14]. The purpose of this
paper is to investigate the technical issue of the existence and properties of
(a suitably generalised concept of) Green operators $E^{\pm}_{P+A(\lambda)}$
for $P+A(\lambda)$ under suitable conditions on $P$ and $A(\lambda)$ for
$\lambda\in{\mathbb{C}}$. A particular focus will be on the analytic
dependence of the resulting Green operators on $\lambda$ within suitable
locally convex spaces and on a suitable domain in ${\mathbb{C}}$. These
results are useful even in situations where the operators $P+A(\lambda)$ are
local Green-hyperbolic operators. For example, they have been applied in [12]
in the context of measurement schemes for observables in QFT. The Green
operators we study have properties similar to those of Green-hyperbolic
operators, with the following generalised support property:

$\operatorname{supp}E_{P+A(\lambda)}^{\pm}f\subset\begin{cases}J^{\pm}(\operatorname{supp}f),&J^{\pm}(\operatorname{supp}f)\cap
K=\varnothing,\\\ J^{\pm}(\operatorname{supp}f\cup
K),&\text{otherwise},\end{cases}$

for compactly supported $f$, where $J^{+/-}(S)$ are the causal future/past of
a set $S$. This support property characterises what we call _$K$ -nonlocal
Green operators_, set out precisely in Definition 2.4 below.

The main result of this paper is Theorem 3.1, which sets out conditions on $P$
and $A(\lambda)$ under which suitable Green operators for $P+A(\lambda)$
exist. The principal hypothesis on $P$ is that it is a Green-hyperbolic
operator whose Green operators have extensions to continuous maps between the
Sobolev spaces $H_{0}^{s}(M)$ and $H^{s+\beta}_{\rm loc}(M)$ for all
sufficiently large $s$ and some fixed $\beta$. This hypothesis is valid for
second order normally hyperbolic operators with $\beta=1$ (see [10, Theorem
6.5.3]). The main hypothesis on $A(\lambda)$ is that each $A(\lambda)$ is a
continuous linear self-map of $C^{\infty}(M)$ with continuous extensions
mapping between Sobolev spaces $H^{s}_{\rm loc}(M)$ to $H^{s+\gamma}_{K}(M)$
for all sufficiently large $s$ and some fixed $\gamma>-\beta$. This implies
that the compositions $A(\lambda)E^{\pm}$ are compact maps between
$H^{s}_{0}(M)$ and $H^{s}_{K}(M)$ and it is required that they depend
holomorphically on $\lambda$ in the topology of bounded convergence on linear
maps between these spaces. A brief summary of the various topological spaces
and topologies used in this work is provided in Appendix A.

Given the above assumptions, the analytic Fredholm theorem can be used to find
inverses $\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}$ on $H^{s}_{K}(M)$ for all
sufficiently large $s\in{\mathbb{R}}$ and all complex $\lambda$ in an open
neighbourhood of zero, whose (possibly empty) complement is a discrete subset
$S$ of ${\mathbb{C}}$. Exceptional values $\lambda\in S$ occur precisely when
there exist nontrivial _smooth_ solutions to $(P+A(\lambda))\phi=0$, whose
support is either past- or future-compact, i.e., nontrivial solutions that
vanish identically at early or late times, representing spontaneously
appearing or disappearing disturbances; they are clearly excluded if any sort
of energy estimate is available. For $\lambda\in{\mathbb{C}}\setminus S$, one
may use the inverses to construct $K$-nonlocal Green operators for
$P+A(\lambda)$. It is also proved that the resulting $K$-nonlocal Green
operators are holomorphic on ${\mathbb{C}}\setminus S$, with respect to the
topology of bounded convergence on linear maps between $C_{0}^{\infty}(M)$ and
$C^{\infty}(M)$. The power series expansion of the Green operators about
$\lambda=0$ corresponds to a Born expansion of the Green operators. We also
show how classical Møller operators [11, 17] can be constructed to relate the
free and interacting dynamics.

In the situation where our main hypotheses are also satisfied for the formal
duals of $P$ and $A(\lambda)$, we apply Fredholm index theory to show that the
dimension of the space of spontaneously appearing (resp., disappearing)
solutions for $P+A(\lambda)$ is equal to the dimension of the space of
spontaneously disappearing (resp., appearing) solutions for its formal dual,

$\dim\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}=\dim\ker\big{(}\leftidx{{}^{t}}{P}{}+\leftidx{{}^{t}}{A}{}(\lambda)\big{)}|_{C^{\infty}_{\mathrm{fc}/\mathrm{pc}}},$
(1.1)

for all $\lambda\in{\mathbb{C}}$. Here, the subscripts pc and fc denote past-
compact and future-compact support, respectively. Consequently, the spaces of
appearing and disappearing solutions of a formally self-dual operator have
equal dimension.

We mention that Dappiaggi and Finster [7] have recently studied nonlocal
equations arising from ‘causal variational principles’. These differ from the
equations we consider because the nonlocality need not be confined to a
compact spacetime region. Their existence proofs are also different: they
assert conditions under which a variant of the classical energy estimates can
be formulated and then follow essentially the classical arguments used to
establish existence for Cauchy problems of normally hyperbolic partial
differential equations.

The present paper is structured as follows: in Section 2, we recall the
definition and main properties of Green-hyperbolic operators and develop the
appropriate notion of $K$-nonlocal Green operators. Section 3 contains the
statement of our main result, Theorem 3.1 and illustrates it with examples of
how it may be used and of the necessity of some of its hypotheses. Section 4
provides an application of our result to the LU factorisation and solution of
certain systems of nonlocal equations. The main result is proved in Section 5,
by a sequence of results initially in Sobolev spaces and then for smooth
functions, while (1.1) is proved in Section 6 using Fredholm theory and the
results of Section 5. Appendix A collects some necessary background on the
topological spaces and topologies appearing in the text.

##  2 Green-hyperbolic operators

Preliminaries. We begin by recalling the general setting of Green-hyperbolic
operators [2]. Let $M$ be a smooth finite-dimensional manifold, allowing the
possibility that $M$ has finitely many connected components with possibly
different dimensions, and let $g$ be a smooth Lorentzian metric on $M$ of
signature $+--\cdots$. With these structures, $M$ is automatically Hausdorff
and paracompact [15]. We assume that $(M,g)$ is time-orientable and that a
time-orientation has been chosen. To minimise notation, we denote the
Lorentzian spacetime formed by the manifold, metric and time orientation with
the single symbol $M$. The volume measure induced by the metric will be
denoted $\mu$. On other points of notation, the symbol $\subset$ will always
allow for the possibility of equality, while ${\mathbb{N}}_{0}$ and
${\mathbb{N}}$ denote the natural numbers with or without zero, respectively.

As usual, the causal future/past of a point $x\in M$ is denoted $J^{\pm}(x)$
and comprises all points (including $x$) that may be reached from $x$ along
smooth future/past-directed curves. (Throughout this paper, we tacitly order
alternatives labelled by $\pm$ or $\mp$ so that the alternative labelled by
the upper symbol comes first.) If $S\subset M$ then one writes
$J^{\pm}(S)=\cup_{x\in S}J^{\pm}(x)$, and $J(S)=J^{+}(S)\cup J^{-}(S)$. The
spacetime is globally hyperbolic if it contains no causal curves and
$J^{+}(K)\cap J^{-}(K)$ is compact for all compact sets $K\subset M$ [6].
Globally hyperbolic spacetimes can be foliated into smooth spacelike Cauchy
surfaces [5]; it is also the case that $J^{\pm}(K)$ are closed whenever $K$ is
compact. We adopt the following terminology: $S\subset M$ is _spacelike
compact_ if $S$ is closed and $S\subset J(K)$ for some compact $K$; $S$ is
_future/past-compact_ if $S\cap J^{\pm}(x)$ is compact for all $x\in M$; $S$
is _strictly future/past-compact_ if $S\subset J^{\mp}(K)$ for some compact
$K$.

If $B$ is a vector bundle with finite-dimensional fibres, then
$\Gamma^{\infty}(B)$ will denote the corresponding space of smooth sections
and
$\Gamma_{0/\mathrm{pc}/\mathrm{fc}/\mathrm{spc}/\mathrm{sfc}/\mathrm{sc}}^{\infty}(B)$
will be the spaces of smooth sections with compact/past-compact/future-
compact/strictly past-compact/strictly future-compact/spacelike-compact
support. The space of smooth sections with support contained in some a closed
subset $A\subset M$ is denoted $\Gamma^{\infty}_{A}(M)$. Further details on
the topologies of these spaces are summarised in Appendix A. We will only
consider bundles with finite-dimensional fibres that are (without any real
loss) vector spaces over ${\mathbb{C}}$. The bilinear (not sesquilinear!)
pairing between sections of dual bundle $B^{*}$ and sections of $B$ is denoted
with angle brackets
$\langle\cdot,\cdot\rangle\colon\Gamma^{\infty}(B^{*})\times\Gamma^{\infty}(B)\to
C^{\infty}(M)$.

Green-hyperbolicity. Suppose that $B_{1}$ and $B_{2}$ are bundles over $M$ and
that $P\colon\Gamma^{\infty}(B_{1})\to\Gamma^{\infty}(B_{2})$ is a linear
partial differential operator. Then there is a formal dual
$\leftidx{{}^{t}}{P}{}\colon\Gamma^{\infty}(B_{2}^{*})\to\Gamma^{\infty}(B_{1}^{*})$
given by

$\int_{M}\mu\,\big{\langle}\leftidx{{}^{t}}{P}{}f,\phi\big{\rangle}=\int_{M}\mu\,\langle
f,P\phi\rangle$

for all $\phi\in\Gamma^{\infty}(B_{1})$ and $f\in\Gamma^{\infty}(B_{2}^{*})$
whose supports intersect compactly; $\leftidx{{}^{t}}{P}{}$ is also a linear
partial differential operator. If they exist, linear maps
$E^{\pm}\colon\Gamma_{0}^{\infty}({B_{2}})\to\Gamma^{\infty}(B_{1})$ obeying

  * (G1)

$E^{\pm}Pf=f$ for all $f\in\Gamma_{0}^{\infty}({B_{1}})$,

  * (G2)

$PE^{\pm}f=f$ for all $f\in\Gamma_{0}^{\infty}({B_{2}})$,

  * (G3)

$\operatorname{supp}E^{\pm}f\subset J^{\pm}(\operatorname{supp}f)$ for all
$f\in\Gamma_{0}^{\infty}({B_{2}})$

are called advanced ($-$) and retarded ($+$) Green operators for $P$.111Bär
reverses the usage of ‘advanced’ and ‘retarded’; we adopt the more standard
convention.

######  Definition 2.1.

$P$ is said to be _Green-hyperbolic_ if both $P$ and $\leftidx{{}^{t}}{P}{}$
admit advanced and retarded Green operators.

It is a remarkable fact that this purely algebraic definition – only requiring
linearity of $E^{\pm}$ and with no presumption of uniqueness or continuity –
has the following consequence, which is a summary of results in [2, Sections 3
and 4] and deserves to be called the _first main theorem of Green-
hyperbolicity_.

######  Theorem 2.2.

Let $P\colon\Gamma^{\infty}(B_{1})\to\Gamma^{\infty}(B_{2})$ be a Green-
hyperbolic operator and consider:

  1. $(i)$

the restriction
$P\colon\Gamma^{\infty}_{\mathrm{spc}/\mathrm{sfc}}(B_{1})\to\Gamma^{\infty}_{\mathrm{spc}/\mathrm{sfc}}(B_{2})$,

  2. $(ii)$

the restriction
$P\colon\Gamma^{\infty}_{\mathrm{pc}/\mathrm{fc}}(B_{1})\to\Gamma^{\infty}_{\mathrm{pc}/\mathrm{fc}}(B_{2})$,
or

  3. $(iii)$

the extension
$P\colon{\mathscr{D}}^{\prime}_{\mathrm{pc}/fc}(B_{1})\to{\mathscr{D}}^{\prime}_{\mathrm{pc}/fc}(B_{2})$
$($see below$)$.

Then, in each case, $P$ has continuous inverses, denoted $\tilde{E}^{\pm}$,
$\overline{E}^{\pm}$ and $\widehat{E}^{\pm}$ in cases $(i)$, $(ii)$, $(iii)$
respectively, and which are successive extensions of $E^{\pm}$. Each of these
inverses has the support property $(G3)$, replacing
$\Gamma_{0}^{\infty}(B_{2})$ by the appropriate domain. In particular,
$E^{\pm}$ are uniquely determined by $P$ and are continuous.

In $(iii)$ the space of distributional sections ${\mathscr{D}}^{\prime}(B)$ of
a bundle $B$ over $M$ is defined as the topological dual of
$\Gamma_{0}^{\infty}(B^{*}\otimes\Omega)$, where $\Omega$ is the bundle of
weight-$1$ densities over $M$. As sections of $B$ and $B^{*}\otimes\Omega$ can
be paired to give a density, there is an obvious embedding of
$\Gamma^{\infty}(B)$ in ${\mathscr{D}}^{\prime}(B)$, with
$\phi\in\Gamma^{\infty}(B)$ corresponding to the distribution
$f\mapsto\int_{M}\langle\phi,f\rangle$ acting on
$f\in\Gamma_{0}^{\infty}(B^{*}\otimes\Omega)$. The map $P$ in $(iii)$ is the
restriction of the dual map
$\big{(}\mu(\leftidx{{}^{t}}{P}{})\mu^{-1}\big{)}^{\prime}$ to elements of
${\mathscr{D}}^{\prime}(B_{1})$ with past- or future-compact support. (Recall
that the formal dual is defined relative to the specific density $\mu$.) In
[2], Bär defines distributional sections of $B$ using the topological dual of
$\Gamma_{0}^{\infty}(B^{*})$, which would be the space of distribution
densities in our terminology. The metric density provides an isomorphism
between the spaces and operators that he considers and those that we do. As we
have in mind potential applications where more than one metric might be in
use, we have elected not to make a fixed identification between distributions
and distribution densities.

The _second main theorem of Green-hyperbolicity_ provides two exact sequences
that are highly useful in applications, combining (and very mildly
extending)222The extension is the right-most arrow, asserting surjectivity of
$P$ onto spacelike compact (distributional) sections. But any such smooth
section can be split as $f=f^{+}+f^{-}$ where $f^{\pm}$ has past/future-
compact support, whereupon
$f=P\big{(}\overline{E}^{+}f^{+}+\overline{E}^{-}f^{-}\big{)}$; the argument
is identical for distributions, replacing $\overline{E}^{\pm}$ by
$\widehat{E}^{\pm}$. Theorems 3.22 and 4.3 of [2].

######  Theorem 2.3.

Let $P\colon\Gamma^{\infty}(B_{1})\to\Gamma^{\infty}(B_{2})$ be Green-
hyperbolic and define the advanced-minus-retarded operators
$E=E^{-}-E^{+}\colon\Gamma^{\infty}_{0}(B_{2})\to\Gamma^{\infty}_{\mathrm{sc}}(B_{1})$
and $\widehat{E}=\widehat{E}^{-}-\widehat{E}^{+}$, using the notation of
Theorem 2.2. Then there are two exact sequences

${0}$${\Gamma_{0}^{\infty}({B_{1}})}$${\Gamma_{0}^{\infty}({B_{2}})}$${\Gamma^{\infty}_{\mathrm{sc}}(B_{1})}$${\Gamma^{\infty}_{\mathrm{sc}}(B_{2})}$${0}$${0}$${{\mathscr{D}}^{\prime}_{0}(B_{1})}$${{\mathscr{D}}^{\prime}_{0}(B_{2})}$${{\mathscr{D}}^{\prime}_{\mathrm{sc}}(B_{1})}$${{\mathscr{D}}^{\prime}_{\mathrm{sc}}(B_{2})}$${0,}$$\scriptstyle{P}$$\scriptstyle{E}$$\scriptstyle{P}$$\scriptstyle{P}$$\scriptstyle{\widehat{E}}$$\scriptstyle{P}$

in which the downward arrows are the natural embeddings of smooth into
distributional sections.

A direct consequence is that the solution space
$\operatorname{Sol}(P)=\\{\phi\in\Gamma^{\infty}_{sc}(B)\colon P\phi=0\\}$ is
given by $\operatorname{Sol}(P)=E\Gamma_{0}^{\infty}({B})$. The special
properties of Green-hyperbolic operators are not confined to the statement of
Theorems 2.2 and 2.3. As shown in [2], products and direct sums of Green-
hyperbolic operators are Green-hyperbolic, and indeed any operator whose
square is Green-hyperbolic is Green-hyperbolic. Turning to physical
applications, suppose that a bundle $B$ admits a nondegenerate bilinear form,
or equivalently a base-point preserving vector bundle isomorphism
$\mathcal{I}\colon B\to B^{*}$, and an antilinear base-point preserving
involution $\mathcal{C}\colon B\to B$. Then $P\colon\Gamma(B)\to\Gamma(B)$ is
said to be _formally self-adjoint_ if
$P=\mathcal{I}^{-1}\leftidx{{}^{t}}{P}{}\mathcal{I}$, and real if
$P=\mathcal{C}P\mathcal{C}$. If a formally self-adjoint operator $P$ admits
advanced and retarded Green operators then $P$ is Green-hyperbolic and
moreover $\operatorname{Sol}(P)$ admits a symplectic form

$\sigma(Ef_{1},Ef_{2})=\langle\mathcal{I}f_{1},Ef_{2}\rangle,\qquad
f_{i}\in\Gamma_{0}^{\infty}({B}).$

If $P$ is also real then there is an associated bosonic QFT described by a
unital $*$-algebra of observables, generated by symbols $\Phi(f)$,
$f\in\Gamma_{0}^{\infty}(B)$, and subject to the relations:

  * •

$f\mapsto\Phi(f)$ is complex linear (linearity),

  * •

$\Phi(f)^{*}=\Phi(\mathcal{C}f)$ (Hermiticity),

  * •

$\Phi(Pf)=0$ (field equation),

  * •

$[\Phi(f),\Phi(h)]={\rm i}\sigma(Ef,Eh){\boldsymbol{1}}$ (canonical
commutation relations)

for all $f,h\in\Gamma_{0}^{\infty}(B)$. This essentially functorial
quantisation is one of the main applications for the theory of Green-
hyperbolic operators. Under more restrictive circumstances first order Green-
hyperbolic operators can also admit fermionic quantisation [3].

Modified Green-hyperbolic operators. Turning to the subject of this paper,
suppose $P\colon\Gamma^{\infty}(B_{1})\to\Gamma^{\infty}(B_{2})$ is Green-
hyperbolic, with Green operators $E^{\pm}$. Let
$A\colon\Gamma^{\infty}(B_{1})\to\Gamma^{\infty}(B_{2})$ be linear, with range
contained in $\Gamma^{\infty}_{K}(B_{2})$ for some compact, topologically
regular $K\subset M$.

Although $P+A$ is not necessarily a differential operator, we wish to state
conditions analogous to (G1)–(G3) that can characterise suitable Green
operators. To gain some insight, let us assume for the moment that for each
$h\in\Gamma_{0}^{\infty}({B_{2}})$, the equation $(P+A)\phi=h$ has unique
solutions with past/future-compact support given by $\phi=E_{P+A}^{\pm}h$
where
$E_{P+A}^{\pm}\colon\Gamma_{0}^{\infty}({B_{2}})\to\Gamma^{\infty}(B_{1})$ are
linear maps with ranges necessarily contained in
$\Gamma^{\infty}_{pc/fc}(B_{1})$. Then we may write

$PE_{P+A}^{\pm}f=f-AE_{P+A}^{\pm}f=PE_{P}^{\pm}\big{(}f-AE_{P+A}^{\pm}f\big{)},\qquad
f\in\Gamma_{0}^{\infty}({B_{2}}),$

using our assumptions on $P$ and $A$. As $P$ is invertible on
$\Gamma^{\infty}_{\mathrm{pc}/\mathrm{fc}}(B_{1})$ by Theorem 2.2 $(ii)$, we
have

$E_{P+A}^{\pm}f=E_{P}^{\pm}\big{(}f-AE_{P+A}^{\pm}f\big{)}$

with support determined using condition (G3) for $P$,

$\operatorname{supp}E_{P+A}^{\pm}f\subset
J^{\pm}\big{(}\operatorname{supp}\big{(}f-AE_{P+A}^{\pm}f\big{)}\big{)}\subset
J^{\pm}(\operatorname{supp}f\cup K).$

By imposing further conditions on $A$ we may refine the information available.
Specifically, suppose that $\phi|_{K}\equiv 0$ implies that $A\phi\equiv 0$.
Observe that

$(P+A)E_{P}^{\pm}f=f+AE_{P}^{\pm}f\in\Gamma_{0}^{\infty}({B_{1}})$

by (G2) for $E_{P}^{\pm}$ and the definition of $A$; by assumption on
$E^{\pm}_{P+A}$ we now have

$E_{P}^{\pm}f=E_{P+A}^{\pm}\big{(}f+AE_{P}^{\pm}f\big{)}$

and deduce that $E_{P+A}^{\pm}f=E_{P}^{\pm}f$ for all
$f\in\Gamma_{0}^{\infty}({B_{2}})$ such that
$J^{\pm}(\operatorname{supp}f)\cap K$ is empty (because $AE_{P}^{\pm}f$
vanishes). Together with our earlier observation, $E^{\pm}_{P+A}$ satisfy the
modified support property

  * (G3′)

for all $f\in\Gamma_{0}^{\infty}({B_{2}})$,

$\displaystyle\operatorname{supp}E_{P+A}^{\pm}f\subset\begin{cases}J^{\pm}(\operatorname{supp}f),&J^{\pm}(\operatorname{supp}f)\cap
K=\varnothing,\\\ J^{\pm}(\operatorname{supp}f\cup
K),&\text{otherwise.}\end{cases}$

With this intuition established, we can drop the assumption that $(P+A)\phi=h$
has unique solutions with past/future-compact support. The standing
assumptions are now

  * (A1)

$P\colon\Gamma^{\infty}(B_{1})\to\Gamma^{\infty}(B_{2})$ is Green-hyperbolic,

  * (A2)

$A\colon\Gamma^{\infty}(B_{1})\to\Gamma^{\infty}(B_{2})$ is linear, with range
contained in $\Gamma^{\infty}_{K}(B_{2})$,

  * (A3)

for $\phi\in\Gamma^{\infty}(B_{1})$, $\phi|_{K}\equiv 0$ implies $A\phi\equiv
0$.

We now make the following definition.

######  Definition 2.4.

Subject to assumptions (A1)–(A3), linear maps
$E_{P+A}^{\pm}\colon\Gamma_{0}^{\infty}({B_{2}})\to\Gamma^{\infty}(B_{1})$ are
said to be _retarded/advanced $K$-nonlocal Green operators_ for $P+A$ if they
satisfy (G1) and (G2) (with $P+A$ replacing $P$ and $E_{P+A}^{\pm}$ replacing
$E^{\pm}$) and (G3${}^{\prime})$. If both $P+A$ and
$\leftidx{{}^{t}}{P}{}+\leftidx{{}^{t}}{A}{}$ admit retarded and advanced
$K$-nonlocal Green operators then $P+A$ is called _$K$ -nonlocally Green-
hyperbolic_.

As with Green-hyperbolic operators, the above definition implies considerably
more and indeed the analogue of Theorem 2.2 holds with (G3) replaced by
(G3${}^{\prime})$. These results will be proved in [14]. Our main goal here
will be to give sufficient conditions for the existence of $K$-nonlocal Green
operators. For simplicity of presentation, we restrict to operators acting on
spaces of smooth functions, rather than bundle sections. We will also
establish some continuity results for the $K$-nonlocal Green operators, both
from $C_{0}^{\infty}(M)$ to $C^{\infty}(M)$ and between various Sobolev
spaces, and address the holomorphicity of the $K$-nonlocal Green operators
with respect to a parameter.

At a formal level it is straightforward to see what the Green operators of
$P+A$ should be, if they exist: if $\phi\in
C^{\infty}_{\mathrm{pc}/\mathrm{fc}}(M)$ solves $(P+A)\phi=f\in
C_{0}^{\infty}\big{(}{M}\big{)}$, then

$\phi=E_{P}^{\pm}(f-A\phi)=E^{\pm}_{P}g,$

where $g=f-A\phi=f-AE_{P}^{\pm}g$, so $g=\big{(}I+AE_{P}^{\pm}\big{)}^{-1}f$.
Formally, therefore, the Green operators for $P+A$ are

$E_{P+A}^{\pm}=E_{P}^{\pm}\big{(}I+AE^{\pm}_{P}\big{)}^{-1},$ (2.1)

and the technical task is to make this formula rigorous, where possible, and
to establish that the resulting operators $E_{P+A}^{\pm}$ are indeed
$K$-nonlocal Green operators for $P+A$. Note that the inversion of
$I+AE^{\pm}$ must be performed in $C_{0}^{\infty}\big{(}{M}\big{)}$. We will
accomplish this by first inverting in Sobolev spaces, where Hilbert space
techniques can be used, and then boosting the result up to
$C_{0}^{\infty}\big{(}{M}\big{)}$.

As can be seen from (2.1), the operator $\big{(}I+AE_{P}^{\pm}\big{)}^{-1}$
acts as a bridge between the free dynamics represented by $P$ and the
interacting dynamics of $P+A$. This link can be expressed precisely in terms
of classical Møller operators – see Corollary 3.2.

##  3 Main result and remarks

Let $M$ be a globally hyperbolic spacetime with at most finitely many
connected components, and let $K$ be a fixed, topologically regular, compact
subset of $M$. For brevity, we write $C^{\infty}$ for $C^{\infty}(M)$ and so
on; for a closed set $A$, $C^{\infty}_{A}$ denotes the space of smooth
functions with support in $A$, and the same convention is used for other
spaces of distributions and Sobolev spaces. If $X$ and $Y$ are locally convex
topological spaces, ${\mathcal{L}}_{b}(X,Y)$ denotes the space of linear maps
between them, equipped with the topology of bounded convergence (see Appendix
A); for linear maps between normed spaces, this coincides with the usual
operator norm topology. As usual, we write ${\mathcal{L}}_{b}(X)$ for
${\mathcal{L}}_{b}(X,X)$.

The main result of this paper is the following.

######  Theorem 3.1.

Let $P$ be a Green-hyperbolic operator and suppose $A(\lambda)$,
$\lambda\in{\mathbb{C}}$, is a family of linear maps $A(\lambda)\colon
C^{\infty}\to C^{\infty}$ whose ranges are contained in $C^{\infty}_{K}$ and
with $A(0)=0$. Suppose that, for some $\beta,\gamma\in{\mathbb{R}}$ with
$\delta=\beta+\gamma>0$, and some $s_{*}\in{\mathbb{R}}$,

  1. $(a)$

the Green operators $E^{\pm}$ of $P$ extend to linear maps
${\mathscr{E}}^{\prime}\to{\mathscr{D}}^{\prime}$ with continuous restrictions
mapping $H^{s}_{0}\to H^{s+\beta}_{\rm loc}$ for all $s\geq s_{*}$;

  2. $(b)$

each map $A(\lambda)$ extends to a linear map
${\mathscr{D}}^{\prime}\to{\mathscr{D}}^{\prime}_{K}$ with continuous
restrictions mapping $H^{s}_{\rm loc}\to H^{s+\gamma}_{K}$ for all $s\geq
s_{*}+\beta$ $($consequently, the continuous maps $A(\lambda)E^{\pm}\colon
H_{0}^{s}\to H_{K}^{s+\delta}$ induce compact maps $A(\lambda)E^{\pm}\colon
H_{0}^{s}\to H_{K}^{s}$ due to the Sobolev embedding theorems and $\delta>0)$;

  3. $(c)$

for all $s\geq s_{*}$, the compact map $H^{s}_{0}\to H^{s}_{K}$ induced by the
compositions $A(\lambda)E^{\pm}$ is holomorphic in $\lambda\in{\mathbb{C}}$
with respect to the topology of ${\mathcal{L}}_{b}(H^{s}_{0},H^{s}_{K})$;

  4. $(d)$

if $f\in C^{\infty}$ vanishes identically on $K$ then $Af=0$.

Then

  1. $(A)$

$\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}$ has finite
dimension for each $\lambda\in{\mathbb{C}}$;

  2. $(B)$

the sets

$S^{\pm}=\big{\\{}\lambda\in{\mathbb{C}}\colon\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}\neq
0\big{\\}}\qquad\text{and}\qquad S=S^{+}\cup S^{-}$

are discrete subsets of ${\mathbb{C}}$, whose complements are open
neighbourhoods of zero;

  3. $(C)$

$P+A(\lambda)$ has advanced and retarded $K$-nonlocal Green operators for all
$\lambda\in{\mathbb{C}}\setminus S$, that are holomorphic in $\lambda$ on this
domain with respect to the topology of
${\mathcal{L}}_{b}(C_{0}^{\infty},C^{\infty})$; these Green operators are
given explicitly as

$\tilde{E}_{\lambda}^{\pm}=E^{\pm}\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1},$

where the inverse on the right-hand side exists and is holomorphic in
$\lambda\in{\mathbb{C}}\setminus S$ with respect to the topology of
${\mathcal{L}}_{b}(C_{0}^{\infty})$;

  4. $(D)$

the Green operators possess continuous extensions mapping $H^{s}_{0}\to
H^{s+\beta}_{\rm loc}$ for all $s\geq s_{*}$ and which are holomorphic in
$\lambda\in{\mathbb{C}}\setminus S$ with respect to the topology of
${\mathcal{L}}_{b}\big{(}H^{s}_{0},H^{s+\beta}_{\rm loc}\big{)}$;

  5. $(E)$

if $(d)$ is replaced by

    1. $(d^{\prime})$

$\operatorname{supp}A(\lambda)f\subset\operatorname{supp}f$ for all $f\in
C^{\infty}$, and $\lambda\in{\mathbb{C}}$

then $P+A(\lambda)$ has advanced and retarded Green operators in the usual
sense for $\lambda\in{\mathbb{C}}\setminus S$.

Of course, in the situation where the hypotheses also apply to the formal
duals of $P$ and $A(\lambda)$ then the upshot is that $P+A(\lambda)$ and
$\leftidx{{}^{t}}{P}{}+\leftidx{{}^{t}}{A}{}(\lambda)$ both admit advanced and
retarded $K$-nonlocal Green operators for all $\lambda\in{\mathbb{C}}$ in an
open $0$-neighbourhood with discrete complement given by the union of $S$ with
the corresponding set for $\leftidx{{}^{t}}{P}{}$. For the non-exceptional
values, $P+A(\lambda)$ is $K$-nonlocally Green-hyperbolic. The general theory
of such operators and their properties is developed in detail in [14]. We note
in particular that when the ranges of both $A(\lambda)$ and
$\leftidx{{}^{t}}{A}{}(\lambda)$ are contained in $C^{\infty}_{K}$ then
assumption $(d)$ holds automatically for both operators.

In many situations the holomorphicity assumption $(c)$ is easily verified,
e.g., where $A(\lambda)$ is a polynomial in $\lambda$. Precomposing with the
continuous embedding $H^{s}_{K}\to H^{s}_{0}$, it follows that the
restrictions of $A(\lambda)E^{\pm}$ to $H^{s}_{K}$, for $s\geq s_{*}$, are
holomorphic with respect to the norm topology of ${\mathcal{L}}(H^{s}_{K})$.
On the other hand, postcomposing with the embedding $H^{s}_{K}\to H^{s}_{0}$,
we also have holomorphicity of $A(\lambda)E^{\pm}$ with respect to
${\mathcal{L}}_{b}(H^{s}_{0})$ (e.g., see Lemma A.2).

An immediate application of Theorem 3.1 is:

######  Corollary 3.2.

Let $P$ be a second-order normally hyperbolic operator on $M$ and let
$K\subset M$ be compact. Let $\rho\colon{\mathbb{C}}\to C^{\infty}(K\times K)$
be a polynomial and define $A(\lambda)\colon C^{\infty}(M)\to
C_{K}^{\infty}(M)$ by

$(A(\lambda)\phi)(x)=\int_{M}\rho(\lambda)(x,y)\phi(y)\mu_{y},\qquad\phi\in
C^{\infty}(M).$

Then $P$ and $A(\lambda)$ satisfy the hypotheses of Theorem 3.1. In
particular, $P+\lambda A_{\rho}$ has advanced and retarded $K$-nonlocal Green
operators for all $\lambda$ in an open $0$-neighbourhood with a discrete
complement in ${\mathbb{C}}$.

###### Proof.

$P$ is Green-hyperbolic, and its Green operators extend to maps $E^{\pm}\colon
H^{s}_{0}\to H^{s+1}_{\rm loc}$, i.e., they improve smoothness by one order,
by [10, Theorem 6.5.3] (see [21] for a bundle version). One easily checks that
the technical conditions on $A(\lambda)$ are certainly met by integral
operators with kernels in $C^{\infty}_{K\times K}(M\times M)$. ∎

Another consequence is the existence of classical Møller operators [11, 17]
relating the solution spaces of $P$ and $P+A(\lambda)$.

######  Corollary 3.3.

Under the hypotheses of Theorem 3.1, let ${\mathscr{S}}(\lambda)=\\{\phi\in
C^{\infty}\colon(P+A(\lambda))\phi=0\\}$. Then the maps
$\mathsf{r}_{\lambda}:C^{\infty}\to C^{\infty}$ given by

$\mathsf{r}_{\lambda}\phi=\phi-E^{+}\big{(}I+A(\lambda)E^{+}\big{)}^{-1}A(\lambda)\phi,\qquad\phi\in
C^{\infty},$ (3.1)

are holomorphic in $\lambda$ on ${\mathbb{C}}\setminus S$ with respect to the
topology of ${\mathcal{L}}_{b}(C^{\infty})$. For each
$\lambda\in{\mathbb{C}}\setminus S$, $\mathsf{r}_{\lambda}$ is a homeomorphism
of $C^{\infty}$ that restricts to a bijection from ${\mathscr{S}}(0)$ to
${\mathscr{S}}(\lambda)$.

Because $\mathsf{r}_{\lambda}\phi$ and $\phi$ differ only in $J^{+}(K)$, the
causal future of the interaction region, one calls $\mathsf{r}_{\lambda}$ the
_retarded classical Møller operator_ intertwining the dynamics of $P$ and
$P+A(\lambda)$. An advanced Møller operator can be constructed by replacing
$E^{+}$ by $E^{-}$ in (3.1).

###### Proof.

By part $(C)$ of Theorem 3.1, it is clear that
$\mathsf{r}_{\lambda}\in{\mathcal{L}}(C^{\infty})$ and holomorphic in
$\lambda$ on ${\mathbb{C}}\setminus S$ with respect to the
${\mathcal{L}}_{b}(C^{\infty})$ topology. A direct calculation, using
invertibility of $I+A(\lambda)E^{+}$ on $C_{0}^{\infty}$, shows that
$\mathsf{r}_{\lambda}$ is invertible on $C^{\infty}$ with continuous inverse
given by

$\mathsf{r}_{\lambda}^{-1}\phi=\phi+E^{+}A(\lambda)\phi,\qquad\phi\in
C^{\infty}.$

Finally, if $\phi\in{\mathscr{S}}(0)$, then

$(P+A(\lambda))\mathsf{r}_{\lambda}\phi=A(\lambda)\phi-\big{(}I+A(\lambda)E^{+}\big{)}\big{(}I+A(\lambda)E^{+}\big{)}^{-1}A(\lambda)\phi=0$

and, if $\phi\in{\mathscr{S}}(\lambda)$, then

$P\mathsf{r}_{\lambda}^{-1}\phi=-A(\lambda)\phi+A(\lambda)\phi=0$

so $\mathsf{r}_{\lambda}{\mathscr{S}}(0)\subset{\mathscr{S}}(\lambda)$ and
$\mathsf{r}_{\lambda}^{-1}{\mathscr{S}}(\lambda)\subset{\mathscr{S}}(0)$, from
which we deduce that $\mathsf{r}_{\lambda}$ restricts to a bijection from
${\mathscr{S}}(0)$ to ${\mathscr{S}}(\lambda)$. ∎

Theorem 3.1 is proved in the next section, after a number of further remarks
and examples. First, we note that Theorem 3.1 is far from the most general
statement that could be made. For instance, at the cost of more notation but
no new ideas, it generalises immediately to operators acting on sections in
finite-dimensional vector bundles over $M$. This can also be shown by applying
Theorem 3.1 in its current form, using a method of LU factorisation described
in Section 4.

Second, generalisations to multi-variable modifications
$P+\sum_{j=1}^{r}A_{j}(\lambda_{j})$ are possible. In this case, the discrete
exceptional sets are replaced by sets in ${\mathbb{C}}^{r}$ that, locally, are
vanishing sets of holomorphic functions in $r$ variables. See [16] for the
appropriate multi-variable Fredholm theorem and [22] for an exposition.

Third, the significance of holomorphicity in the topology of bounded
convergence is that compositions of such functions are also holomorphic in the
topology of bounded convergence, with derivatives given by the usual Leibniz
rule. This is explained in more detail in Appendix A.3.

Fourth, Theorem 3.1 shows that the obstruction to the existence of
$K$-nonlocal Green functions for $P+\lambda A$ is provided by nontrivial
smooth solutions to $(P+\lambda A)\phi=0$ with past- or future-compact support
(which, if present, span finite-dimensional spaces). Any sort of energy
estimate would be sufficient to exclude nontrivial solutions of this type,
which indicate unphysical behaviour in a closed system. Mathematical examples
are easily constructed, however.

######  Example 3.4.

Let $P$ be Green-hyperbolic on $M$ with Green operators $E^{\pm}$. Fix any
nontrivial $f,h\in C^{\infty}_{K}$ and define $A\colon C^{\infty}\to
C_{0}^{\infty}$ by

$A\phi=-\left(\int h\phi\,\mu\right)f,$

noting that the range of $A$ is contained in $C^{\infty}_{K}$. If $\phi\in
C^{\infty}_{\mathrm{pc}/\mathrm{fc}}$ obeys $(P+\lambda A)\phi=0$, then
$P\phi=-(\lambda\int h\phi\,\mu)f$ and we deduce that $\phi$ is a constant
multiple of $E^{\pm}f$; hence it must also be that $(P+\lambda A)E^{\pm}f=0$
or equivalently $f=\lambda\nu^{\pm}f$ where $\nu^{\pm}=\int hE^{\pm}f\mu$.
Thus a necessary condition for the existence of nontrivial
$\phi\in\ker(P+\lambda A)|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}$ is that
$\lambda\nu^{\pm}=1$, and it is easily seen that this condition is sufficient.
Summarising,

$\ker(P+\lambda
A)|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}=\begin{cases}0,&\lambda\nu^{\pm}\neq
1,\\\ {\mathbb{C}}E^{\pm}f,&\lambda\nu^{\pm}=1.\end{cases}$

Next, note that $\leftidx{{}^{t}}{A}{}$ takes the same form as $A$ but with
$f$ and $h$ exchanged. Therefore, $\ker\leftidx{{}^{t}}{(P+\lambda
A)}{}|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}$ is nontrivial if and only if
$\lambda\leftidx{{}^{t}}{\nu}{}^{\pm}=1$, where
$\leftidx{{}^{t}}{\nu}{}^{\pm}=\int fE_{t}^{\pm}h\mu$ and $E_{t}^{\pm}$ are
the Green operators of $\leftidx{{}^{t}}{P}{}$. But $E_{t}^{\pm}$ is the
formal dual of $E^{\mp}$, so in fact $\leftidx{{}^{t}}{\nu}{}^{\pm}=\nu^{\mp}$
and

$\ker\leftidx{{}^{t}}{(P+\lambda
A)}{}|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}=\begin{cases}0,&\nu^{\mp}\lambda\neq
1,\\\ {\mathbb{C}}E^{\pm}h,&\nu^{\mp}\lambda=1\end{cases}$

and consequently

$\dim\ker(P+\lambda
A)|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}=\dim\ker\big{(}\leftidx{{}^{t}}{P}{}+\lambda\leftidx{{}^{t}}{A}{}\big{)}\big{|}_{C^{\infty}_{\mathrm{fc}/\mathrm{pc}}}.$

Thus there are at most two values of $\lambda$ for which $P+\lambda A$ can
fail to be $K$-nonlocal Green-hyperbolic. In the case where $P$ obeys the
hypothesis $(a)$ of Theorem 3.1 (e.g., if $P$ is the Klein–Gordon operator)
then $P+\lambda A$ is $K$-nonlocally Green-hyperbolic for all
$\lambda\in{\mathbb{C}}\setminus\big{\\{}(\nu^{\pm})^{-1}\big{\\}}$.

The example above illustrates a general result based on Fredholm index theory,
which was prompted by an insightful question posed to the author by Bär, and
is proved in Section 6.

######  Theorem 3.5.

Suppose that the hypotheses of Theorem 3.1 are met for both $P$ and
$A(\lambda)$, and $\leftidx{{}^{t}}{P}{}$ and
$\leftidx{{}^{t}}{A}{}(\lambda)$, with $\beta\geq 0$ and
$s_{*}\leq-\beta<\gamma$. Then, for all $\lambda\in{\mathbb{C}}$,

$N^{\pm}(\lambda):=\dim\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}=\dim\ker\big{(}\leftidx{{}^{t}}{P}{}+\leftidx{{}^{t}}{A}{}(\lambda)\big{)}\big{|}_{C^{\infty}_{\mathrm{fc}/\mathrm{pc}}}.$

In particular, $N^{+}(\lambda)=N^{-}(\lambda)$ in the case where $P$ and
$A(\lambda)$ are formally self-dual.

At first sight it is quite surprising that the spaces of spontaneously
appearing and disappearing solutions have equal dimension for the self-dual
case. At its core is the fundamental fact about Lorentzian causality that
$x\in J^{\pm}(y)$ if and only if $y\in J^{\mp}(x)$, and its consequence that
the advanced and retarded Green operators of a Green-hyperbolic operator $P$
are formal duals of the retarded and advanced Green operators of the formal
dual $\leftidx{{}^{t}}{P}{}$. This by itself is not enough for the result
above, which also makes essential use of the that $A(\lambda)E^{\pm}$ is
compact, because it improves regularity.

The last remark prompts one to consider situations where $A(\lambda)E^{\pm}$
does not improve regularity.

######  Example 3.6.

Consider the Green-hyperbolic operator $P=\partial_{u}\partial_{v}$ with
respect to $(u,v)$ coordinates on ${\mathbb{R}}^{2}$, regarding vectors with
nonnegative components in these coordinates as future-pointing and causal. We
may write $P=\partial\otimes\partial$ in an obvious tensor product notation.
Let $f$, $g$ and $h$ be smooth real-valued functions on ${\mathbb{R}}$, so
that $f$, $g^{\prime}$ and $h$ have support contained in $[-2,2]$,
$\operatorname{supp}g\subset[-2,\infty)$, $f\equiv 1$ on a neighbourhood of
$[-1,1]$ and ${\big{\langle}h\,\big{|}\,g\big{\rangle}}=1$ in the usual
$L^{2}({\mathbb{R}})$ inner product. Define $T=-{|g^{\prime}\rangle}{\langle
h|}$ and note that

$g^{\prime}+Tg=0.$

Setting $K=[-2,2]\times[-2,2]$, the operator

$A=(f\partial)\otimes T$

maps $C^{\infty}\big{(}{\mathbb{R}}^{2}\big{)}$ continuously to
$C_{0}^{\infty}\big{(}{K}\big{)}$, vanishes on $\phi\in C^{\infty}(M)$ with
$\phi|_{K}\equiv 0$, and extends to a continuous map $H^{s}_{\rm loc}\to
H^{s-1}_{K}$ for any $s\in{\mathbb{R}}$. Thus $AE^{\pm}\colon H^{s}_{K}\to
H^{s}_{K}$. It is now easily seen that the equation

$P\varphi+A\varphi=0$

is solved by any distribution $\varphi=\upsilon\otimes g$, where
$\upsilon\in{\mathscr{D}}^{\prime}({\mathbb{R}})$ is supported in $[-1,1]$.
Such $\varphi$ have past compact support, contained in
$[-1,1]\times[-2,\infty)$.

This example shows that the regularity-improving nature of $AE^{\pm}$ is
responsible for both the smoothness of past-/future-compact solutions to
$(P+A)\phi=0$ and the finite-dimensionality of the corresponding solution
spaces.

In the light of this example it is clear that further conditions would be
needed to deal with modifications of first order derivative operators. For
example, if $D$ is a Dirac operator then there is a companion operator
$\tilde{D}$ so that $P=D\tilde{D}$ and $\tilde{P}=\tilde{D}D$ are second order
Green-hyperbolic with Green operators $E^{\pm}$ and $\tilde{E}^{\pm}$
respectively that improve regularity by one order. Then
$G^{\pm}=\tilde{D}E^{\pm}$ and $\tilde{G}^{\pm}=D\tilde{E}^{\pm}$ are Green
operators for $D$ and $\tilde{D}$ respectively. Similarly,
$G_{t}^{\pm}=\leftidx{{}^{t}}{\tilde{D}}{}E_{t}^{\pm}$ and
$\tilde{G}_{t}^{\pm}=\leftidx{{}^{t}}{D}{}\tilde{E}_{t}^{\pm}$ are Green
operators for $\leftidx{{}^{t}}{D}{}$ and $\leftidx{{}^{t}}{\tilde{D}}{}$,
where $E_{t}^{\pm}$ and $\tilde{E}_{t}^{\pm}$ denote the Green operators for
$\leftidx{{}^{t}}{P}{}$ and $\leftidx{{}^{t}}{\tilde{P}}{}$. Thus $D$ and
$\tilde{D}$ are Green-hyperbolic, but we cannot assume that their Green
operators improve regularity.

Now consider a $K$-nonlocal modification $D+A$ of $D$, suppressing
$\lambda$-dependence in the following. If $A$ and $\leftidx{{}^{t}}{A}{}$ have
range in $C^{\infty}_{K}$ and _improve_ regularity, then Theorem 3.1 applies
and shows that $D+A$ is $K$-nonlocally Green-hyperbolic provided that there
are no nontrivial past-compact or future-compact solutions to $(D+A)\phi=0$ or
$\big{(}\leftidx{{}^{t}}{D}{}+\leftidx{{}^{t}}{A}{}\big{)}\phi=0$. More
generally, a natural strategy to find Green operators for the modified
operator is to seek a $K$-nonlocal operator $\tilde{A}$ so that

$P_{A}=(D+A)\big{(}\tilde{D}-\tilde{A}\big{)}=D\tilde{D}+A\tilde{D}-D\tilde{A}-A\tilde{A}$

and

$\tilde{P}_{A}=\big{(}\tilde{D}-\tilde{A}\big{)}(D+A)=\tilde{D}D+\tilde{D}A-\tilde{A}D-\tilde{A}A$

are $K$-nonlocally Green-hyperbolic. If $A$ is regularity-preserving and one
may find a regularity-preserving $\tilde{A}$ such that $A\tilde{D}-D\tilde{A}$
and $\tilde{D}A-\tilde{A}D$ lose _strictly less_ than one order of regularity
(for all $\lambda\in{\mathbb{C}}$), then Theorem 3.1 could be applied to
$P_{A}$ and $\tilde{P}_{A}$ and their Green operators (when they exist) may be
used to construct Green operators for $D+A$ and $\tilde{D}+\tilde{A}$ as
before. Lacking such a choice, one potentially falls victim to the behaviour
in Example 3.6, unless Theorem 3.1 can be extended to situations in which
$AE^{\pm}$ does not improve regularity. As these questions are best pursued in
the context of specific models, we leave the discussion here and turn to the
proof of Theorem 3.1.

##  4 Application: LU factorisation method for systems

LU factorisation is a standard technique in linear algebra for solving system
of linear equations $Mx=y$ for $x,y\in{\mathbb{C}}^{N}$. In situations where
an invertible matrix $M=LU$ with $L$ lower-triangular and $U$ upper-
triangular, the solution is given by solving the triangular systems $Lz=y$,
$Ux=z$. The method can be generalised in many ways, see, e.g., [20]. Here, we
discuss its use to solve systems of Green-hyperbolic equations, or
$K$-nonlocal generalisations thereof. To see why this cannot be done
straightforwardly, consider a system $\mathcal{P}\Phi=F$ where
$\mathcal{P}\colon C^{\infty}\big{(}M;{\mathbb{C}}^{N}\big{)}\to
C^{\infty}\big{(}M;{\mathbb{C}}^{N}\big{)}$ is a $N\times N$ matrix of linear
self-maps of $C^{\infty}(M)$, where $M$ is a globally hyperbolic spacetime as
usual and $N\geq 2$. We adopt a block form

$\mathcal{P}=\begin{pmatrix}P&R\\\ S&Q\end{pmatrix},$

where $Q$ is an $(N-1)\times(N-1)$ block, fixing the dimensions of the other
blocks accordingly, so $P=\mathcal{P}_{11}$ is a self-map of $C^{\infty}(M)$.
If $P$ is a differential operator with retarded and advanced Green operators
$E^{\pm}_{P}$ and $RC^{\infty}\big{(}M;{\mathbb{C}}^{N-1}\big{)}\subset
C_{0}^{\infty}\big{(}{M;{\mathbb{C}}}\big{)}$, one may factorise $\mathcal{P}$
on $C_{0}^{\infty}\big{(}{M;{\mathbb{C}}^{N}}\big{)}$ as

$\mathcal{P}=\begin{pmatrix}1&0\\\
SE_{P}^{\pm}&1\end{pmatrix}\begin{pmatrix}P&R\\\
0&Q-SE_{P}^{\pm}R\end{pmatrix}.$ (4.1)

This is the first step towards an LU factorisation: the first factor is indeed
lower-triangular, but the second is not generally upper-triangular. The
problem is that even if all the individual matrix elements of $\mathcal{P}$
are differential operators, the factorised form involves the typically
nonlocal operators $SE^{\pm}_{P}$ and $SE^{\pm}_{P}R$. Therefore to proceed
with this strategy one should rather phrase the problem from the start in
terms of suitable nonlocal operators. (Modulo smoothing operators, one could
employ pseudodifferential operators if the leading diagonal operators were
elliptic, but here we require something more.)

We describe how an exact LU factorisation can be achieved using Theorem 3.1.
Let $K$ be a fixed compact, topologically regular subset of $M$, and let
$s_{*}\in{\mathbb{R}}$. Fix $\beta,\gamma\in{\mathbb{R}}$ with
$\beta+\gamma>0$. Let $\mathscr{A}$ be the space of maps
$A\colon{\mathbb{C}}\to{\mathcal{L}}(C^{\infty}(M))$ so that, for all
$\lambda\in{\mathbb{C}}$,

  * •

$A(\lambda)C^{\infty}(M)\subset C^{\infty}_{K}(M)$,

  * •

$A(\lambda)f\equiv 0$ if $f\in C^{\infty}(M)$ vanishes identically on $K$,

  * •

$A(\lambda)$ extends to a linear map
${\mathscr{D}}^{\prime}(M)\to{\mathscr{D}}^{\prime}_{K}(M)$ with continuous
restrictions mapping $H^{s}_{\rm loc}\to H^{s+\gamma}_{K}$ for all $s\geq
s_{*}+\beta$,

and so that $\lambda\to A(\lambda)$ is holomorphic with respect to the
topology of ${\mathcal{L}}_{b}\big{(}H^{s}_{\rm loc},H_{K}^{s+\gamma}\big{)}$.
Consider a system $\mathcal{P}$ depending on a parameter $\lambda$ so that
every off-diagonal component $\mathcal{P}_{ij}$, $i\neq j$, of $\mathcal{P}$
is a map in $\mathscr{A}$, and the diagonal components take the form

$\mathcal{P}_{ii}=P_{i}+A_{i},$

where each $P_{i}$ is a $\lambda$-independent Green-hyperbolic operator and
each $A_{i}$ is a map in $\mathscr{A}$. Assume finally that the Green
operators of the $P_{i}$ extend to maps $H^{s}_{0}\to H^{s+\beta}_{\rm loc}$
for all $s\geq s_{*}$.

By Theorem 3.1, $P(\lambda)=P_{1}+A_{1}(\lambda)$ has retarded and advanced
$K$-nonlocal Green operators for all $\lambda$ in an open $0$-neighbourhood
with discrete complement in ${\mathbb{C}}$. We may therefore factorise
$\mathcal{P}$ as in (4.1). A key point is that the
$(N-1)\times(N-1)$-dimensional systems

$\mathcal{P}^{\pm}(\lambda)=Q(\lambda)-S(\lambda)E_{P(\lambda)}^{\pm}R(\lambda)$

obey the same assumptions as the original system, noting that the matrix
components of $SE_{P}^{\pm}R$ determine continuous maps $H^{s}_{0}\to
H_{K}^{s+\beta+\delta}\hookrightarrow H_{K}^{s+\beta}$. Theorem 3.1 implies
that the leading diagonal component of $\mathcal{P}^{+}(\lambda)$ (resp.
$\mathcal{P}^{-}(\lambda)$) has retarded (resp. advanced) $K$-nonlocal Green
operators for all $\lambda$ outside a possibly enlarged exceptional set, so we
may factor each of $\mathcal{P}^{\pm}(\lambda)$. Repeating the process, this
leads to two LU factorisations of $\mathcal{P}(\lambda)$, which differ only in
whether advanced or retarded Green operators are used in their construction.
At each stage in the process we may gain more exceptional points, but the
overall exceptional set is still discrete and excludes zero. From now on, we
suppress the parameter $\lambda$ in the notation.

For non-exceptional $\lambda$, we may now use the LU factorisation to obtain
retarded and advanced Green operators for $\mathcal{P}$. We proceed
inductively in $N$. When $N=1$ we are precisely in the situation of Theorem
3.1. Now suppose that Green operators can be constructed for
$(N-1)$-dimensional systems of the type considered, where $N\geq 2$. To
establish the inductive step we return to the factorisation (4.1), in which we
may now assume that $Q-SE_{P}^{\pm}R$ has retarded/advanced Green operators
and that $P$ has both retarded and advanced Green operators. We claim that

$E^{\pm}_{\mathcal{P}}=\begin{pmatrix}E_{P}^{\pm}&-E_{P}^{\pm}RE_{Q-SE_{P}^{\pm}R}^{\pm}\\\
0&E_{Q-SE_{P}^{\pm}R}^{\pm}\end{pmatrix}\begin{pmatrix}1&0\\\
-SE_{P}^{\pm}&1\end{pmatrix}$ (4.2)

are retarded/advanced Green operators for $\mathcal{P}$. To check this, we
first observe that the lower-triangular factor in (4.1), which we denote $L$,
is invertible on $C_{0}^{\infty}\big{(}{M;{\mathbb{C}}^{N}}\big{)}$ with
inverse

$L^{-1}=\begin{pmatrix}1&0\\\ -SE_{P}^{\pm}&1\end{pmatrix}.$

Then, we compute on the one hand, that

$\mathcal{P}E^{\pm}_{\mathcal{P}}F=\begin{pmatrix}P&R\\\
S&Q\end{pmatrix}\begin{pmatrix}E_{P}^{\pm}&-E_{P}^{\pm}RE_{Q-SE_{P}^{\pm}R}^{\pm}\\\
0&E_{Q-SE_{P}^{\pm}R}^{\pm}\end{pmatrix}L^{-1}F=\begin{pmatrix}1&0\\\
SE_{P}^{\pm}&1\end{pmatrix}L^{-1}F=F$

and on the other, that

$E^{\pm}_{\mathcal{P}}\mathcal{P}F=\begin{pmatrix}E_{P}^{\pm}&-E_{P}^{\pm}RE_{Q-SE_{P}^{\pm}R}^{\pm}\\\
0&E_{Q-SE_{P}^{\pm}R}^{\pm}\end{pmatrix}L^{-1}L\begin{pmatrix}P&R\\\
0&Q-SE_{P}^{\pm}R\end{pmatrix}F=F$

for any $F\in C_{0}^{\infty}\big{(}{M;{\mathbb{C}}^{N}}\big{)}$ in each case.
Finally, the support property (G3′) is clear, provided one takes a consistent
choice of $+$ or $-$ in (4.2). Thus $\mathcal{P}$ has retarded and advanced
$K$-nonlocal Green operators, which is the inductive step, and shows that all
finite-dimensional systems of the type considered possess advanced and
retarded $K$-nonlocal Green operators; furthermore, these will vary
holomorphically in $\lambda$, in the topology of
${\mathcal{L}}_{b}\big{(}C_{0}^{\infty}\big{(}{M;{\mathbb{C}}^{N}}\big{)},C^{\infty}\big{(}M;{\mathbb{C}}^{N}\big{)}\big{)}$
outside the exceptional set. In particular, this justifies the treatment of
such systems in a recent paper on measurement in QFT [12], where interactions
between a ‘system’ QFT and one or more ‘probe QFT’s are analysed.

##  5 Proof of Theorem 3.1

The proof of Theorem 3.1 starts by establishing an inversion result that will
later be applied to $I+A(\lambda)E^{\pm}$, where $E^{\pm}$ are the Green
operators of $P$. This is proved first in Sobolev spaces $H^{s}_{K}$, using
the analytic Fredholm theorem, and then extended to $C^{\infty}_{K}$. The main
part of the proof then uses this information to define Green operators for
$P+A(\lambda)$.

######  Theorem 5.1.

Let $K\subset M$ be a fixed topologically regular compact set. Suppose
$Y(\lambda)$, $\lambda\in{\mathbb{C}}$, is a family of linear maps
$Y(\lambda)\colon{\mathscr{D}}_{K}^{\prime}\to{\mathscr{D}}^{\prime}_{K}$,
each restricting to continuous maps $Y_{s}(\lambda)\colon H^{s}_{K}\to
H^{s+\delta}_{K}$ for all $s\geq s_{*}\in{\mathbb{R}}$ and some fixed
$\delta>0$. Assume also that $Y(0)=0$. Let
$\hat{Y}_{s}(\lambda)\in{\mathcal{L}}(H^{s}_{K})$ denote the compact maps
obtained by composing $Y_{s}(\lambda)$ with the embedding
$H^{s+\delta}_{K}\hookrightarrow H^{s}_{K}$. Suppose that
$\lambda\to\hat{Y}_{s}(\lambda)$ is holomorphic on ${\mathbb{C}}$ in
${\mathcal{L}}_{b}(H^{s}_{K})$ for all $s\geq s_{*}$. Then

  1. $(a)$

The set

$S=\big{\\{}\lambda\in{\mathbb{C}}\colon\ker(I+Y(\lambda))|_{C^{\infty}_{K}}\neq
0\big{\\}}$ (5.1)

is a discrete subset of ${\mathbb{C}}$, and ${\mathbb{C}}\setminus S$ is an
open $0$-neighbourhood. Furthermore, $\ker(I+Y(\lambda))|_{C^{\infty}_{K}}$ is
finite-dimensional, and equal to $\ker(I+\hat{Y}_{s}(\lambda))$ for each
$s\geq s_{*}$.

  2. $(b)$

For all $\lambda\in{\mathbb{C}}\setminus S$ and all $s\geq s_{*}$, the map
$I+\hat{Y}_{s}(\lambda)$ is continuously invertible for all $s\geq s_{*}$, and
$\lambda\mapsto\big{(}I+\hat{Y}_{s}(\lambda)\big{)}^{-1}$ is holomorphic in
${\mathbb{C}}\setminus S$. Moreover,
$\big{(}I+\hat{Y}_{s}(\lambda)\big{)}^{-1}$ is the restriction of
$\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}$ to $H^{s}_{K}$ for each $s\geq
s_{*}$, $\lambda\in{\mathbb{C}}\setminus S$.

  3. $(c)$

$(i)$ Suppose that $f\colon{\mathbb{C}}\setminus S\to H^{s_{*}}_{K}$ is
holomorphic and there is a compact subset ${K_{f}\subset K}$ so that
$\operatorname{supp}\big{(}\hat{Y}_{s_{*}}(\lambda)\big{)}^{r}f(\lambda)\subset
K_{f}$ for all $r\in{\mathbb{N}}_{0}$ and $\lambda\in{\mathbb{C}}\setminus S$.
Then
$\operatorname{supp}\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}f(\lambda)\subset
K_{f}$ for all $\lambda\in{\mathbb{C}}\setminus S$.

$(ii)$ If $Y(\lambda)C^{\infty}_{K}\subset C^{\infty}_{K^{\prime}}$ for some
compact $K^{\prime}\subset K$, then
$\operatorname{supp}\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}f\subset
K^{\prime}$ for all $\lambda\in{\mathbb{C}}\setminus S$ and all $f\in
C^{\infty}_{K}$, and $\ker\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}$ is a
finite-dimensional subspace of $C^{\infty}_{K^{\prime}}$ for all
$\lambda\in{\mathbb{C}}$.

NB. In $(c)$, ‘support’ is to be understood as distributional support.

###### Proof.

$(a)$ Compactness of $\hat{Y}_{s}(\lambda)$ follows from the Sobolev embedding
theorems. The function $\lambda\mapsto I+\hat{Y}_{s_{*}}(\lambda)$ is an
analytic function on ${\mathbb{C}}$ with values in the Fredholm operators on
$H^{s_{*}}_{K}$, and which is invertible for $\lambda=0$ because
$\hat{Y}_{s_{*}}(0)=0$. By the analytic Fredholm theorem [26, Theorem VI.14],
$I+\hat{Y}_{s_{*}}(\lambda)$ is invertible for all $\lambda\in{\mathbb{C}}$
with the exception of the (possibly empty) set $S_{s_{*}}$ of
$\lambda\in{\mathbb{C}}$ for which
$\ker\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}$ is nontrivial (and necessarily
finite-dimensional by [26, Theorem VI.15]). Furthermore, $S_{s_{*}}$ is a
discrete subset of ${\mathbb{C}}$, whose complement is an open
$0$-neighbourhood, and the inverse
$\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}$ is meromorphic on
${\mathbb{C}}$ and holomorphic on ${\mathbb{C}}\setminus S_{s_{*}}$.

If $f\in\ker\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}$ then
$f=-\hat{Y}_{s_{*}}(\lambda)f\in H^{s_{*}+\delta}_{K}$ obeys
$f\in\ker\big{(}I+\hat{Y}_{s_{*}+\delta}(\lambda)\big{)}$; iterating,
$f\in\bigcap_{s\geq s_{*}}H^{s}_{K}=C^{\infty}_{K}$ and
$f\in\ker(I+Y(\lambda))|_{C^{\infty}_{K}}$. As the converse inclusion is
trivial, we deduce that
$\ker\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}=\ker(I+Y(\lambda))|_{C^{\infty}_{K}}$
for all $\lambda\in{\mathbb{C}}$ and hence that $S_{s_{*}}=S$ defined in
(5.1). In particular, for each fixed $\lambda\in{\mathbb{C}}$, all the kernels
$\ker\big{(}I+\hat{Y}_{s}(\lambda)\big{)}$ are equal for $s\geq s_{*}$.

$(b)$ The first statement is immediate by the Fredholm theorem and part (a);
the second follows because $I+\hat{Y}_{s}(\lambda)$ coincides with
$I+\hat{Y}_{s_{*}}(\lambda)$ on $H^{s}_{K}$ and both operators are invertible
for $\lambda\in{\mathbb{C}}\setminus S$.

$(c)$ $(i)$ Let $\chi\in C_{0}^{\infty}$ vanish on $K_{f}$. Regarding each
$\big{(}\hat{Y}_{s_{*}}(\lambda)\big{)}^{n}f(\lambda)$ as a distribution with
support in $K_{f}$, we see (e.g., by [18, Theorem 2.3.3]) that
$\big{(}\hat{Y}_{s_{*}}(\lambda)^{n}f(\lambda)\big{)}(\chi)=0$ for all
$n\in{\mathbb{N}}_{0}$ and so the holomorphic function on
${\mathbb{C}}\setminus S$ defined by
$\lambda\mapsto\big{(}\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}f(\lambda)\big{)}(\chi)$
vanishes in a neighbourhood of the origin on which the resolvent may be
expanded as a convergent geometric series in powers of
$\hat{Y}_{s_{*}}(\lambda)$, recalling that $\hat{Y}_{s_{*}}(\lambda)\to 0$ as
$\lambda\to 0$. By holomorphicity, it follows that
$\big{(}\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}f(\lambda)\big{)}(\chi)=0$
for all $\lambda\in{\mathbb{C}}\setminus S$. Allowing $\chi$ to vary, we
conclude that
$\operatorname{supp}\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}f(\lambda)\subset
K_{f}$.

$(c)$ $(ii)$ The assumption implies that of $(c)$ $(i)$ for all $f\in
C^{\infty}_{K}$ (regarded as constant functions from ${\mathbb{C}}\setminus S$
to $H^{s_{*}}_{K}$) with $K_{f}=K^{\prime}$, so
$\operatorname{supp}\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}f\subset
K^{\prime}$ for all $\lambda\in{\mathbb{C}}\setminus S$ and all $f\in
C^{\infty}_{K}$. Finally, if
$g\in\ker\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}=\ker(I+Y(\lambda))|_{C^{\infty}_{K}}$
(already known to be finite-dimensional) then $g\in C^{\infty}_{K}$ and
$g=-Y(\lambda)g\in C^{\infty}_{K^{\prime}}$. ∎

The hypotheses of Theorem 5.1 entail that each $Y(\lambda)$ restricts to a
continuous endomorphism of $C^{\infty}_{K}$. To see this, note that for any
$k\in{\mathbb{N}}_{0}$ and any integer $s>k+n/2$, where $n$ is the maximum
dimension of any component of $M$, there are continuous maps

$C^{s}_{K}\longrightarrow H^{s}_{K}\stackrel{{\scriptstyle
Y(\lambda)}}{{\longrightarrow}}H^{s}_{K}\longrightarrow C^{k}_{K},$

where the unlabelled arrows are Sobolev embeddings. Thus for all
$k\in{\mathbb{N}}_{0}$ there is $s\in{\mathbb{N}}_{0}$ and a constant $C$ such
that $\|Y(\lambda)f\|_{K,k}\leq C\|f\|_{K,s}$ for all $f\in C^{\infty}_{K}$.
This observation allows the following conclusion to be drawn.

######  Corollary 5.2.

In the notation of Theorem 5.1, and for $\lambda\in{\mathbb{C}}\setminus S$,
$I+Y(\lambda)$ restricts to a homeomorphism of $C^{\infty}_{K}$, with inverse
to be denoted $R(\lambda)$. The map $\lambda\mapsto R(\lambda)$ is holomorphic
on ${\mathbb{C}}\setminus S$ in the topology of
${\mathcal{L}}_{b}\big{(}C^{\infty}_{K}\big{)}$. If, additionally, for some
holomorphic function $f\colon{\mathbb{C}}\setminus S\to C^{\infty}_{K}$ there
is a compact subset $K_{f}\subset K$ so that
$\operatorname{supp}Y(\lambda)^{r}f(\lambda)\subset K_{f}$ for all
$r\in{\mathbb{N}}_{0}$ and $\lambda\in{\mathbb{C}}\setminus S$, then
$\operatorname{supp}R(\lambda)f(\lambda)\subset K_{f}$ for all
$\lambda\in{\mathbb{C}}\setminus S$. If $Y(\lambda)C^{\infty}_{K}\subset
C^{\infty}_{K^{\prime}}$ for some compact $K^{\prime}\subset K$, then
$\ker(I+Y(\lambda))|_{C^{\infty}_{K}}$ is a finite-dimensional subspace of
$C^{\infty}_{K^{\prime}}$ for all $\lambda\in{\mathbb{C}}$.

###### Proof.

Suppose $f\in C^{\infty}_{K}$ and $\lambda\notin S$. Then, for all $s\geq
s_{*}$, we have $f\in H^{s}_{K}$ and
$\big{(}I+\hat{Y}_{s}(\lambda)\big{)}^{-1}f=\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}f$,
and consequently

$\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}f\in\bigcap_{s\geq
s_{*}}H^{s}_{K}=C^{\infty}_{K}.$

Thus $\big{(}I+\hat{Y}_{s_{*}}(\lambda)\big{)}^{-1}$ restricts to linear self-
map of $C^{\infty}_{K}$ which (because $Y(\lambda)$ and
$\hat{Y}_{s_{*}}(\lambda)$ agree on $C^{\infty}_{K}$) is a linear inverse to
the restriction of $I+Y(\lambda)$. Accordingly, $I+Y(\lambda)$ restricts to a
continuous bijection of $C^{\infty}_{K}$, and since the latter is a Fréchet
space the inverse mapping theorem implies that the inverse $R(\lambda)$ is
continuous. As every $C^{\infty}_{K}$ semi-norm is dominated by a Sobolev norm
and vice versa, convergence in the operator norm of every $H^{s}_{K}$, for
$s\geq s_{*}$, implies convergence in
${\mathcal{L}}_{b}\big{(}C^{\infty}_{K}\big{)}$, by Corollary A.5. It follows
that $\lambda\mapsto R(\lambda)$ is holomorphic on ${\mathbb{C}}\setminus S$
in the topology of ${\mathcal{L}}_{b}\big{(}C^{\infty}_{K}\big{)}$. The
remaining statements are immediate from parts $(a)$ and $(c)$ of Theorem 5.1
and the fact that distributional support of a smooth function is exactly its
usual support. ∎

We need the following elementary observation, which will be used for
$F=C_{K}^{\infty}$, $G=C_{0}^{\infty}$.

######  Lemma 5.3.

Let $F$ and $G$ be topological vector spaces and suppose that the diagram

${F}$${F}$${G}$${G}$$\scriptstyle{S}$$\scriptstyle{\iota}$$\scriptstyle{\iota}$$\scriptstyle{T}$$\scriptstyle{\hat{T}}$

of continuous linear maps commutes. If ${\rm id}_{F}+S$ is continuously
invertible, then ${\rm id}_{G}+T$ is continuously invertible with inverse

$({\rm id}_{G}+T)^{-1}={\rm id}_{G}-T+\iota S({\rm id}_{F}+S)^{-1}\hat{T}.$

###### Proof.

Assuming that ${\rm id}_{F}+S$ is continuously invertible, we compute

$\displaystyle({\rm id}_{G}+T)\big{(}{\rm id}_{G}-T+\iota S({\rm
id}_{F}+S)^{-1}\hat{T}\big{)}={\rm id}_{G}-T^{2}+\iota({\rm id}_{F}+S)S({\rm
id}_{F}+S)^{-1}\hat{T}={\rm id}_{G}$

and

$\displaystyle\big{(}{\rm id}_{G}-T+\iota S({\rm
id}_{F}+S)^{-1}\hat{T}\big{)}({\rm id}_{G}+T)$ $\displaystyle={\rm
id}_{G}-T^{2}+\iota S({\rm id}_{F}+S)^{-1}({\rm id}_{F}+S)\hat{T}={\rm
id}_{G}.$

using in both cases the identities $\iota S=T\iota$ and hence $\iota
S\hat{T}=T\iota\hat{T}=T^{2}$, and also
$\hat{T}T=\hat{T}\iota\hat{T}=S\hat{T}$. Therefore ${\rm id}_{G}+T$ has a
linear inverse, given by a manifestly continuous expression. ∎

We come to the proof of the main result.

###### Proof of Theorem 3.1.

The proof involves several steps, and uses some technical lemmas from the
Appendix.

_$1.$ Preliminary observations._ Define
$Y^{\pm}(\lambda)\in{\mathcal{L}}(C_{0}^{\infty},C^{\infty})$ by

$Y^{\pm}(\lambda)=A(\lambda)E^{\pm}.$

Owing to assumptions $(a)$ and $(b)$, we may extend $Y^{\pm}(\lambda)$ to
linear maps ${\mathscr{E}}^{\prime}\to{\mathscr{D}}^{\prime}_{K}$ with
continuous restrictions mapping $H^{s}_{0}\to H^{s+\delta}_{K}$ for all $s\geq
s_{*}$; hence there are also continuous restrictions
$Y_{s}^{\pm}(\lambda)\colon H^{s}_{K}\to H^{s+\delta}_{K}$ for $s\geq s_{*}$,
and by the Sobolev embedding theorems, compact maps
$\hat{Y}^{\pm}_{s}(\lambda)\colon H^{s}_{K}\to H^{s}_{K}$ for $s\geq s_{*}$
which are holomorphic with respect to the operator norm topology, as noted
after the statement of Theorem 3.1.

In fact, if $K^{\prime}$ is any compact topologically regular set, the same
argument shows that $Y^{\pm}(\lambda)$ define compact maps depending
holomorphically on $\lambda$ in the topology of
${\mathcal{L}}_{b}\big{(}H^{s}_{K\cup K^{\prime}}\big{)}$ for all
${s\\!\geq\\!s_{*}}$. By Corollary A.5, it follows that $Y^{\pm}(\lambda)$ are
holomorphic in $\lambda$ with respect to the
${\mathcal{L}}_{b}\big{(}C^{\infty}_{K\cup K^{\prime}}\big{)}$ topology. As
$C^{\infty}_{K}$ and $C^{\infty}_{K^{\prime}}$ are continuously embedded
topological subspaces of $C^{\infty}_{K\cup K^{\prime}}$, and
$\operatorname{Ran}Y^{\pm}(\lambda)\subset C^{\infty}_{K}$, we may use Lemma
A.2 $(a)$, $(c)$ to deduce that the $Y^{\pm}(\lambda)$ are holomorphic in
$\lambda$ with respect to the topology of
${\mathcal{L}}_{b}\big{(}C^{\infty}_{K^{\prime}},C^{\infty}_{K}\big{)}$. By
Lemma A.6, it follows that $A(\lambda)E^{\pm}$ are also holomorphic with
respect to ${\mathcal{L}}_{b}\big{(}C_{0}^{\infty},C^{\infty}_{K}\big{)}$ and
${\mathcal{L}}_{b}(C_{0}^{\infty})$; using Lemma A.2 $(a)$, we also have
holomorphicity with respect to
${\mathcal{L}}_{b}\big{(}C^{\infty}_{K},C_{0}^{\infty}\big{)}$.

_$2.$ Finite-dimensionality of
$\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}$._ Next, observe
that $P$ induces bijections between
$\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}$ and
$\ker\big{(}I+Y^{\pm}(\lambda)\big{)}\big{|}_{C^{\infty}_{K}}$, with inverses
given by the restrictions of $E^{\pm}$. For if $(P+A(\lambda))\phi=0$ with
$\phi\in C^{\infty}_{\mathrm{pc}/\mathrm{fc}}$ then $P\phi=-A(\lambda)\phi\in
C^{\infty}_{K}$ and $\phi=E^{\pm}P\phi$, so
$P\phi\in\ker\big{(}I+A(\lambda)E^{\pm}\big{)}\big{|}_{C^{\infty}_{K}}$;
conversely, if $\big{(}I+A(\lambda)E^{\pm}\big{)}h=0$ with $h\in
C^{\infty}_{K}$, then $PE^{\pm}h=h=-A(\lambda)E^{\pm}h$, so
$E^{\pm}h\in\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}$. Thus

$S^{\pm}:=\big{\\{}\lambda\in{\mathbb{C}}\colon\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}\neq
0\big{\\}}=\big{\\{}\lambda\in{\mathbb{C}}\colon\ker\big{(}I+Y^{\pm}(\lambda)\big{)}\big{|}_{C^{\infty}_{K}}\neq
0\big{\\}}.$

In combination with Theorem 5.1 $(a)$, we also have

$\dim\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}=\dim\ker\big{(}I+Y_{s}^{\pm}(\lambda)\big{)}\big{|}_{C^{\infty}_{K}}=\dim\ker\big{(}I+\hat{Y}_{s}^{\pm}(\lambda)\big{)}<\infty$
(5.2)

(the latter kernel taken in $H^{s}_{K}$) for all $s\geq s_{*}$. Part $(A)$ of
the Theorem is thus proved.

_$3.$ Construction of holomorphic candidate $K$-nonlocal Green operators._
Applying Theorem 5.1 and Corollary 5.2 to $Y^{\pm}(\lambda)$, one finds that
$S^{\pm}$ are discrete subsets of ${\mathbb{C}}$, whose complements in
${\mathbb{C}}$ are open $0$-neighbourhoods, and the operators
$I+Y^{\pm}(\lambda)$ are continuously invertible on $C^{\infty}_{K}$ for
$\lambda\in{\mathbb{C}}\setminus S^{\pm}$, with inverses that are holomorphic
on ${\mathbb{C}}\setminus S^{\pm}$ in the topology of
${\mathcal{L}}_{b}\big{(}C^{\infty}_{K}\big{)}$. All these properties hold
also for $S=S^{+}\cup S^{-}$, so part (B) is proved.

Fixing any $\lambda\in{\mathbb{C}}\setminus S$, the diagram of continuous maps

${C^{\infty}_{K}}$${C^{\infty}_{K}}$${C_{0}^{\infty}}$${C_{0}^{\infty}}$$\scriptstyle{T_{K}}$$\scriptstyle{T_{0}}$$\scriptstyle{\hat{T}_{0}}$

commutes, where the unlabelled arrows are the canonical inclusions, $T_{0}$
and $T_{K}$ are the respective restrictions of $Y^{\pm}(\lambda)$ to
$C_{0}^{\infty}$, and $C_{K}^{\infty}$ and $\hat{T}_{0}$ exists because
$A(\lambda)C^{\infty}\subset C^{\infty}_{K}$. It follows from Lemma 5.3 that
$I+Y^{\pm}(\lambda)$ are continuously invertible on $C_{0}^{\infty}$. By abuse
of notation we write the inverses as $\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}$;
Lemma 5.3 now gives the identity

$\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}=I-A(\lambda)E^{\pm}+A(\lambda)E^{\pm}\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}A(\lambda)E^{\pm},$
(5.3)

on $C_{0}^{\infty}(K)$, where we suppress notation for inclusions and
restrictions. Note that the inverse on the right-hand side is taken in
${\mathcal{L}}\big{(}C^{\infty}_{K}\big{)}$, while the left-hand side is an
inverse in ${\mathcal{L}}(C^{\infty}_{0})$. Because the former inverse is
holomorphic in ${\mathcal{L}}_{b}\big{(}C^{\infty}_{K}\big{)}$, the Leibniz
rule (see Corollary A.3) implies that the left-hand side is holomorphic in
${\mathcal{L}}_{b}(C_{0}^{\infty})$; here, we have also used the
holomorphicity of $A(\lambda)E^{\pm}$ in
${\mathcal{L}}_{b}\big{(}C_{0}^{\infty},C^{\infty}_{K}\big{)}$,
${\mathcal{L}}_{b}(C_{0}^{\infty})$ and
${\mathcal{L}}_{b}\big{(}C^{\infty}_{K},C_{0}^{\infty}\big{)}$ established in
step 1 of the proof. It follows that the operators

$\tilde{E}^{\pm}_{\lambda}=E^{\pm}\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}\in{\mathcal{L}}(C_{0}^{\infty},C^{\infty}),$

which are the candidate Green operators for $P+A(\lambda)$, are holomorphic in
$\lambda$ on ${\mathbb{C}}\setminus S$ with respect to the topology of
${\mathcal{L}}_{b}(C_{0}^{\infty},C^{\infty})$. To prove part (C) it is now
enough to check that $\tilde{E}^{\pm}_{\lambda}$ are indeed $K$-nonlocal Green
operators.

_$4.$ Verification that $\tilde{E}^{\pm}_{\lambda}$ are $K$-nonlocal Green
operators._ Given any $f\in C_{0}^{\infty}$,

$g=\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}f$

is the unique element of $C_{0}^{\infty}$ obeying

$g+A(\lambda)E^{\pm}g=f,$ (5.4)

whereupon we deduce that $\operatorname{supp}g\subset\operatorname{supp}f\cup
K$ and that

$\varphi=E^{\pm}g=E^{\pm}\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}f=\tilde{E}^{\pm}_{\lambda}f$

satisfies

$P\varphi+A(\lambda)\varphi=g+A(\lambda)E^{\pm}g=f,\qquad\operatorname{supp}\varphi\subset
J^{\pm}(\operatorname{supp}g)\subset J^{\pm}(K\cup\operatorname{supp}f).$
(5.5)

In the special case $f=(P+A(\lambda))h$, for $h\in C_{0}^{\infty}$, we have

$f=(P+A(\lambda))E^{\pm}Ph=Ph+A(\lambda)E^{\pm}Ph$

and the unique solution to (5.4) is clearly $g=Ph$. Thus

$Ph=g=\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}f=\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}(P+A(\lambda))h;$

consequently $E^{\pm}\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}(P+A(\lambda))h=h$.
In combination with (5.5), we have shown

$(P+A(\lambda))\tilde{E}^{\pm}_{\lambda}f=f=\tilde{E}^{\pm}_{\lambda}(P+A(\lambda))f,\qquad\text{and}\qquad\operatorname{supp}\tilde{E}^{\pm}_{\lambda}f\subset
J^{\pm}(K\cup\operatorname{supp}f)$

for all $f\in C_{0}^{\infty}$. Now suppose more specifically that
$J^{\pm}(\operatorname{supp}f)\cap K=\varnothing$ for $f\in C^{\infty}_{0}$.
Then $A(\lambda)E^{\pm}f=0$, due to assumption $(d)$, and (5.4) is solved by
$g=f$, so $\tilde{E}^{\pm}_{\lambda}f=E^{\pm}f$ has support contained in
$J^{\pm}(\operatorname{supp}f)$. Accordingly,

$\operatorname{supp}\tilde{E}^{\pm}_{\lambda}f\subset\begin{cases}J^{\pm}(\operatorname{supp}f),&J^{\pm}(\operatorname{supp}f)\cap
K=\varnothing,\\\
J^{\pm}(K\cup\operatorname{supp}f),&\text{otherwise,}\end{cases}$

so $\tilde{E}^{\pm}_{\lambda}$ are $K$-nonlocal Green operators for
$P+A(\lambda)$. Part $(C)$ is complete.

_$5.$ Continuous extension._ Due to (5.3), one has the formula

$\tilde{E}_{\lambda}^{\pm}=E^{\pm}-E^{\pm}A(\lambda)E^{\pm}+E^{\pm}A(\lambda)E^{\pm}\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}A(\lambda)E^{\pm},$

in which the three terms on the right-hand side have continuous extensions as
maps from $H^{s}_{0}$ to $H^{s+\beta}_{\rm loc}$, $H^{s+\beta+\delta}_{\rm
loc}$ and $H^{s+\beta+2\delta}_{\rm loc}$ respectively. Because $\delta>0$, we
deduce that $\tilde{E}^{\pm}_{\lambda}$ extends continuously to a map
$H^{s}_{0}\to H_{\rm loc}^{s+\beta}$ as required. By the Leibniz rule
(Corollary A.3), this extension is holomorphic in $\lambda$ with respect to
the topology of ${\mathcal{L}}_{b}\big{(}H^{s}_{0},H_{\rm
loc}^{s+\beta}\big{)}$. This proves part (D).

_$6.$ Support non-increasing modifications._ Finally, suppose condition
$(d^{\prime})$ holds, so that one has
$\operatorname{supp}A(\lambda)f\subset\operatorname{supp}f$ as well as
$\operatorname{supp}A(\lambda)f\subset K$ for all $f\in C^{\infty}$. Then
$(d)$ also holds, either as a consequence of Peetre’s theorem [25] or by the
following direct argument: if $f\in C^{\infty}$ vanishes identically on $K$,
then the carrier of $A(\lambda)f\in C^{\infty}_{K}$ satisfies both

$\operatorname{carr}A(\lambda)f\subset\operatorname{int}(K)$

and

$\operatorname{carr}A(\lambda)f\subset\operatorname{supp}A(\lambda)f\subset\operatorname{supp}f=\overline{\operatorname{carr}f}\subset\overline{M\setminus
K}=M\setminus\operatorname{int}(K)$

and we deduce that $\operatorname{carr}A(\lambda)f$ is empty, i.e.,
$A(\lambda)f$ vanishes identically. Therefore, all the conclusions reached
previously still hold.

It further follows that

$\operatorname{supp}Y^{\pm}(\lambda)f=\operatorname{supp}A(\lambda)E^{\pm}f\subset
K\cap\operatorname{supp}E^{\pm}f\subset K\cap
J^{\pm}(\operatorname{supp}f)\qquad\text{for all}\ f\in C_{0}^{\infty};$

iterating, we have in particular that

$\operatorname{supp}\big{(}Y^{\pm}(\lambda)\big{)}^{r}f\subset
K^{\pm}_{f}:=K\cap J^{\pm}(\operatorname{supp}f)\qquad\text{for all}\
r\in{\mathbb{N}},\ f\in C_{0}^{\infty}\ \text{and}\
\lambda\in{\mathbb{C}}\setminus S.$

Therefore $\big{(}Y^{\pm}(\lambda)\big{)}^{r}Y^{\pm}(\lambda)f\in
C^{\infty}_{K^{\pm}_{f}}\subset C^{\infty}_{K}$ for all
$r\in{\mathbb{N}}_{0}$, $f\in C_{0}^{\infty}$,
$\lambda\in{\mathbb{C}}\setminus S$. Using Corollary 5.2, applied to the
function $\lambda\mapsto Y^{\pm}(\lambda)f\in C^{\infty}_{K}$, it follows that
both
$\operatorname{supp}\big{(}I+Y^{\pm}(\lambda)\big{)}^{-1}Y^{\pm}(\lambda)f$
and

$\operatorname{supp}Y^{\pm}(\lambda)\big{(}I+Y^{\pm}(\lambda)\big{)}^{-1}Y^{\pm}(\lambda)f=\operatorname{supp}\big{(}I-(I+Y^{\pm}(\lambda))^{-1}\big{)}Y^{\pm}(\lambda)f$

are contained in $K^{\pm}_{f}$ for all $\lambda\in{\mathbb{C}}\setminus S$.
Using the identity (5.3), it now follows that

$\operatorname{supp}\big{(}I+A(\lambda)E^{\pm}\big{)}^{-1}f\subset\operatorname{supp}f\cup\left(K\cap
J^{\pm}(\operatorname{supp}f)\right)\subset J^{\pm}(\operatorname{supp}f)$

and hence $\operatorname{supp}\tilde{E}^{\pm}_{\lambda}f\subset
J^{\pm}\big{(}J^{\pm}(\operatorname{supp}f)\big{)}=J^{\pm}(\operatorname{supp}f)$,
which is the support property (G3) for standard Green operators, completing
the proof of part $(E)$ and therefore the whole theorem. ∎

##  6 Proof of Theorem 3.5

The aim of this section is to prove a relation between the spontaneously
appearing and disappearing solutions for the operators $P+A(\lambda)$ and
$\leftidx{{}^{t}}{P}{}+\leftidx{{}^{t}}{A}{}(\lambda)$ stated in Theorem 3.5.

Starting with some preliminaries, let $E^{\pm}$ and $E_{t}^{\pm}$ be the Green
operators for $P$ and $\leftidx{{}^{t}}{P}{}$. Because $s_{*}\leq-\beta\leq
0$, $E^{\pm}$ and $E_{t}^{\pm}$ have extensions from $C^{\infty}_{K}$ to
continuous maps in both ${\mathcal{L}}\big{(}H^{0}_{0},H^{\beta}_{\rm
loc}\big{)}$ and ${\mathcal{L}}\big{(}H^{\gamma}_{0},H^{\delta}_{\rm
loc}\big{)}$, while $A(\lambda)$ and $\leftidx{{}^{t}}{A}{}(\lambda)$ have
extensions from $C^{\infty}$ to continuous maps in both
${\mathcal{L}}\big{(}L^{2}_{\rm loc},H^{\gamma}_{K}\big{)}$ (and consequently
${\mathcal{L}}\big{(}L^{2}_{K},H^{\gamma}_{K}\big{)}$) and
${\mathcal{L}}\big{(}H^{\beta}_{\rm loc},H^{\delta}_{K}\big{)}$. We also write
$Y^{\pm}(\lambda)=A(\lambda)E^{\pm}$ and
$Y_{t}^{\pm}=\leftidx{{}^{t}}{A}{}(\lambda)E_{t}^{\pm}$ as compact operators
on any $H^{s}_{K}$ (the value of $s$ will be clear from context, and we
suppress embedding maps between various $H^{s}_{K}$ spaces).

We generally abuse notation by using the same notation for $A(\lambda)$
whether it operates on $H^{s}_{K}$ or $H^{s}_{\rm loc}$ and regardless of $s$,
but our arguments will take proper account of the domains concerned. We make
use of two technical facts in the $s=0$ case.

######  Lemma 6.1.

Under the stated assumptions on $A(\lambda)$ and $Y^{\pm}(\lambda)$:

  1. $(a)$

the identity

$A(\lambda)f=A(\lambda)(f|_{K})$

holds for all $f\in L^{2}_{\rm loc}$, where we understand the map
$A(\lambda)\in{\mathcal{L}}\big{(}L^{2}_{\rm loc},H^{\gamma}_{K}\big{)}$ on
the left-hand side and
$A(\lambda)\in{\mathcal{L}}\big{(}L^{2}_{K},H^{\gamma}_{K}\big{)}$ on the
right-hand side;

  2. $(b)$

for the operators
$Y_{t}^{\pm}(\lambda)\in{\mathcal{L}}\big{(}L^{2}_{K},L^{2}_{K}\big{)}$, we
have

$\big{(}Y_{t}^{\pm}(\lambda)\big{)}^{*}f=(\Gamma E^{\mp}A(\lambda)\Gamma
f)|_{K},\qquad f\in L^{2}_{K},$

where $\Gamma$ denotes complex conjugation.

###### Proof.

We suppress the $\lambda$ dependence in this proof. For part $(a)$, first
choose $f_{n}\in C^{\infty}$ with $f_{n}\to f$ in $L^{2}_{\rm loc}$. Then for
any $g\in C_{0}^{\infty}$ the distributional action of $Af$ on $\mu g$
(recalling that $\mu$ is the volume density) is given by

$(Af)(\mu g)=\lim_{n\to\infty}\int_{M}\mu
g(Af_{n})=\lim_{n\to\infty}\int_{M}\mu(\leftidx{{}^{t}}{A}{}g)f_{n}={\big{\langle}\overline{\leftidx{{}^{t}}{A}{}g}\,\big{|}\,f|_{K}\big{\rangle}},$

where the inner product $\langle\cdot\mid\cdot\rangle$ is that of $L^{2}_{K}$
and we use the fact that $\leftidx{{}^{t}}{A}{}g\in C^{\infty}_{K}$. Now
choose a new sequence $f_{n}\in C^{\infty}_{K}$ with $f_{n}\to f|_{K}$ in
$L^{2}_{K}$. Then

$(Af)(\mu
g)={\big{\langle}\overline{\leftidx{{}^{t}}{A}{}g}\,\big{|}\,f|_{K}\big{\rangle}}=\lim_{n\to\infty}\int_{M}\mu\big{(}\leftidx{{}^{t}}{A}{}g\big{)}f_{n}=\lim_{n\to\infty}\int_{M}\mu
g(Af_{n})=(Af|_{K})(\mu g).$

As $g$ was arbitrary, $Af$ and $Af|_{K}$ define the same distribution and
hence the same element of $H^{\gamma}_{K}$.

For part $(b)$, we compute for $f,h\in C^{\infty}_{K}$ that

$\displaystyle{\big{\langle}\big{(}Y_{t}^{\pm}\big{)}^{*}f\,\big{|}\,h\big{\rangle}}$
$\displaystyle={\big{\langle}f\,\big{|}\,Y_{t}^{\pm}h\big{\rangle}}=\int_{M}\mu\overline{f}(\leftidx{{}^{t}}{A}{})E_{t}^{\pm}h=\int_{M}\mu\big{(}E^{\mp}A\overline{f}\big{)}h={\big{\langle}(\Gamma
E^{\mp}A\Gamma f)|_{K}\,\big{|}\,h\big{\rangle}}.$

Thus we have $\big{(}Y_{t}^{\pm}(\lambda)\big{)}^{*}f=(\Gamma E^{\mp}A\Gamma
f)|_{K}$ for all $f\in C^{\infty}_{K}$ and hence for all $f\in L^{2}_{K}$ by
continuity.∎

###### Proof of Theorem 3.5.

Let
$\leftidx{{}^{t}}{N}{}^{\pm}(\lambda)=\dim\ker(P+A(\lambda))|_{C^{\infty}_{\mathrm{pc}/\mathrm{fc}}}$.
By the remark (5.2) in the proof of Theorem 3.1, applied to $Y_{t}^{\pm}$ in
the case $s=0$, we have
$\leftidx{{}^{t}}{N}{}^{\pm}(\lambda)=\dim\ker\big{(}I+Y_{t}^{\pm}(\lambda)\big{)}$
in $L^{2}_{K}$. As $Y_{t}^{\pm}$ are compact and holomorphic in the suppressed
parameter $\lambda$, Fredholm theory implies that the index

$\operatorname{Index}\big{(}Y_{t}^{\pm}(\lambda)\big{)}=\dim\ker\big{(}I+Y_{t}^{\pm}(\lambda)\big{)}-\dim\ker\big{(}I+\big{(}Y_{t}^{\pm}(\lambda)\big{)}^{*}\big{)}$

is independent of $\lambda$ and therefore vanishes on considering $\lambda=0$
– see, e.g., [8, Theorem 4.3.12] – so

$\leftidx{{}^{t}}{N}{}^{\pm}(\lambda)=\dim\ker\big{(}I+\big{(}Y_{t}^{\pm}(\lambda)\big{)}^{*}\big{)}.$

Below, we will show that $A\Gamma$ induces an antilinear injection between the
kernels of $I+\big{(}Y_{t}^{\pm}(\lambda)\big{)}^{*}$ and $I+Y^{\mp}(\lambda)$
in $L^{2}_{K}$; swapping the roles of $P$, $A(\lambda)$ and
$\leftidx{{}^{t}}{P}{}$, $\leftidx{{}^{t}}{A}{}(\lambda)$, we find that the
spaces have equal (finite) dimension. Consequently,

$\leftidx{{}^{t}}{N}{}^{\pm}(\lambda)=\dim\ker(I+Y^{\mp}(\lambda))=N^{\mp}(\lambda),$

again using remark (5.2) in the proof of Theorem 3.1, which is the desired
result.

It remains to prove that $A\Gamma$ provides the required antilinear injection.
Suppose that there is $f\in L^{2}_{K}\setminus\\{0\\}$ with $\Gamma
f=-\big{(}Y_{t}^{\pm}(\lambda)\big{)}^{*}\Gamma f$ for some fixed $\lambda$.
Then by Lemma 6.1 $(b)$

$f=-(E^{\mp}A(\lambda)f)|_{K}$

and since $f\neq 0$ it also follows that $A(\lambda)f\neq 0$ (otherwise
$f=-E^{\mp}A(\lambda)f|_{K}=0$). Furthermore,

$A(\lambda)f=-A(\lambda)(E^{\mp}A(\lambda)f)|_{K}=-A(\lambda)E^{\mp}A(\lambda)f=-Y^{\mp}(\lambda)A(\lambda)f$

holds in $H^{\gamma}_{K}$, so $A(\lambda)f\in\ker(I+Y^{\mp}(\lambda))$ in
$H^{\gamma}_{K}$, and therefore also in $L^{2}_{K}$ (since $\gamma>0\geq
s_{*}$ the kernels are equal by Theorem 5.1 $(a)$). Accordingly,
$A(\lambda)\Gamma$ is an antilinear injection from the $L^{2}_{K}$ kernel of
$I+\big{(}\leftidx{{}^{t}}{Y}{}{}^{\pm}(\lambda)\big{)}^{*}$ to the
$L^{2}_{K}$ kernel of $I+Y^{\mp}(\lambda)$. ∎

##  Appendix A Some topological vector spaces

We briefly rehearse the definition and main properties of the various $C^{k}$
and Sobolev spaces encountered in the text, broadly following [2, 4], before
turning to some properties of the topology of bounded convergence that are
also needed. No originality is claimed for the material given here.

###  A.1 Spaces of smooth and $\boldsymbol{C^{k}}$ functions

Let $M$ be a smooth manifold and let $C^{k}(M)$,
$k\in{\mathbb{N}}_{0}\cup\\{\infty\\}$, be the vector space of complex-valued
$k$-times continuously differentiable functions on $M$. For each
$k\in{\mathbb{N}}_{0}$ and compact $K\subset M$, one has a seminorm

$\|f\|_{K,k}=\max_{0\leq r\leq k}\max_{x\in K}|(\nabla^{r}f)(x)|_{r}$

on $C^{k}(M)$, where $\nabla$ is an arbitrarily chosen connection on $M$ and
$|\cdot|_{r}$ an arbitrarily chosen norm making $T^{*}M^{\otimes r}$ a
(finite-dimensional) Banach bundle; different choices result in equivalent
seminorms. The collection of seminorms $\|\cdot\|_{K,k}$ as $K$ runs over
compact subsets of $M$ and $k\in{\mathbb{N}}_{0}$ provides a Fréchet topology
on $C^{\infty}(M)$; similarly, we obtain a Fréchet topology on $C^{k}(M)$,
$k\in{\mathbb{N}}_{0}$, using the seminorms $\|\cdot\|_{K,k}$.

If $A$ is closed, we define $C^{\infty}_{A}(M)=\\{f\in
C^{\infty}(M)\colon\operatorname{supp}f\subset A\\}$ with the relative
topology. Thus the topology is defined by the seminorms $\|\cdot\|_{K,k}$ as
$K$ runs over compact subsets $K\subset A$ and $k\in{\mathbb{N}}_{0}$; if $A$
is compact, it is sufficient to use the seminorms $\|\cdot\|_{A,k}$,
$k\in{\mathbb{N}}_{0}$. As $C^{\infty}_{A}(M)$ is closed subspace of a Fréchet
space, it is also Fréchet. Defining $C^{k}_{A}(M)$ as the analogous subspace
of $C^{k}(M)$, the topology is generated by seminorms $\|\cdot\|_{K,k}$ for
compact $K\subset A$, or just the single seminorm $\|\cdot\|_{A,k}$ (which is
a norm on $C^{k}_{A}(M)$) in the case that $A$ is compact.

A _support system_ [2] is a subset $\mathscr{A}$ of the set of all closed
subsets on $M$, which is closed under finite unions and has the property that
for each $A\in\mathscr{A}$ it holds that $(i)$
$A\subset\textrm{int}\,(A^{\prime})$ for some $A^{\prime}\in\mathscr{A}$ and
$(ii)$ if $A^{\prime}$ is a closed subset of $M$ with $A^{\prime}\subset A$
then $A^{\prime}\in\mathscr{A}$. Any support system is a directed system with
respect to inclusion and we write

$C^{\infty}_{\mathscr{A}}(M)=\bigcup_{A\in\mathscr{A}}C^{\infty}_{A}(M)$

with the locally convex inductive limit topology, so that a convex set
$U\subset C^{\infty}_{\mathscr{A}}(M)$ is a neighbourhood of $0$ if and only
if $U\cap C^{\infty}_{A}(M)$ is a neighbourhood of $0$ in $C^{\infty}_{A}(M)$
for every $A\in\mathscr{A}$; because $\mathscr{A}$ is directed, one also has
that a convex set $O$ is open if and only if $O\cap A$ is open in each
$C^{\infty}_{A}(M)$.

Examples of support systems include the set of compact sets, leading to the
space of compactly supported functions $C_{0}^{\infty}\big{(}{M}\big{)}$, and
the sets of (strictly) future/past/spatially-compact sets, giving rise to
$C^{\infty}_{\mathrm{sfc}/\mathrm{spc}/\mathrm{fc}/\mathrm{pc}/\mathrm{sc}}(M)$.

A linear map $T$ from $C^{\infty}_{\mathscr{A}}(M)$ to a locally convex
topological vector space $Y$ is continuous if and only if all its restrictions
$T_{A}\colon C^{\infty}_{A}(M)\to Y$ are continuous ($A\in\mathscr{A}$). In
particular, for a linear map $T\colon C^{\infty}_{\mathscr{A}}(M)\to
C^{\infty}_{\mathscr{B}}(M)$ to be continuous, it suffices that each
restriction $T_{A}$ has range contained in some $C^{\infty}_{B}(M)$ for some
$B\in\mathscr{B}$ (depending on $A$) and determines a continuous map
$C^{\infty}_{A}(M)\to C^{\infty}_{B}(M)$, thus implying that $T_{A}\colon
C^{\infty}_{A}(M)\to C^{\infty}_{\mathscr{B}}(M)$ is continuous. We record the
following application:

######  Lemma A.1.

If $T$ is a continuous linear self-map of $C^{\infty}(M)$ such that
$\operatorname{supp}Tf\subset K\cup\operatorname{supp}f$ for all $f\in
C^{\infty}(M)$, where $K$ is a fixed compact set, then $T$ restricts to any of
$C^{\infty}_{0/\mathrm{sfc}/\mathrm{spc}/\mathrm{fc}/\mathrm{pc}/\mathrm{sc}}(M)$
as a continuous map.

###### Proof.

If $\mathscr{A}$ is one of the relevant support systems then $A\cup
K\in\mathscr{A}$ for each $A\in\mathscr{A}$. As the restriction $T_{A}$ of $T$
to $C^{\infty}_{A}(M)$ has its range in $C^{\infty}_{A\cup K}(M)$, and both
these function spaces are subspaces of $C^{\infty}(M)$ with the relative
topology, continuity of $T_{A}\colon C^{\infty}_{A}(M)\to C^{\infty}_{A\cup
K}(M)$ follows from that of $T$. Letting $A$ vary in $\mathscr{A}$, the result
is proved. ∎

Letting $\Omega_{\alpha}$ be the bundle of densities of weight $\alpha$, we
can define spaces $\Gamma^{\infty}(\Omega_{\alpha})$,
$\Gamma_{A}^{\infty}(\Omega_{\alpha})$ and
$\Gamma_{0}^{\infty}({\Omega_{\alpha}})$ of smooth sections of
$\Omega_{\alpha}$ in a similar way; any smooth nowhere vanishing density $\mu$
on $M$ provides topological isomorphisms between these spaces and their zero-
weight analogues, simply by multiplication by appropriate powers of $\mu$. The
space of distributions on $M$, ${\mathscr{D}}^{\prime}(M)$ is the topological
dual of $\Gamma_{0}^{\infty}({\Omega_{1}})$, equipped with the weak-$*$
topology. In particular, $C^{\infty}(M)$ is canonically embedded as a subspace
of ${\mathscr{D}}^{\prime}(M)$. More generally,
${\mathscr{D}}^{\prime}(M,\Omega_{\alpha})$ is defined as the dual of
$\Gamma_{0}^{\infty}({\Omega_{1-\alpha}})$. We also write
${\mathscr{E}}^{\prime}(M)$ for the topological dual of
$\Gamma^{\infty}(\Omega_{1})$, i.e., the distributions of compact support.

###  A.2 Sobolev spaces

####  A.2.1 Compact manifolds

The spaces $\mathbf{H^{s}(M)}$. On a _compact_ manifold $M$, choose an
auxiliary Riemannian metric and define $L^{2}(M)$ in the usual way, using the
volume element induced by the metric and writing the inner product as
$\langle\cdot\mid\cdot\rangle$ (linear in the second slot). Let
$T=(-\triangle+I)^{1/2}$, where $\triangle$ is the Laplace–Beltrami operator,
initially defined on $C^{\infty}(M)$ and then extended (uniquely) to a self-
adjoint negative operator on $L^{2}(M)$. Its complex powers $T^{s}$ (as
defined by functional calculus) are classical pseudodifferential operators of
order $s$, $T^{s}\in\Psi^{s}(M)$ [27] (see [1] for an axiomatically based
proof) and $T^{s}$ is compact for $\operatorname{Re}s<0$ (most easily seen
using the spectral properties of $-\triangle$ on compact manifolds). The
domain of $T^{s}$ contains $C^{\infty}(M)$ for all $s$.

For $s\in{\mathbb{R}}$, the Sobolev space $H^{s}(M)$ is defined as the
completion of $C^{\infty}(M)$ with respect to the norm

$\|f\|_{H^{s}(M)}=\|T^{s}f\|_{L^{2}(M)}$

and, as a topological vector space, is independent of the choice of auxiliary
Riemannian metric involved in its construction (of course, the specific norm
$\|\cdot\|_{H^{s}(M)}$ and its compatible Hilbert space inner product are
metric-dependent). Indeed, any positive second-order elliptic operator that is
essentially self-adjoint on $C^{\infty}(M)$ could be used in place of
$-\triangle$. Evidently $T^{s}$ extends to an isometry from $H^{s}(M)$ to
$L^{2}(M)$, which may be used to embed $H^{s}(M)$ in
${\mathscr{D}}^{\prime}(M)$ so that $u\in H^{s}(M)$ corresponds to the
distribution
$f\mapsto{\big{\langle}T^{-s}\overline{f/\nu}\,\big{|}\,T^{s}u\big{\rangle}}$,
$f\in\Gamma_{0}^{\infty}({\Omega_{1}})$, where $\nu$ is the density
corresponding to the volume element of the auxiliary metric used to define
$L^{2}(M)$. Restricted to $u\in C^{\infty}(M)$, this embedding is consistent
with the embedding of $C^{\infty}(M)$ in ${\mathscr{D}}^{\prime}(M)$ already
mentioned. As $T^{q}$ is compact on $L^{2}(M)$ for $q<0$, it follows that
$H^{s}(M)\subset H^{t}(M)$ for all $s>t$, with a compact inclusion map. More
generally. $P\colon H^{s+m}(M)\to H^{s}(M)$ is continuous for any partial
differential operator of order $m$ with smooth coefficients (or indeed any
pseudodifferential operator $P\in\Psi^{m}(M)$), because
$T^{-m}P\in\Psi^{0}(M)$ extends to a bounded operator on $L^{2}(M)$.

Duality. For any $s\in{\mathbb{R}}$, suppose that $\ell\in H^{s}(M)^{\prime}$,
so there is a constant $c$ so that $|\ell(u)|\leq c\|u\|_{H^{s}(M)}$ for all
$u\in H^{s}(M)$. Then $|\ell(T^{-s}f)|\leq c\|f\|_{L^{2}(M)}$ for $f\in
L^{2}(M)$ and consequently there is $w\in L^{2}(M)$ so that $\ell(u)=\langle
w\,|\,T^{s}u\rangle_{L^{2}(M)}$ for all $u\in H^{s}(M)$. Choose a sequence
$w_{n}\to w$ in $L^{2}(M)$ with $w_{n}\in C^{\infty}(M)$ and note that
$T^{s}w_{n}\in C^{\infty}(M)$ is a Cauchy sequence with respect to the
$H^{-s}(M)$ norm, converging to some $v\in H^{-s}(M)$ for which
$T^{-s}v=\lim_{n}w_{n}=w$. Thus $\ell(u)=\langle T^{-s}v\,|\,T^{s}u\rangle$
for all $u\in H^{s}(M)$ and $|\ell(u)|\leq\|v\|_{H^{-s}(M)}\|u\|_{H^{s}(M)}$.
Noting that

$\frac{\ell(T^{-s}w_{n})}{\|T^{-s}w_{n}\|_{H^{s}(M)}}=\frac{\langle w\mid
w_{n}\rangle_{L^{2}(M)}}{\|w_{n}\|_{L^{2}(M)}}\to\|w\|_{L^{2}(M)}=\|v\|_{H^{-s}(M)},$

we see that $\|\ell\|=\|v\|_{H^{-s}(M)}$, and we have established that the
space $H^{-s}(M)$ is anti-isomorphic to $H^{s}(M)^{\prime}$ with the operator
norm topology (i.e., the strong dual) with respect to the sesquilinear pairing
$\langle v,u\rangle=\langle T^{-s}v\,|\,T^{s}u\rangle$, $v\in H^{-s}(M)$,
$u\in H^{s}(M)$. To be precise: every choice of Riemannian metric on $M$
induces an anti-isomorphism of this type, and the pairing depends on the
choice made.

Embedding theorems. The relationship between $C^{k}$ and Sobolev spaces is
given as follows. In one direction, the formula $-\triangle=\delta d$ (on
$0$-forms) gives the estimates

$0\leq{\big{\langle}f\,\big{|}\,(-\triangle)^{2r}f\big{\rangle}}=\|(-\triangle)^{r}f\|^{2}_{L^{2}(M)}\leq
C\|f\|_{M,2r}^{2}$

and

$0\leq{\big{\langle}f\,\big{|}\,(-\triangle)^{2r+1}f\big{\rangle}}=\|d(-\triangle)^{r}f\|^{2}_{\Lambda^{1}(M)}\leq
C\|f\|_{M,2r+1}^{2},$

where $\Lambda^{1}(M)$ is the Hilbert space of square-integrable $1$-forms
(with respect to the auxiliary Riemannian metric) and $f\in C^{\infty}(M)$.
From this we find

$\|f\|_{H^{k}(M)}^{2}={\big{\langle}f\,\big{|}\,(-\triangle+I)^{k}f\big{\rangle}}=\sum_{j=0}^{k}\binom{k}{j}{\big{\langle}f\,\big{|}\,(-\triangle)^{j}f\big{\rangle}}\leq
C\|f\|_{M,k}^{2}$

for all $k\in{\mathbb{N}}_{0}$. Thus $C^{k}(M)$ is continuously embedded in
$H^{s}(M)$ for all $s\geq k$.

On the other hand, now let $n$ be the maximum dimension of any component of
$M$. For $\operatorname{Re}s>n/2$, the integral kernel $Z_{s}(p,q)$ of the
operator $(-\triangle+I)^{-s}=T^{-2s}$ on $L^{2}(M)$ is continuous, $Z_{s}\in
C(M\times M)$ and can also be written in terms of the spectral decomposition
of $(-\triangle+I)$ as

$Z_{s}(p,q)=\sum_{r}\frac{e_{r}(p)\overline{e_{r}(q)}}{\lambda_{r}^{s}},$

where $e_{r}\in L^{2}(M)$ are a basis of smooth orthonormal eigenfunctions for
$-\triangle+I$ with corresponding eigenvalues $\lambda_{r}$ – see [27]. In
particular,

$v_{p}=\sum_{r}\frac{\overline{e_{r}(p)}e_{r}}{\lambda_{r}^{s/2}}$

belongs to $L^{2}(M)$ with $\|v_{p}\|^{2}=Z_{s}(p,p)$ by the Pythagoras
theorem. For $\operatorname{Re}s>n/2$, it follows that $\|e_{r}\|_{\infty}\leq
C_{s}\big{|}\lambda_{r}^{s/2}\big{|}$, where $C_{s}=\sup_{p\in
M}|Z_{s}(p,p)|^{1/2}$. Moreover, if $f\in C^{\infty}(M)$,

$|\langle e_{r}\mid
f\rangle|=\lambda_{r}^{-k}\big{|}{\big{\langle}e_{r}\,\big{|}\,T^{k}f\big{\rangle}}\big{|}\leq\lambda^{-k}_{r}\|f\|_{H^{k}(M)},\qquad
k\in{\mathbb{N}},$

so ${\big{\langle}e_{r}\,\big{|}\,f\big{\rangle}}$ decays faster than any
inverse power of $\lambda_{r}$ and
${\big{\langle}v_{p}\,\big{|}\,T^{s}f\big{\rangle}}=\sum_{r}e_{r}(p){\big{\langle}e_{r}\,\big{|}\,f\big{\rangle}}=f(p)$
for all (not merely almost all) $p\in M$. Consequently,

$\|f\|_{\infty}=\sup_{p\in M}|\langle v_{p}\mid f\rangle|\leq
C_{s}\|f\|_{H^{s}(M)},\qquad f\in C^{\infty}(M).$

By density of $C^{\infty}(M)$ in $H^{s}(M)$, it follows that there is a
continuous embedding $H^{s}(M)\to C(M)$ if $s>n/2$; considering
$\|Pf\|_{\infty}$ in a similar way for differential operators $P$, one sees
that $H^{s}(M)$ is continuously embedded in $C^{k}(M)$ for all $s>k+n/2$. It
follows that $\cap_{s\in{\mathbb{R}}}H^{s}(M)=C^{\infty}(M)$.

If $u\in{\mathscr{D}}^{\prime}(M)=\Gamma_{0}^{\infty}({\Omega_{1}})^{\prime}$
is a distribution then, owing to compactness of $M$, there is a partial
differential operator $P$ so that $|u(\nu f)|\leq\|Pf\|_{\infty}\leq
C_{s}\|Pf\|_{H^{s}(M)}$ for any $s>n/2$ and all $f\in C^{\infty}(M)$. Thus,
with $t=s+\operatorname{ord}(P)$, $u(\nu
f)=\big{\langle}\overline{f},w\big{\rangle}={\big{\langle}T^{t}\overline{f}\,\big{|}\,T^{-t}w\big{\rangle}}_{L^{2}(M)}$
for some $w\in H^{-t}(M)$, which is evidently compatible with the embedding of
$H^{-t}(M)$ in ${\mathscr{D}}^{\prime}(M)$ described earlier. In this sense,

${\mathscr{D}}^{\prime}(M)=\bigcup_{s\in{\mathbb{R}}}H^{s}(M).$

####  A.2.2 General manifolds

For a general (not-necessarily compact) smooth manifold $M$ with at most
finitely many components, we proceed as follows. If $K\subset M$ is compact
and topologically regular,333Note that if $K$ is any compact subset, then the
closure of the interior of $K$ is compact and topologically regular. we may
diffeomorphically identify $K$ with a compact subset $\hat{K}$ of a compact
manifold $N$ (for example, by ‘doubling’ a compact set that contains $K$ in
its interior and has a smooth boundary [4]; for $K$ contained in a coordinate
chart one could take $N$ to be a torus). Then the Sobolev space $H^{s}_{K}(M)$
is defined as the completion of $C^{\infty}_{K}(M)$ with respect to the pull
back of (some choice of) the $H^{s}(N)$ norm. The space $H^{s}_{K}(M)$ can be
identified with a subspace of ${\mathscr{D}}^{\prime}_{K}(M)$, the
distributions with support contained in $K$. As a topological vector space, it
is independent of the choices used in its construction. However, by making
such a choice one may endow $H^{s}_{K}(M)$ with a norm, denoted
$\|\cdot\|_{H^{s}_{K}(M)}$, and indeed a compatible Hilbert space structure.
Estimates of the form $\|f\|_{H^{k}_{K}(M)}\leq C\|f\|_{K,k}$, $f\in
C^{\infty}_{K}(M)$, and hence continuity of the embedding $C^{k}_{K}\to
H^{k}_{K}$, carry over from the compact case for each fixed
$k\in{\mathbb{N}}_{0}$, as does compactness of the embedding $H^{s}_{K}(M)\to
H^{t}_{K}(M)$ for $s>t$ and the continuous embedding of $H^{s}_{K}(M)$ in
$C^{k}_{K}(M)$ for $s>k+n/2$, where again $n$ is the maximum dimension of any
component of $M$. Consequently one has
$\cap_{s\in{\mathbb{R}}}H^{s}_{K}(M)=C^{\infty}_{K}(M)$.

Next, we define (abbreviating ‘compact and topologically regular’ by c.t.r.)

$H_{0}^{s}(M)=\bigcup_{\text{c.t.r. $K\subset M$}}H^{s}_{K}(M),$

with the locally convex inductive limit topology. As $M$ admits countable
compact exhaustions (and consequently, countable c.t.r. exhaustions),
$H_{0}^{s}(M)$ may be realised as a countable strict inductive limit – i.e.,
it is a $LF$ space. A fact to be used later is that, by [28, Proposition 14.6]
(see also Proposition 4 in [9]) a subset $B$ of $H_{0}^{s}(M)$ is bounded if
and only if $B$ is a bounded subset of $H^{s}_{K}(M)$ for some compact $K$. It
is easily shown that

${\mathscr{E}}^{\prime}(M)=\bigcup_{s\in{\mathbb{R}}}H_{0}^{s}(M).$

Finally, the local Sobolev space $H^{s}_{\rm loc}(M)$ is defined as

$H^{s}_{\rm loc}(M)=\big{\\{}u\in{\mathscr{D}}^{\prime}(M)\colon\text{$\chi
u\in H^{s}_{0}(M)$ for all $\chi\in C_{0}^{\infty}({M})$}\big{\\}},$

and is equipped with the Fréchet topology induced by the seminorms
$\|\chi\cdot\|_{H^{s}_{K}(M)}$ as $K$ runs over compact topologically regular
subsets of $M$ and $\chi$ runs over $C^{\infty}_{K}(M)$. The inclusion
$H^{s}_{0}(M)\hookrightarrow H^{s}_{\rm loc}(M)$ is continuous for all $s$ and
indeed we have

$H^{s}_{0}(M)=H^{s}_{\rm loc}(M)\cap{\mathscr{E}}^{\prime}(M).$

Thus, if $M$ is compact, $H^{s}_{\rm loc}(M)$ and $H^{s}_{0}(M)$ coincide as
topological vector spaces. The construction above produces the same spaces as
the chart-based approach taken in [19].

The Sobolev embedding theorems already mentioned entail the existence of
continuous embeddings of $C^{k}(M)$ in $H^{k}_{\rm loc}(M)$ for all
$k\in{\mathbb{N}}_{0}$ and of $H^{s}_{\rm loc}(M)$ in $C^{k}(M)$ for all
$s>k+n/2$, where $n$ is the maximum dimension of any component of $M$.
Consequently, $\cap_{s\in{\mathbb{R}}}H^{s}_{\rm loc}(M)=C^{\infty}(M)$.

###  A.3 The topology of bounded convergence

A general reference for the following is Trèves [28, Chapter 32].

If $E$ and $F$ are Hausdorff locally convex topological spaces then the
topology of bounded convergence on ${\mathcal{L}}(E,F)$, the space of all
continuous linear maps $E\to F$, is defined by the neighbourhood base of zero,
consisting of sets

${\mathscr{U}}(B;V)=\\{T\in{\mathcal{L}}(E,F)\colon T(B)\subset V\\}$

as $B$ runs over the bounded subsets of $E$ and $V$ runs over any
neighbourhood base of zero in the topology of $F$. The notation
${\mathcal{L}}_{b}(E,F)$ denotes ${\mathcal{L}}(E,F)$ equipped with the
topology of bounded convergence. Thus a net $T_{\alpha}$ converges to $0$ in
${\mathcal{L}}_{b}(E,F)$ if and only if, for every bounded $B\subset E$ and
neighbourhood base set $V$, $T_{\alpha}$ eventually maps $B$ into $V$. This
topology makes ${\mathcal{L}}(E,F)$ Hausdorff and locally convex (inherited
from $F$ [28, p. 336]). We record some basic facts.

######  Lemma A.2.

Let $E$, $F$, $G$ be Hausdorff locally convex topological spaces. $(a)$ If
$T\in{\mathcal{L}}(E,F)$ and the net $S_{\alpha}\to S$ in
${\mathcal{L}}_{b}(F,G)$ then $S_{\alpha}T\to ST$ in ${\mathcal{L}}_{b}(E,G)$;
$(b)$ if the net $T_{\alpha}\to T$ in ${\mathcal{L}}_{b}(E,F)$ and
$S\in{\mathcal{L}}(F,G)$ then $ST_{\alpha}\to ST$ in ${\mathcal{L}}_{b}(E,G)$;
$(c)$ if $T_{\alpha}\to T$ in ${\mathcal{L}}_{b}(E,F)$ and
$\operatorname{Ran}T_{\alpha}\subset\hat{F}$, where $\hat{F}$ is a topological
subspace of $F$, then $T_{\alpha}\to T$ in ${\mathcal{L}}_{b}(E,\hat{F})$;
$(d)$ if $F$ is barrelled and $T_{n}\to T$ and $S_{n}\to S$ are convergent
_sequences_ in ${\mathcal{L}}_{b}(E,F)$ and ${\mathcal{L}}_{b}(F,G)$, then
$S_{n}T_{n}\to ST$ in ${\mathcal{L}}_{b}(E,G)$.

Note that Hilbert, Banach, Fréchet and LF spaces are all barrelled [28,
Chapter 33].

###### Proof.

$(a)$ It is enough to prove this in the case $S=0$; taking any bounded $B$ in
$E$ with $0\in B$, and any $0$-neighbourhood $V$ in $G$, note that $T(B)$ is
bounded in $F$ and deduce that $S_{\alpha}T(B)$ is eventually contained in
$V$. Thus $S_{\alpha}T\to 0$ in ${\mathcal{L}}_{b}(E,G)$. $(b)$ Without loss,
suppose $T=0$ and with $B$ and $V$ as before, we note that $S^{-1}(V)$ is a
$0$-neighbourhood in $F$ and again deduce that $ST_{\alpha}(B)$ is eventually
contained in $V$, so $ST_{\alpha}\to 0$.

$(c)$ Again, it is enough to treat $T=0$. Take any bounded $B\subset E$ and
$0$-neighbourhood $\hat{V}$ in $\hat{F}$. Then $\hat{V}=\hat{F}\cap V$ for an
$0$-neighbourhood $V$ in $F$ because $\hat{F}$ carries the subspace topology.
We know that, eventually, $T_{\alpha}(B)\subset V$, and as
$\operatorname{Ran}T_{\alpha}\subset\hat{F}$, we have, eventually, that
$T_{\alpha}(B)\subset V\cap\hat{F}=\hat{V}$. Thus $T_{\alpha}\to 0$ in
${\mathcal{L}}_{b}(E,\hat{F})$.

$(d)$ Suppose first that $T=0$ and let $0\in B$ be any bounded subset of $E$
and $V$ any $0$-neighbourhood in $G$. Then the $S_{n}$, together with $S$,
form a bounded subset in ${\mathcal{L}}_{b}(F,G)$ (which, we recall, is
Hausdorff). As $F$ is barrelled, they form an equicontinuous set of maps by
the Banach–Steinhaus theorem [28, Theorem 33.1]. Thus there exists a
$0$-neighbourhood $W$ in $F$ such that $S_{n}(W)\subset V$ for all $n$. As
$T_{n}(B)\subset W$ for all sufficiently large $n$, we have
$S_{n}T_{n}(B)\subset W$ for such $n$. Thus $S_{n}T_{n}\to 0$ in this case. If
$T\neq 0$, we now know that $S_{n}(T_{n}-T)\to 0$, and as $S_{n}T\to ST$ we
find $S_{n}T_{n}\to ST$ as required. ∎

Typical applications of Lemma A.2 $(a)$, $(b)$ are to show, for instance, that
a holomorphic function from a domain in ${\mathbb{C}}$ to
${\mathcal{L}}_{b}\big{(}H^{s}_{0}(M),H^{s}_{K}(M)\big{)}$ is also holomorphic
as a function to ${\mathcal{L}}_{b}\big{(}H^{s}_{K}\big{)}$ and
${\mathcal{L}}_{b}(H^{s}_{0}(M))$ due to continuous embedding of
$H_{K}^{s}(M)$ in $H^{s}_{0}(M)$. Part $(d)$ has the following use.

######  Corollary A.3.

Let $E$, $F$, $G$ be Hausdorff locally convex topological spaces, with $F$
being barrelled. If $\lambda\mapsto T(\lambda)$ and $\lambda\mapsto
S(\lambda)$ are functions from a domain in ${\mathbb{C}}$ to
${\mathcal{L}}(E,F)$ and ${\mathcal{L}}(F,G)$ respectively, then $(a)$ if $T$
and $S$ are both continuous in the topology of bounded convergence $($at
$\mu$$)$, so is $(ST)(\lambda)=S(\lambda)T(\lambda)$ $($at $\mu$$)$; $(b)$ if
$T$ and $S$ are differentiable at $\mu$ in the topology of bounded convergence
then so is $ST$, with a derivative given by the Leibniz rule
$(ST)^{\prime}(\mu)=S^{\prime}(\mu)T(\mu)+S(\mu)T^{\prime}(\mu)$.

###### Proof.

As ${\mathbb{C}}$ is first-countable, questions of continuity and
differentiability may be reduced to sequential considerations, and the result
follows using Lemma A.2 $(d)$ and the standard proof of the Leibniz formula. ∎

It is also useful to have some sufficient conditions for convergence in
${\mathcal{L}}_{b}(E,F)$ topology where $E$ and $F$ are Fréchet or countable
strict inductive limits thereof (LF spaces). If $F$ is Fréchet, then the
bounded subsets are precisely the subsets $B$ so that $\sup_{f\in
B}\rho_{j}(f)<\infty$ for every seminorm $\rho_{j}$ defining the topology of
$F$, while sets $V_{j,\epsilon}=\\{f\in F\colon\rho_{j}(f)<\epsilon\\}$, for
arbitrary $\epsilon>0$ and defining seminorm $\rho_{j}$, provide a basis of
neighbourhoods of zero. On the other hand, if $E$ is an LF space with defining
sequence $E_{n}$ of Fréchet spaces, then the bounded subsets $B$ of $E$
comprise precisely those subsets that are bounded subsets of some $E_{n}$ [28,
Proposition 14.6], while a neighbourhood base is provided by convex sets
$V\subset E$ so that each $V\cap E_{n}$ is an open neighbourhood of zero in
$E_{n}$. We recall that continuous linear maps between topological vector
spaces preserve boundedness [28, Proposition 14.2]. The following results are
used in the text.

######  Lemma A.4.

Suppose $T_{\alpha}\in{\mathcal{L}}_{b}(F)$ is a net of operators on a Fréchet
space $F$ with defining seminorms $\rho_{j}$. If, for each $j$, there exists
$k(j)$, such that for all $\epsilon>0$, it is eventually true that
$\rho_{j}(T_{\alpha}f)<\epsilon\rho_{k(j)}(f)$ for all $f\in F$, then
$T_{\alpha}\to 0$ in ${\mathcal{L}}_{b}(F)$.

###### Proof.

Given any bounded set $B$ and neighbourhood $V_{j,\epsilon}$, we set
$C=\sup_{f\in B}\rho_{k(j)}(f)$ and apply the given property to $\epsilon/C$
to find that, eventually,

$\rho_{j}(T_{\alpha}f)<\epsilon C^{-1}\rho_{k(j)}(f)\leq\epsilon,\qquad\forall
f\in B.$

We have shown that, eventually,
$T_{\alpha}\in{\mathscr{U}}(B;V_{j,\epsilon})$; as $j$ and $\epsilon$ were
arbitrary, $T_{\alpha}\to 0$ in ${\mathcal{L}}_{b}(F)$. ∎

######  Corollary A.5.

Consider a net $T_{\alpha}\in{\mathcal{L}}\big{(}C^{\infty}_{K}\big{)}$ such
that each $T_{\alpha}$ extends to an operator in ${\mathcal{L}}(H^{s}_{K})$
$($which we also denote $T_{\alpha}$$)$ for all $s\geq s_{*}$. Suppose also
that $T_{\alpha}\to 0$ in every ${\mathcal{L}}(H^{s}_{K})$, $s\geq s_{*}$.
Then $T_{\alpha}\to 0$ in ${\mathcal{L}}_{b}\big{(}C^{\infty}_{K}\big{)}$.

###### Proof.

Any $C^{\infty}_{K}$ seminorm is bounded by an $H^{s}_{K}$ norm and vice
versa, so, for any $C^{\infty}_{K}$-seminorm $\|\cdot\|_{K,j}$ we may estimate

$\|T_{\alpha}f\|_{K,j}\leq c\|T_{\alpha}f\|_{H^{s(j)}_{K}}\leq
c\|T_{\alpha}\|_{{\mathcal{L}}(H^{s(j)}_{K})}\|f\|_{H^{s(j)}_{K}}\leq
c^{\prime}\|T_{\alpha}\|_{{\mathcal{L}}(H^{s(j)}_{K})}\|f\|_{K,k(s(j))}$

for suitable choices of $s(j)$ and $k(s(j))$, uniformly in $f\in
C^{\infty}_{K}$ and $\alpha$. As $T_{\alpha}\to 0$ in
${\mathcal{L}}\big{(}H^{s(j)}_{K}\big{)}$, for any $\epsilon>0$ it is
eventually true that $\|T_{\alpha}f\|_{K,j}\leq\epsilon\|f\|_{K,k(s(j))}$ for
all $f\in C^{\infty}_{K}$. Hence $T_{\alpha}\to 0$ in
${\mathcal{L}}_{b}\big{(}C^{\infty}_{K}\big{)}$ by Lemma A.4. ∎

######  Lemma A.6.

Suppose $T_{\alpha}$ is a net of operators on the LF space
$E=\bigcup_{n\in{\mathbb{N}}}E_{n}$. If for some fixed $n$, one has
$\operatorname{Ran}(T_{\alpha})\subset E_{n}$ and $T_{\alpha}|_{E_{m}}\to 0$
in ${\mathcal{L}}_{b}(E_{m},E_{n})$ for all $m$, then $T_{\alpha}\to 0$ in
both ${\mathcal{L}}_{b}(E,E_{n})$ and ${\mathcal{L}}_{b}(E)$.

###### Proof.

Consider any neighbourhood $V$ of zero in the standard neighbourhood base of
$E_{n}$ and any bounded set $B\subset E$, necessarily obeying $B\subset E_{m}$
for some $m$. Then $T_{\alpha}$ eventually maps $B$ into $V$; as $B$ and $V$
were arbitrary, $T_{\alpha}\to 0$ in ${\mathcal{L}}_{b}(E,E_{n})$. Post-
composing with the continuous embedding of $E_{n}$ in $E$, $T_{\alpha}\to 0$
in ${\mathcal{L}}_{b}(E)$ as well. ∎

### Acknowledgements

It is a pleasure to thank Rainer Verch for useful discussions at various
stages of this work and Maximilian Ruep for a careful reading and comments on
a draft of the manuscript. I particularly thank Christian Bär for asking the
question that prompted Theorem 3.5 and also Lashi Bandara and other
participants of the conference ‘Global analysis on manifolds’ held in Bär’s
honour (Freiburg, September 2022) for useful remarks and conversations. I also
thank the referees for valuable suggestions.

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