# Realisations of elliptic operators on compact manifolds with boundary

Lashi Bandara  ,  Magnus Goffeng  and  Hemanth Saratchandran  Lashi Bandara,
Institut für Mathematik, Universität Potsdam, D-14476, Potsdam OT Golm,
Germany http://www.math.uni-potsdam.de/ bandara<EMAIL_ADDRESS>Magnus Goffeng, Centre for Mathematical Sciences, University of Lund, Box 118,
221 00 LUND, Sweden
https://www.lunduniversity.lu.se/lucat/user/e604494124f99d9bb6048e890306f7a4
<EMAIL_ADDRESS>Hemanth Saratchandran The University of Adelaide
Adelaide, South Australia, 5005, Australia
https://researchers.adelaide.edu.au/profile/hemanth.saratchandran
<EMAIL_ADDRESS>
###### Abstract.

This paper investigates realisations of elliptic differential operators of
general order on manifolds with boundary following the approach of Bär-
Ballmann to first order elliptic operators. The space of possible boundary
values of elements in the maximal domain is described as a Hilbert space
densely sandwiched between two mixed order Sobolev spaces. The description
uses Calderón projectors which, in the first order case, is equivalent to
results of Bär-Bandara using spectral projectors of an adapted boundary
operator. Boundary conditions that induce Fredholm as well as regular
realisations, and those that admit higher order regularity, are characterised.
In addition, results concerning spectral theory, homotopy invariance of the
Fredholm index, and well-posedness for higher order elliptic boundary value
problems are proven.

###### Key words and phrases:

Elliptic differential operator, Fredholm boundary conditions, boundary
regularity, Calderón projector

###### 2010 Mathematics Subject Classification:

35J58, 35J56, 58J05, 58J32

###### Contents

  1. 1 Introduction
  2. 2 Elliptic differential operators and boundary conditions
  3. 3 Calderón projectors and the Douglis-Nirenberg calculus
  4. 4 The Cauchy data space of an elliptic differential operator
  5. 5 Characterisations of regularity via graphical decompositions
  6. 6 Higher order boundary regularity
  7. 7 Differential operators with positive principal symbols and their Weyl laws
  8. 8 Rigidity results for the index of elliptic differential operators
  9. 9 Symbol computations with Calderón projectors in the first order case
  10. A Lemmas on dimensions and subspaces
  11. B Bär-Ballmann’s approach from an abstract perspective

##  1\. Introduction

Elliptic boundary value problems have a long history emerging from classical
problems in physics and engineering. Their mathematical description dates back
to classical works such as [grisvard, horIII, lionsmagenes, grubb68, grubb74,
grubb77, schechter59, vishik]. Elliptic operators – their index theory and
spectral theory – is broadly used in mathematics connecting seemingly disjoint
areas of mathematics, for instance through noncommutative geometry and
geometric analysis. The spectral theory of elliptic boundary value problems
has primarily been focused on Laplace type operators (for an overview see
[ivrii16]) or Dirac type operators (see for instance [bruninglesch99,
bruninglesch01, bosswojc, Grubb03]), but has also been studied in larger
generality (e.g. in [geygru74, gerdsgreenbook, grubb74, grubb77a, grubb84]).
The index theory of elliptic boundary value problems is also a well studied
area. For instance, the index theory of Dirac operators with the spectral APS
boundary condition was computed by Atiyah-Patodi-Singer [APS], and has since
been well studied, see for instance [G99, grubb92, melroseAPS, bruninglesch99,
bosswojc]. The index theory for boundary value problems in the Boutet de
Monvel calculus was computed by [boutetdemonvel] relating to ideas of
Agranovich-Dynin. This formula covers the case of local elliptic boundary
conditions. Boutet de Monvel’s results have been extensively studied, see
[elmaretal, fedosovindex, fedoind, rempelschulze], and were generalised to an
extended Boutet de Monvel calculus by Schulze-Seiler [schulzeseiler].

In this paper we take a new approach to higher order elliptic operators on
manifolds with boundary. The approach is based on work of Bär-Ballmann [BB].
The paper [BB] consists of a coherent overview of first order boundary value
problems (admitting a self-adjoint adapted boundary operator) based on an
analytic description of the Cauchy data space as well as a novel graphical
decomposition of regular realisations of first order operators. The results
were later extended to general first order problems by the first listed author
and Bär in [BBan]. Although this approach was first fully utilised and made
explicit in [BB], the ideas have been around since the ’60s and was implicitly
contained in a paper by Seeley [seeley65, top of page 782]. They were visible
already in Agranovich-Dynin’s classical works on index theory for elliptic
boundary value problems of general order. Similar approaches predating Bär-
Ballmann’s have been studied in the abstract in [bossfury], been applied to
Dirac-Schrödinger operators in [ballbruncarr] as well as having been used in
the study of Maxwell’s equations on Lipschitz domains [HR2019, BuCoSh].
Related ideas can also be seen in for instance [APS, bruninglesch01, grubb68,
grubb77, grubb92, G99]. Since its appearance, the coherence of Bär-Ballmann’s
work [BB] has paved the way for numerous applications [MR4011805, MR4000837,
MR3981455, MR3908762, MR3850258].

The aim of this paper is to present a perspective similar to [BB], providing
an overview of the theory for general order elliptic boundary value problems
from the vantage point of the Cauchy data space. As such, the main novelty
lies in the framework and the methods. Indeed, several of the results in this
paper are, as stand alone results, at best mild generalisations of known
results and are unlikely to surprise experts in the field of boundary value
problems. A key feature in several of the results is that they can treat all
(or at least the regular extensions with the occasional restriction to the
pseudo-local case) closed extensions of an elliptic differential operator on
an equal footing with the special classes of boundary conditions studied
classically. We have attempted to the best of our abilities to indicate where
the results we are generalising can be found in the abundance of literature on
boundary value problems.

The approach we take lies close in spirit to the ideas of noncommutative
geometry [connesbook] whose methods have proven their worth in index theory,
relating to recent work on boundaries in noncommutative geometry [FGMR]. It
takes a more abstract perspective than the semiclassical methods ordinarily
deployed to study boundary value problems, e.g. in [grisvard, horIII,
lionsmagenes, grubb77, schechter59, vishik], and lies closer to methods for
order one elliptic operators seen in [G99, grubb92, bruninglesch99, bosswojc,
bossfury, BB, BBan] rather than the abstract methods of, for instance,
[grubb68, grubb74, malamud, behrndtetal]. The level of abstraction makes the
method feasible for further development of index theory and applications to a
larger class of problems than elliptic differential operators on manifolds
with boundary. We anticipate this includes elliptic operators on singular
manifolds, hypoelliptic operators (cf. the subelliptic boundary conditions in
[epsteinsub]) or geometric operators in Lorentzian geometry (cf. [MR4011805]).
These facts raises optimism for demystifying the general index formulas
appearing in the (extended) Boutet de Monvel calculus relying on abstract
homotopies [boutetdemonvel, rempelschulze], non-explicit inverses to
isomorphisms in $K$-theory [elmaretal] or Chern characters of operator valued
symbols [fedoind, fedosovindex].

The method of Bär-Ballmann for first order operators can be summarised as
describing boundary value problems by means of the Cauchy data space, i.e. the
range of the corresponding trace map onto a mixed Sobolev space on the
boundary. For higher order operators we apply the same ideas to the full trace
map (taking into account all boundary traces up to the order of the operator)
into a mixed Sobolev space on the boundary. The range of the full trace map –
the Cauchy data space – characterises the realisation defining the boundary
value problem when topologising it so that the full trace map is a quotient.
The analytic details involved in this procedure produces precise information
about regularity and Fredholm properties of the realisations in terms of
properties of the boundary condition relative to the range of the full trace
mapping on the maximal domain of the elliptic operator. The general theme in
this framework is to reduce properties of realisations, and their proofs, to
neat functional analysis arguments and Fredholm theory.

The description of the possible realisations of an elliptic operator as
subspaces of a Hilbert space of functions on the boundary relies on a precise
description of the Cauchy data space. Abstractly, the Cauchy data space is
isomorphic to the quotient of the maximal domain by the minimal domain. By
classical results of Lions-Magenes [lionsmagenes63, lionsmagenes] the Cauchy
data space is a concrete subspace of a mixed order Sobolev space on the
boundary. For an elliptic first order differential operator this is described
in detail in [BBan] using an adapted boundary operator. To describe the Cauchy
data space of a general order elliptic differential operator, we employ the
Calderón projection of Seeley [seeley65]. Although our setup is based on the
work of Bär-Ballmann, similar ideas date back much further as discussed above.

We emphasise that this paper, despite allowing for elliptic differential
operators of general order, by no means supersedes [BB, BBan]. A significant
novelty distinguishing [BBan] from this paper is that [BBan] links analysis on
the Cauchy data space to the $\mathrm{H}^{\infty}$-functional calculus, which
in [BBan] forms the key to obtaining the estimates required to topologise the
Cauchy data space. In contrast, in this paper we use Seeley’s work on Calderón
projectors to analyse the Cauchy data space.

###  1.1. Main results and overview of paper

Let $M$ be a compact manifold with boundary $\Sigma:=\partial M$. We fix a
smooth measure $\mu$ on $M$. In our convention, $\Sigma\subset M$ and the
interior of $M$ is denoted by $\mathring{M}=M\setminus\Sigma$. We also choose
a smooth interior pointing vectorfield $\vec{T}$ transversal to
$\Sigma=\partial M$. The smooth measure on $\Sigma$ induced by $\mu$ and
$\vec{T}$ will be denoted by $\nu$.

In this paper, we are mainly concerned with elliptic differential operators
$D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ acting between hermitian
vector bundles $(E,h^{E})\to M$ and $(F,h^{F})\to M$. The hermitian metric
will be implicit in the notations. We let $m$ denote the order of $D$. The
maximal realisation $D_{\rm max}$ of $D$ on ${\rm L}^{2}$ is given by
$\mathrm{dom}(D_{\max}):=\\{f\in{\rm L}^{2}(M;E):Df\in{\rm L}^{2}(M;F)\\}$
where $D$ acts in a distributional sense on ${\rm L}^{2}(M;E)$ as a
consequence of the presence of a unique formal adjoint $D^{\dagger}:{\rm
C}^{\infty}(M;F)\to{\rm C}^{\infty}(M;E)$. The minimal realisation $D_{\rm
min}$ is defined from the domain obtained from closing ${\rm C}^{\infty}_{\rm
c}(\mathring{M};E)$ in the graph norm of $D$; it is readily seen that
$\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm 0}(M;E)$ (cf. Proposition 2.1,
on page 2.1). Many of the general results we prove only rely on abstract
properties of the minimal and maximal realisations using the machinery of
Appendix B (see page B). For $s\in\mathbb{R}$, we use the notation

${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m}):=\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm s-j}(\Sigma;E).$

As described in Subsection 3.1, the space
${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})$ is easiest thought of as a
graded Sobolev space with respect to a certain grading on
$E\otimes\mathbb{C}^{m}$.

A (closed) realisation of $D$ is a (closed) extension $D_{e}$ of $D_{\rm min}$
such that $D_{e}\subseteq D_{\rm max}$. In other words, a realisation of $D$
is an extension of $D$ acting on ${\rm H}^{\rm m}_{\rm 0}(M;E)$ to a domain on
${\rm L}^{2}(M;E)$ where it acts in the ordinary distributional sense.

The following theorem summarises how the Bär-Ballmann machinery applies in the
setting of higher order elliptic differential operators on manifolds with
boundary.

######  Theorem 1.

Let $D$ be an elliptic differential operator of order $m>0$ as in the
preceding paragraph.

  1. (i)

The full trace mapping $\gamma:{\rm H}^{\rm
m}(M;E)\to{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
$u\mapsto(\partial_{x_{n}}^{j}u|_{\Sigma})_{j=0}^{m-1}$ extends to a
continuous trace mapping $\gamma:\mathrm{dom}(D_{\rm
max})\to{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ with
kernel $\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm 0}(M;E)$ and range
being the _Cauchy data space_

$\check{\mathrm{H}}(D)=P_{\mathcal{C}}{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\bigoplus(1-P_{\mathcal{C}}){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

where $P_{\mathcal{C}}$ is a Calderón projection (for more details on
$P_{\mathcal{C}}$, see Section 3). Equipping $\check{\mathrm{H}}(D)$ with the
Hilbert space topology making the projectors onto
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
respectively, into partial isometries, the canonical isomorphism
$\check{\mathrm{H}}(D)\cong\mathrm{dom}(D_{\rm max})/\mathrm{dom}(D_{\rm
min})$ is a Banach space isomorphism.

  2. (ii)

There is a one-to-one correspondence between (closed) realisations of $D$ and
(closed) subspaces $B\subseteq\check{\mathrm{H}}(D)$ as follows. A (closed)
subspace $B\subseteq\check{\mathrm{H}}(D)$ uniquely determines a (closed)
realisation $D_{\rm B}$ of $D$ defined by

$\mathrm{dom}(D_{\rm B})=\\{f\in\mathrm{dom}(D_{\rm max}):\gamma(f)\in B\\},$

and any (closed) realisation $\hat{D}$ uniquely determines a (closed) subspace
$B:=\gamma\mathrm{dom}(\hat{D})$ with $D_{\rm B}=\hat{D}$. Moreover,

$(D_{\rm B})^{*}=D^{\dagger}_{{\rm B}^{*}},$

where $B^{*}$ is the adjoint boundary condition (for more details see
Subsection 4.3, starting on page 4.3, and Appendix B.2, starting on page B.2).

  3. (iii)

A realisation $D_{\rm B}$ of $D$ is semi-regular, that is we have the domain
inclusion

$\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm m}(M;E),$

if and only if
$B\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$. A
realisation $D_{\rm B}$ is regular (i.e. $D_{\rm B}$ and $(D_{\rm B})^{*}$ are
semi-regular) if and only if
$B\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and
$B^{*}\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
where $B^{*}$ is the adjoint boundary condition.

The reader can find this theorem spread out in the bulk of the paper as
follows. Part i can be found in Theorem 3.18 (on page 3.18). Part ii is proven
in much larger generality in Appendix B, more precisely in Proposition B.5 on
page B.5. Part iii is proven in Proposition 2.5, see page 2.5. The precise
definition of regularity can be found in Definition 2.3 (page 2.3). In [BB,
BBan], regular boundary conditions are called elliptic, we discuss our choice
of terminology further in Remark 2.4 (page 2.4). We remark that Theorem 1
concerns local statements at the boundary and therefore111Using for instance
Lemma 4.1 on page 4.1 or Proposition 4.3 on page 4.3 extends to the case that
$M$ is noncompact with compact boundary when $D$ is complete.

Based on Theorem 1, a boundary condition for $D$ is defined to be a closed
subspace $B\subseteq\check{\mathrm{H}}(D)$ and we write $D_{\rm B}$ for the
associated realisation. We say that a boundary condition $B$ is (semi-)
regular if $D_{\rm B}$ is (semi-) regular.

Theorem 1 has direct consequences to the well-posedness of PDEs. If $D$ is an
elliptic differential operator of order $m>0$ with a semi-regular boundary
condition $B$ such that $\ker(D_{\rm B})=0$, then the partial differential
equation

$Du=f,\qquad\gamma(u)\in B$

is well-posed for $f\in\mathrm{ran}(D_{\rm B})=\ker(D_{{\rm
B}^{*}}^{\dagger})^{\perp}$. For more details, see Proposition 2.20 on page
2.20. In particular, if $D$ is a Dirac type operator (so $m=1$), the partial
differential equation

$Du=f,\qquad\gamma(u)=0$

is well-posed for $f\in\mathrm{ran}(D_{\rm min})$. See more in Corollary 2.22
on page 2.22.

The previous theorem has the following consequence on the spectral theory of
the maximal and minimal realisations of an elliptic operator. We include this
result because we have noticed some confusion surrounding it in the community.
The result is known to experts in the field (cf. the discussion on
[grubbdistop, page 60-61]).

######  Theorem 2.

Let $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;E)$ be an elliptic
differential operator of order $m>0$ acting between sections on a Hermitian
vector bundle $E\to M$ over a compact manifold with boundary with $\dim(M)>1$.
The spectrum of $D_{\rm max}$ on ${\rm L}^{2}(M;E)$ is

$\mathrm{spec}(D_{\max})=\mathrm{spec}_{\mathrm{pt}}(D_{\max})=\mathbb{C}.$

That is, the spectrum of $D_{\max}$ is purely discrete and each generalised
eigenspace is of infinite dimension. Moreover, the spectrum of $D_{\rm min}$
on ${\rm L}^{2}(M;E)$ is

$\mathrm{spec}(D_{\min})=\mathbb{C}.$

If $m=1$ and $D$ is a Dirac type operator then
$\mathrm{spec}(D_{\min})=\mathrm{spec}_{\mathrm{res}}(D_{\min})=\mathbb{C}$;
i.e., it consists solely of residual spectrum.

The reader can find the first part of this theorem on general order operators
as Proposition 3.12 (see page 3.12) in the bulk of the text. The statement
concerning $\mathrm{spec}(D_{\min})$ for Dirac-type operators can be found in
Corollary 3.14 (see page 3.14). The property distinguishing Dirac type
operators from general elliptic operators is the so-called _unique
continuation property_ which undermines the existence of eigenvalues for
$D_{\rm min}$. See the discussion in Remark 3.13 on page 3.13.

We now turn to some results on the index theory of realisations of elliptic
differential operators. We say that a boundary condition
$B\subseteq\check{\mathrm{H}}(D)$ is Fredholm if the associated realisation
$D_{\rm B}$ is Fredholm. We also introduce the notation
$\mathcal{C}_{D}\subseteq\check{\mathrm{H}}(D)$ for the Hardy space – the
image of $\ker(D_{\rm max})$ under the full trace mapping. Note that
$\mathcal{C}_{D}=P_{\mathcal{C}}\check{\mathrm{H}}(D)=P_{\mathcal{C}}{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
by Theorem 1.

######  Theorem 3.

Let $D$ be an elliptic differential operator of order $m>0$ as above. A
boundary condition $B\subseteq\check{\mathrm{H}}(D)$ is Fredholm if and only
if $(B,\mathcal{C}_{D})$ is a Fredholm pair in $\check{\mathrm{H}}(D)$. In
this case, it holds that

$\operatorname{ind}(D_{\rm
B})=\operatorname{ind}(B,\mathcal{C}_{D})+\dim\ker(D_{\min})-\dim\ker(D_{\min}^{\dagger}).$

Moreover, $\operatorname{ind}(D_{\rm B})$ is a homotopy invariant of $(D,B)$
in the sense of Theorem 8.1 on page 8.1 (see also Theorem 8.5 on page 8.5).

The characterisation of Fredholm boundary conditions and the index formula is
proven by abstract principles and can be found in Theorem 2.12 (see page 2.12)
in the body of the text. Results similar to Theorem 3 for operators of order
$m=1$ can be found in [bossfury, bosswojc].

######  Theorem 4.

Let $D$ be an elliptic differential operator of order $m>0$. Assume that $P$
is a projection on
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and set
$A:=P_{\mathcal{C}}-(1-P)$. Consider the following statements:

  1. (1)

The operator $A$ is Fredholm on
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

  2. (2)

The operator $P$ extends by continuity to $\check{\mathrm{H}}(D)$ and
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$, and $A$
defines a Fredholm operator on $\check{\mathrm{H}}(D)$.

  3. (3)

The operator $P$ extends by continuity to
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and $A$
defines a Fredholm operator on
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

The following holds:

  1. (i)

If 1 and 2 holds, then $P$ is _boundary decomposing_ (see Definition 3.26 on
page 3.26) and in particular

$\|u\|_{\check{\mathrm{H}}(D)}\simeq\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|Pu\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$

  2. (ii)

If 1 and 3 holds, then the space

$B_{P}:=(1-P){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

defines a regular boundary condition for $D$.

Moreover, if $P$ is a zeroth order pseudodifferential projection in the
Douglis-Nirenberg calculus (see more details in Subsection 3.1), then the
following are equivalent:

  1. a)

$P$ is Shapiro-Lopatinskii elliptic with respect to $D$ (in the sense of
Definition 4.16).

  2. b)

The operator $P_{\mathcal{C}}-(1-P)\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is elliptic in the Douglis-Nirenberg
calculus.

  3. c)

The space

$B_{P}:=(1-P){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

is a regular boundary condition for $D$, so in particular, the realisation
$D_{\rm B}$ defined from

$\mathrm{dom}(D_{\rm B}):=\\{u\in\mathrm{dom}(D_{\rm max}):P\gamma
u=0\\}=\\{u\in{\rm H}^{\rm m}(M;E):P\gamma u=0\\},$

is regular.

  4. d)

The space

$B_{P}:=(1-P){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

is a Fredholm boundary condition for $D$, so in particular, the realisation
$D_{\rm B}$ defined from

$\mathrm{dom}(D_{\rm B}):=\\{u\in\mathrm{dom}(D_{\rm max}):P\gamma u=0\\},$

is a Fredholm operator.

If any of the equivalent conditions a)-d) holds, and moreover
$[P,P_{\mathcal{C}}]$ is order $-m$ in the Douglis-Nirenberg calculus, then
$P$ is also boundary decomposing.

The reader can find item i) stated as Theorem 3.30 (see page 3.30) and item
ii) stated as Theorem 4.14 (see page 4.14) in the body of the text. The
equivalence of item a) and b) is found in Proposition 4.18 (see page 4.18),
and the equivalence of item b), c) and d) is found in Theorem 4.15 (see page
4.15). The final conclusion of the theorem appears as Corollary 3.32 (see page
3.32).

######  Remark 5.

The reader should be wary of the fact that the structure of the Cauchy data
space $\check{\mathrm{H}}(D)$ is quite different from that of a Sobolev space.
We give three instances of how this manifests:

  * •

If $P$ is a zeroth order pseudodifferential projection in the Douglis-
Nirenberg calculus and $P_{\mathcal{C}}-(1-P)$ is elliptic then
$B_{P}=(1-P){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ is
a regular boundary condition but if $(1-P_{\mathcal{C}})PP_{\mathcal{C}}$ is
not of order $-m$ (in the Douglis-Nirenberg calculus) then $P$ fails to be
boundary decomposing and even fails to act boundedly on
$\check{\mathrm{H}}(D)$. This follows from Lemma 3.31.

  * •

The obvious projectors on the Cauchy data space defining Dirichlet or Neumann
conditions for a Laplacian are not bounded operators on the Cauchy data space
by Example 3.34, but a more refined projection (projecting along the Hardy
space) produces a bounded projector as in Subsection 6.3. The
pseudodifferential operators in the Douglis-Nirenberg calculus that act
boundedly on the Cauchy data space are characterised in Lemma 3.31.

  * •

We give an example in Section 9 (see page 9) of a formally self-adjoint first
order elliptic operator on the unit disc, with associated adapted boundary
operator $A$ such that the classical pseudodifferential operator
$P_{\mathcal{C}}-\chi^{+}(A)$ on the boundary (i.e. the unit circle) is of
order $-1$ but does not act compactly on $\check{\mathrm{H}}(D)$. Therefore,
the contrast between the approach in this paper to that in [BBan] (where
spectral projectors $\chi^{\pm}(A)$ topologise the Cauchy data space) goes
beyond compact perturbations even in the first order case. The image of
$\chi^{+}(A)$ differs substantially from the image of $P_{\mathcal{C}}$ – the
Hardy space – as seen in Proposition 9.12 containing an example where the two
images’ intersection is a finite-dimensional space of smooth functions.

Much of the work in [BBan] concerning regular boundary conditions relied on
graphical decompositions. The following theorem extends [BBan, Theorem 2.9] to
elliptic differential operators of any order $>0$. The theorem can be found as
Theorem 5.6 (see page 5.6) in the body of the text.

######  Theorem 6.

Let $D$ be an elliptic differential operator of order $m>0$, $\mathcal{P}_{+}$
a boundary decomposing projection (see Definition 3.26 on page 3.26), and $B$
a boundary condition for $D$. The following are equivalent:

  1. (i)

$B\subset\check{\mathrm{H}}(D)$ is an regular boundary condition,

  2. (ii)

$B$ is $\mathcal{P}_{+}$ graphically decomposable (see Definition 5.3 on page
5.3),

  3. (iii)

$B$ is $\mathcal{P}_{+}$ Fredholm decomposable in
${\mathbb{H}}^{{m-\frac{1}{2}}}$ (see Definition 5.4 on page 5.4),

  4. (iv)

$B$ is $\mathcal{P}_{+}$ Fredholm decomposable in $\check{\mathrm{H}}$ (see
Definition 5.5 on page 5.5).

The ideas of Theorem 1 can also be applied to study higher boundary
regularity. Inspired by [BBan], we say that a boundary condition
$B\subseteq\check{\mathrm{H}}(D)$ is $s$-semiregular, for $s\geq 0$, if
whenever $\xi\in B$ satisfies that
$(1-P_{\mathcal{C}})\xi\in\bigoplus_{j=0}^{m-1}{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
then
$\xi\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$. For
$s=0$, $0$-semiregularity is equivalent to semi-regularity. In [BBan], a
notion of $s$-semiregular boundary conditions were introduced to prove higher
boundary regularity for operators of order $m=1$ that implies our notion of
$s-{\frac{1}{2}}$-semiregularity for $s>{\frac{1}{2}}$. The following theorem
can be found as Theorem 6.6 (see page 6.6) in the body of the text.

######  Theorem 7.

Let $D$ be an elliptic operator of order $m>0$, $s\geq 0$ and $B$ a boundary
condition. Then $B$ is $s$-semiregular if and only if $D_{B}$ is
$s$-semiregular, i.e. that whenever $u\in\mathrm{dom}(D_{\rm B})$ satisfies
$D_{\rm B}u\in{\rm H}^{\rm s}(M;F)$ it holds that $u\in{\rm H}^{\rm
s+m}(M;E)$.

Theorem 7 generalises results from [BB, BBan] from first order to general
order. The method of proof can be localised to the boundary, so in the case
that $M$ is noncompact with compact boundary and $D$ is complete, it holds
that $B$ is $s$-semiregular if and only if whenever $u\in\mathrm{dom}(D_{\rm
B})$ satisfies that $D_{\rm B}u\in{\rm H}^{\rm s}_{\rm\rm loc}(M;F)$ it holds
that $u\in{\rm H}^{\rm s+m}_{\rm\rm loc}(M;E)$ (where we use the notation
${\rm H}^{\rm s}_{\rm\rm loc}$ from [BB, BBan]).

######  Theorem 8.

Let $D$ be a formally self-adjoint elliptic operator of order $m>0$ acting on
a vector bundle $E\to M$. Assume that $D$ has positive interior principal
symbol $\sigma_{D}$ and define

$c_{D}:=\left(\frac{1}{n(2\pi)^{n}}\int_{S^{*}M}\mathrm{Tr}_{E}(\sigma_{D}(x,\xi)^{-\frac{n}{m}})\mathrm{d}x\mathrm{d}\xi\right)^{-\frac{m}{n}},$

where $n$ is the dimension of $M$. Then any lower semi-bounded self-adjoint
realisation $D_{\rm B}$ with $\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm
m}(M;E)$ is bounded from below with discrete real spectrum $\lambda_{0}(D_{\rm
B})\leq\lambda_{1}(D_{\rm B})\leq\lambda_{2}(D_{\rm B})\leq\cdots$ satisfying
that

$\lambda_{k}(D_{\rm B})=c_{D}k^{\frac{m}{n}}+o(k^{\frac{m}{n}}),\quad\mbox{as
$k\to+\infty$}.$

Theorem 8 appears as Theorem 7.4 (see page 7.4) in the body of the text. The
result might not be surprising as Weyl laws with error estimates are known
under mild assumptions, for a brief historical overview see Section 7. Our
assumptions are even milder, and we include the result as it showcases yet
another feature of regular realisations that only depend on abstract
principles. The proof of Theorem 8 was suggested to us by Gerd Grubb and
relies on a precise description of the resolvent of $D_{\rm B}$, see Theorem
7.1 on page 7.1, and previous work of Grubb [grubb68, grubb77a]. For
historical note, we mention that for Laplace operators on manifolds with
boundary (see for instance [ivrii16] for a historical overview) and for
elliptic pseudodifferential operators on closed manifolds (see for instance
[horIV, Chapter XXIX] or [shubinsbook, Chapter III]), the remainder terms in
the Weyl law are known more precisely.

###  1.2. Overview of contents

Let us briefly describe the contents of the paper.

In section 2 we set up the theory for elliptic differential operators,
building heavily on an abstract viewpoint presented in Appendix B. The
abstract theory allows us to set up boundary conditions (Subsection 2.2) as in
[BB, BBan] and characterise when they define Fredholm realisations. It allow
us to make some straightforward applications to wellposedness in Subsection
2.4. We also consider a number of examples: in Subsection 2.5 we describe some
classes of boundary conditions, in Subsection 2.6 we consider first order
elliptic operators reconciling with results obtained in [BBan]. In Subsection
2.7, we consider scalar properly elliptic operators reconciling with more
classical theory, e.g. [grubb68, schechter59, lionsmagenes].

In Section 3 we recall Seeley’s work on Calderón projectors and study its
implications for the Cauchy data space and its analytic structure. The
required Douglis-Nirenberg calculus and Hörmander’s description of the
symbolic structure of a Calderón projection is described in Subsection 3.1.
The Calderón projection from [seeley65] and further consequences for boundary
conditions are given in Subsection 3.2. In Subsection 3.3 we introduce the
notion of boundary decomposing projections, the projections that characterise
the analytic structure of the Cauchy data space.

In Section 4 we further analyse the Cauchy data space. The results of
Subsection 4.1 shows that the Cauchy data space can be locally defined and
that the Calderón projection could equally well be replaced with a projection
constructed from a finite number of terms in Hörmander’s construction in
Subsection 3.1. In Subsection 4.2 we give a precise description of the
boundary pairing between the Cauchy data space of an elliptic operator and the
Cauchy data space of its adjoint, allowing us to characterise adjoint boundary
conditions and regularity of pseudolocal boundary conditions in Subsection
4.3.

In Section 5 we characterise regular boundary conditions via graphical
decompositions, in Section 6 we characterise boundary conditions admitting
higher regularity, in Section 7 we prove the Weyl law for elliptic
differential operators with regular boundary conditions, and in Section 8 we
prove a rigidity result for the index of Fredholm realisations of elliptic
differential operators. Following this, Section 9 consists of a computational
exercise comparing a Calderón projection to the spectral projectors used in
[BBan], quantifying their qualitative differences.

###  1.3. Notation

Throughout the paper, we use the analyst’s inequality $a\lesssim b$ to mean
that $a\leq Cb$ for some constant $C$ where the dependence is apparent from
context or explicitly specified. By $a\simeq b$, we mean that $a\lesssim b$
and $b\lesssim a$.

The notation $+$ is used for internal sums of vector subspaces and $\oplus$ is
used for direct sums. For two subspaces $V_{1},V_{2}\subseteq V$ we use the
notation $V_{1}\oplus_{W}V_{2}=V_{1}+V_{2}$ to indicate $W:=V_{1}\cap V_{2}$,
note that $V_{1}\oplus_{W}V_{2}\cong(V_{1}\oplus V_{2})/d(W)$ for the diagonal
embedding $d$. In general, $\oplus$ does not imply that the sum is orthogonal.
We use the notation $\oplus^{\perp}$ for orthogonal direct sums. The term
projection will be used for an idempotent operator on a Hilbert space; in
general, they will not be orthogonal.

All manifolds and their boundaries are assumed to be smooth. For a manifold
$X$ without boundary, we write $\mathcal{D}^{\prime}(X)$ for the Fréchet space
of distributions, i.e. the topological dual of ${\rm C}^{\infty}_{\rm c}(X)$.
For a manifold with boundary $M$ we use the convention that the boundary is
included in $M$. We write $\Sigma:=\partial M$ and
$\mathring{M}:=M\setminus\Sigma$.

For an operator $T:\mathscr{H}\to\mathscr{H}$ over a Hilbert space
$\mathscr{H}$, potentially unbounded, we denote its _domain_ by
$\mathrm{dom}(T)$. The _kernel_ and _range_ are then given by $\ker(T)$ and
$\mathrm{ran}(T)$ respectively. The _graph norm_ of $T$ is
$\|\cdot\|_{T}=\sqrt{\|\cdot\|^{2}+\|T\cdot\|^{2}}$, and the operator $T$ is
said to be _closed_ if $(\mathrm{dom}(T),\|\cdot\|_{T})$ is a Banach space.
Since the graph norm polarises, $(\mathrm{dom}(T),\|\cdot\|_{T})$ is a Hilbert
space if $T$ is closed. If $S,T$ are two operators, then we write $S\subset T$
if $\mathrm{dom}(S)\subset\mathrm{dom}(T)$ and $S=T$ on $\mathrm{dom}(S)$. If
$S\subseteq T$ are two closed operators then
$\mathrm{dom}(S)\subset\mathrm{dom}(T)$ is a closed subspace with respect to
$\|\cdot\|_{T}$.

An operator $S$ is said to be _adjoint_ to $T$ if $\left\langle
Tu,v\right\rangle=\left\langle u,Sv\right\rangle$ for $u\in\mathrm{dom}(T)$
and $v\in\mathrm{dom}(S)$. A densely-defined $T$ has a _unique_ adjoint
$T^{\ast}$ with domain

$\mathrm{dom}(T^{\ast})=\left\\{u\in\mathscr{H}:\exists
C_{u,T}\quad|\left\langle u,Tv\right\rangle|\leq C_{u,T}\|v\|,\,\forall
v\in\mathrm{dom}(T)\right\\}.$

A closed operator $T$ is _Fredholm_ if it has closed range with finite
dimensional $\ker(T)$ and $\ker(T^{\ast})$.

The operator $T$ is said to be _invertible_ if it is injective with dense
range and $T^{-1}:\mathrm{ran}(T)\to\mathscr{H}$ is a bounded map. An
invertible $T$ has a unique extension $T^{-1}:\mathscr{H}\to\mathscr{H}$. The
_spectrum_ of the operator $T$ are the points $\lambda\in\mathbb{C}$ for which
$(\lambda-T)$ is not invertible. The set of all such points are denoted by
$\mathrm{spec}(T)$. There are a number of non-equivalent definitions of
_essential spectrum_ , but for us, this means the points
$\lambda\in\mathrm{spec}(T)$ for which $(\lambda-T)$ fails to be Fredholm. The
_resolvent set_ is the complement of $\mathrm{spec}(T)$ in $\mathbb{C}$ and it
is denoted by $\mathrm{res}(T)$.

A signification notion throughout this paper is that of a _Formally Adjointed
Pair (FAP)_ (cf. Appendix B). This notion was used, without being named, in
[grubb68, Chapter II]. Given Hilbert spaces $\mathscr{H}_{1}$ and
$\mathscr{H}_{2}$, these are a pair of densely-defined and closed operators
$(T_{\min},T_{\min}^{\dagger})$ with
$T_{\min}:\mathscr{H}_{1}\to\mathscr{H}_{2}$ and
$T_{\min}^{\dagger}:\mathscr{H}_{2}\to\mathscr{H}_{1}$ adjoint to each other
(i.e. $T_{\min}\subseteq(T_{\min}^{\dagger})^{*}$ and
$T_{\min}^{\dagger}\subseteq T_{\min}^{*}$). These yield the following
important maximal extensions

$T_{\max}:=(T_{\min}^{\dagger})^{\ast}\quad\mbox{and}\quad
T_{\max}^{\dagger}:=T_{\min}^{\ast}.$

A realisation of a FAP is an extension $T_{\min}\subseteq\hat{T}\subseteq
T_{\max}$. We will see FAPs arising from geometry and elliptic differential
operators.

It is of fundamental importance is to understand extensions, closed or
otherwise, of $T_{\min}$ (and respectively $T_{\min}^{\dagger}$). For this,
the space

$\check{\mathscr{H}}_{T}:=\faktor{\mathrm{dom}(T_{\max})}{\mathrm{dom}(T_{\min})}$

is of fundamental importance. A _Cauchy data space_ is a pair
$(\gamma,\check{\mathrm{H}}(T))$, where
$\gamma:\mathrm{dom}(T_{\max})\to\check{\mathrm{H}}(T)$ is a bounded
surjection with $\ker\gamma=\mathrm{dom}(T_{\min})$. It is clear from the open
mapping theorem that $\gamma$ induces an isomorphism
$\check{\mathscr{H}}(T)\cong\check{\mathrm{H}}(T)$.

###  1.4. Acknowledgements

LB was supported by SPP2026 from the German Research Foundation (DFG). MG was
supported by the Swedish Research Council Grant VR 2018-0350. HS was supported
by the Australian Research Council, through the Australian Laureate Fellowship
FL170100020 held by V. Mathai. The authors would also like to thank Christian
Bär and Andreas Rosén for useful discussions. The authors are most grateful to
Gerd Grubb, who motivated and inspired us through a multitude of comments and
helpful suggestions on earlier versions of the paper and providing us with the
method of proof used in Section 7.

##  2\. Elliptic differential operators and boundary conditions

###  2.1. Setup

Let $M$ be a compact manifold with boundary $\partial M=\Sigma$ carrying a
smooth measure $\mu$. In our convention, $\Sigma\subset M$ and the interior of
$M$ is denoted by $\mathring{M}=M\setminus\Sigma$.

Let $(E,h^{E})\to M$ be a hermitian vector bundle and let ${\rm
C}^{\infty}(M;E)$ denote the smooth sections over $E$. In particular, the
support of such sections are allowed to touch the boundary. The subspace ${\rm
C}^{\infty}_{\rm c}(\mathring{M};E)$ consists of $u\in{\rm C}^{\infty}(M;E)$
such that ${\rm spt}{\text{ }}u\cap\Sigma=\varnothing$. By an abuse of
notation, we write ${\rm C}^{\infty}(\Sigma;E)$ for the space of smooth
sections of $E|_{\Sigma}$, and similarly for other function spaces of
sections. We shall also fix a smooth interior vectorfield $\vec{T}$
transversal to $\partial M=\Sigma$ and let $\nu$ denote the smooth volume
measure on $\Sigma$, induced by $\mu$ and $\vec{T}$. In what is to follow, the
coordinate and derivative, respectively, in this transversal direction will be
denoted $x_{n}$ and $\partial_{x_{n}}$. For $s\in\mathbb{R}$, by ${\rm H}^{\rm
s}(M;E)$ and ${\rm H}^{\rm s}(\Sigma;E)$, we denote the ${\rm L}^{2}$ Sobolev
spaces of $s$ derivatives on $M$ and $\Sigma$, respectively. We let ${\rm
H}^{\rm s}_{\rm 0}(M;E)$ denote the closure of ${\rm C}^{\infty}_{\rm
c}(\mathring{M};E)$ in ${\rm H}^{\rm s}(\hat{M};E)$ for some closed manifold
$\hat{M}$ containing $M$ as an open subset with smooth boundary. Note that
${\rm H}^{\rm s}(M;E)$ can be identified with the quotient $\faktor{{\rm
H}^{\rm s}(\hat{M};E)}{{\rm H}^{\rm s}_{\rm 0}(\hat{M}\setminus M;E)}$.

For $(F,h^{F})\to M$ another hermitian vector bundle, we will be concerned
with elliptic differential operators $D:{\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(M;F)$ of order $m>0$. Further, by $D^{\dagger}:{\rm
C}^{\infty}(M;F)\to{\rm C}^{\infty}(M;E)$, let us denote the formal adjoint,
obtained via integration by parts (cf. [BB]). By $D_{c}$, we denote $D$ with
domain $\mathrm{dom}(D_{c})={\rm C}^{\infty}_{\rm c}(\mathring{M};E)$, and
analogously $D^{\dagger}_{c}$ is defined from
$\mathrm{dom}(D_{c}^{\dagger})={\rm C}^{\infty}_{\rm c}(\mathring{M};F)$. The
maximal and minimal domains of $D$ are then obtained as follows:

$D_{\max}=(D^{\dagger}_{\mathrm{c}})^{\ast}\quad\text{and}\quad
D_{\min}=\overline{D_{c}}.$

It is clear that both these operators are closed and that $D_{\min}\subset
D_{\max}$. Moreover, it holds that $D_{\rm min}^{*}=D_{\rm max}^{\dagger}$ and
$D_{\rm max}^{*}=D_{\rm min}^{\dagger}$. In the language of Appendix B,
$(D_{\rm min},D_{\rm min}^{\dagger})$ is a formally adjointed pair (FAP), see
Definition B.1. The next result shows that the FAP associated with an elliptic
operator of order $m>0$ is a Fredholm FAP (see Definition B.14).

######  Proposition 2.1.

If $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ is an elliptic
differential operator of order $m>0$ then the following holds:

  1. (1)

$\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm 0}(M,E)$;

  2. (2)

$D_{\rm min}$ has “compact resolvent” in the sense that $(1+D_{\rm
min}^{*}D_{\rm min})^{-{\frac{1}{2}}}$ is a compact operator on ${\rm
L}^{2}(M,E)$;

  3. (3)

$D_{\rm min}$ has finite-dimensional kernel, closed range and $D_{\rm max}$
has closed range. Moreover,

${\rm
L}^{2}(M;E)=\ker(D_{\min})\oplus^{\perp}\mathrm{ran}(D_{\max}^{\dagger})\quad\mbox{and}\quad{\rm
L}^{2}(M;F)=\ker(D_{\min}^{\dagger})\oplus^{\perp}\mathrm{ran}(D_{\max}).$

###### Proof.

Item 1 follows from the Gårding inequality showing that the graph norm of $D$
is equivalent to the ${\rm H}^{\rm m}(M;E)$-norm on ${\rm C}^{\infty}_{\rm
c}(\mathring{M};E)$. Item 2 follows from the fact that $(1+D_{\rm
min}^{*}D_{\rm min})^{-{\frac{1}{2}}}$ factors over the domain inclusion
$\mathrm{dom}(D_{\rm min})\hookrightarrow{\rm L}^{2}(M,E)$ which is compact by
item 1 as $m>0$. Item 3 follows from item 2 by Proposition B.15. ∎

Using the transversal vectorfield $\vec{T}$ at the boundary, for
$j=0,1,2,\ldots$, we can define a trace mapping

$\gamma_{j}:{\rm C}^{\infty}(M,E)\to{\rm C}^{\infty}(\Sigma,E),\quad
u\mapsto\partial_{x_{n}}^{j}u|_{\Sigma},$

where $x_{n}$ denotes the variable transversal to the boundary defined from
the vector field $\vec{T}$.

######  Theorem 2.2.

Let $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ be an elliptic
differential operator of order $m>0$. The trace mapping

$\gamma:{\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m}),\quad
u\mapsto(\gamma_{j}u)_{j=0}^{m-1},$

extends to a continuous mapping

$\gamma:\mathrm{dom}(D_{\rm max})\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma;E),$

with dense range and

$\ker\gamma={\rm H}^{\rm m}_{\rm 0}(M;E).$

This result is classical and can be found in [lionsmagenes63, lionsmagenes].
The result is also proven in [seeley65].

Following the notation of Appendix B we use the notation

$\check{\mathscr{H}}_{D}:=\check{\mathscr{H}}_{(D_{\rm min},D_{\rm
min}^{\dagger})}\equiv\faktor{\mathrm{dom}(D_{\rm max})}{\mathrm{dom}(D_{\rm
min})}.$

We note that since $\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm
0}(M;E)=\ker\gamma$, setting

$\check{\mathrm{H}}(D):=\gamma\mathrm{dom}(D_{\max}),$

the map $\gamma$ induces a bounded inclusion

$\check{\mathscr{H}}_{D}\hookrightarrow\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E),$

with range $\check{\mathrm{H}}(D)$. The reader should be aware that this
inclusion is not a Banach space isomorphism onto its image. However, by giving
$\check{\mathrm{H}}(D)$ the induced topology from this inclusion, we obtain
that

$\check{\mathscr{H}}_{D}\cong\check{\mathrm{H}}(D)\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E).$

Since $\check{\mathrm{H}}(D)$ is isomorphic to the Hilbert space quotient
$\faktor{\mathrm{dom}(D_{\rm max})}{\mathrm{dom}(D_{\rm min})}$, the Banach
space $\check{\mathrm{H}}(D)$ is in fact a Hilbert space (although we shall
not make use of a specific choice of inner product). We also note that since
${\rm H}^{\rm m}(M;E)\subseteq\mathrm{dom}(D_{\max})$ we have continuous
inclusions with dense ranges

$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma,E)\subseteq\check{\mathrm{H}}(D)\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E).$

Similarly, we obtain $\check{\mathscr{H}}_{D^{\dagger}}$ and
$\check{\mathrm{H}}(D^{\dagger})$ and continuous inclusions with dense ranges

$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma,F)\subseteq\check{\mathrm{H}}(D^{\dagger})\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,F).$

###  2.2. Boundary conditions

Any elliptic operator $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ as
above defines the FAP $(D_{\rm min},D_{\rm min}^{\dagger})$ in the terminology
of Appendix B. Therefore, realisations of elliptic operators are readily
described from abstract principles as subspaces of the Cauchy data space
$\check{\mathrm{H}}(D)$. We reiterate that realisations and boundary
conditions of elliptic operators form, to say the least, a well understood
topic. Our ambition is solely to provide a new perspective consistent with the
perspective due to Bär-Ballmann in the first order case.

A _generalised boundary condition_ $B$ for $D$ is a subspace of
$\check{\mathrm{H}}(D)$. It is simply called a _boundary condition_ if $B$ is
further required to be closed. We note that by Proposition B.5, there is a
one-to-one correspondence between realisations of an elliptic operator and its
generalised boundary conditions. This correspondence is set up by associating
the realisation $D_{\rm B}$ defined from

$\mathrm{dom}(D_{\rm B}):=\\{f\in\mathrm{dom}(D_{\rm max}):\gamma(f)\in B\\},$

with $B$, and conversely to associate the subspace
$\gamma(\mathrm{dom}(D_{e}))\subseteq\check{\mathrm{H}}(D)$ with a realisation
$D_{e}$ of $D$. We also note that by Proposition B.7, for any generalised
boundary condition $B$ of $D$, it holds that

$(D_{\rm B})^{*}=D_{{\rm B}^{*}}^{\dagger},$

where $B^{\ast}\subset\check{\mathrm{H}}(D^{\dagger})$ is the _adjoint
boundary condition_ defined by

$B^{*}:=\\{\eta\in\check{\mathrm{H}}(D^{\dagger}):\langle
D^{\dagger}f,g\rangle=\langle f,Dg\rangle\;\quad\forall
f\in\gamma^{-1}(\\{\eta\\}),\,g\in\mathrm{dom}(D_{\rm B})\\}.$

In Subsection 4.3 (see page 4.3) we further describe the adjoint boundary
condition as an annihilator of $B$ with respect to an explicit continuous
sesquilinear form
$\check{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D)\to\mathbb{C}$,
compare also to Appendix B.2, on page B.2. Note, $B^{\ast}$ is necessarily
closed. For convenience, we shall simply refer to a realisation or boundary
condition for $D$ when speaking of a realisation or boundary condition for
$(D_{\rm min},D_{\rm min}^{\dagger})$. This terminology is consistent with the
terminology for boundary conditions for first order elliptic differential
operators from [BB, BBan].

######  Definition 2.3.

Let $D$ be an elliptic differential operator of order $m>0$ and $D_{\rm B}$ a
closed realisation. We say that that $D_{\rm B}$ is:

  * •

_semi-regular_ if $\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm m}(M;E)$,

  * •

_regular_ if $D_{\rm B}$ and $(D_{\rm B})^{*}$ are semi-regular.

In light of Proposition B.5, we say that $B$ is (semi-) regular if $D_{\rm B}$
is (semi-) regular.

######  Remark 2.4.

The literature on boundary value problems contains a broad range of refined
notions for ellipticity and regularity. Historically, the term ellipticity has
been used for a local condition at the leading symbol level (including
possible boundary symbols) that implies regularity properties. For the
boundary conditions consider in this paper and in [BB, BBan], there is in
general no notion of boundary symbols. In [BB, BBan], the regularity property
$\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm m}(M;E)$ was therefore termed
semi-ellipticity. To better reflect the terminology in bulk of the literature
and to capture the fact that $\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm
m}(M;E)$ is a regularity property, we use the term semi-regular boundary
condition in this paper.

######  Proposition 2.5.

Let $D$ be an elliptic differential operator of order $m>0$ and $B$ a boundary
condition. Then $D_{\rm B}$ is semi-regular if and only if
$B\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$.

###### Proof.

If $D_{\rm B}$ is semi-regular, then $\gamma(\mathrm{dom}(D_{\rm
B}))\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$
by the trace theorem. We conclude that $B=\gamma(\mathrm{dom}(D_{\rm
B}))\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$.

If $B\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$,
then any $x\in\mathrm{dom}(D_{\rm B})$ can be written as $x=x_{0}+x_{\rm B}$
where $x_{0}\in\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm 0}(M;E)$ and
$x_{\rm B}\in\mathrm{dom}(D_{\rm max})$ is a pre-image of $\gamma(x)\in B$
under $\gamma$. By the trace theorem, we can take $x_{\rm B}\in{\rm H}^{\rm
m}(M;E)$ since $\gamma(x)\in\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma;E)$. We conclude that $\mathrm{dom}(D_{\rm
B})\subseteq\mathrm{dom}(D_{\rm min})+{\rm H}^{\rm m}(M;E)={\rm H}^{\rm
m}(M;E)$. ∎

######  Proposition 2.6.

Let $D$ be an elliptic differential operator of order $m>0$ and $B$ a semi-
regular boundary condition. Then, $\ker(D_{\rm B})$ is finite dimensional and
$\mathrm{ran}(D_{\rm B})$ and $\mathrm{ran}(D_{\rm B}^{\ast})$ are closed.
Moreover,

${\rm L}^{2}(M;E)=\ker(D_{{\rm B}})\oplus^{\perp}\mathrm{ran}(D_{{\rm
B}}^{\ast})\quad\mbox{and}\quad{\rm L}^{2}(M;F)=\ker(D_{{\rm
B}}^{\ast})\oplus^{\perp}\mathrm{ran}(D_{{\rm B}}).$

In particular, if $B$ is regular then $D_{\rm B}$ is Fredholm.

###### Proof.

If $B$ is semi-regular, the pair $\mathcal{T}_{\rm B}:=(D_{\rm
B},D^{\dagger}_{\rm min})$ is a FAP (indeed $D_{\rm B}\subseteq D_{\rm
max}=(D^{\dagger}_{\rm min})^{*}$) and $D^{\dagger}_{\rm min}\subseteq D_{{\rm
B}^{*}}^{\dagger}=(D_{\rm B})^{*}$) with compact domain inclusion
$\mathrm{dom}(D_{\rm B})\hookrightarrow{\rm L}^{2}(M;E)$. The claimed result
now follows from Proposition B.15. ∎

######  Remark 2.7.

We return to the study of regular boundary conditions below in Subsection 4.3
and Subsection 5.

For an elliptic operator $D$, we use the notation

$\mathcal{C}_{D}:=\gamma\ker D_{\rm
max}\subseteq\check{\mathrm{H}}(D)\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E).$

The space $\mathcal{C}_{D}$ is called the Hardy space of $D$. We shall later
see that $\mathcal{C}_{D}\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E)$ is, in fact, closed in
$\check{\mathrm{H}}(D)$. In the setting of Appendix B, more precisely
Definition B.11, the space $\mathcal{C}_{D}$ is denoted by
$\mathcal{C}_{(D_{\rm min},D_{\rm min}^{\dagger})}$.

######  Proposition 2.8.

Let $D$ be an elliptic differential operator of order $m>0$ and $B\subseteq
B^{\prime}$ be boundary conditions.

  1. (i)

It holds that $D_{{\rm B}}\subseteq D_{{\rm B}^{\prime}}$ and there is a short
exact sequence of Hilbert spaces

$0\to\mathrm{dom}(D_{{\rm B}})\to\mathrm{dom}(D_{{\rm
B}^{\prime}})\to\faktor{B^{\prime}}{B}\to 0.$

In particular,

$\displaystyle\faktor{\mathrm{dom}(D_{\rm B})}{\mathrm{dom}(D_{\rm min})}\cong
B,\quad\text{and}\quad$ $\displaystyle\faktor{\mathrm{dom}(D_{\rm
max})}{\mathrm{dom}(D_{\rm B})}\cong\faktor{\check{\mathrm{H}}(D)}{B}$

hold for a boundary condition $B$.

  2. (ii)

The inclusion and restriction mappings fit into a short exact sequence of
Hilbert spaces

$0\to\ker(D_{\rm min})\to\ker(D_{{\rm B}})\to\mathcal{C}_{D}\cap B\to 0.$

In particular, we have the short exact sequence

$0\to\ker(D_{\rm min})\to\ker(D_{\rm
max})\xrightarrow{\gamma}\mathcal{C}_{D}\to 0.$

###### Proof.

The result follows from the Propositions B.10 and B.12. ∎

######  Theorem 2.9.

Let $D$ be an elliptic differential operator of order $m>0$ and $B$ be a
generalised boundary condition. Then the following holds:

  1. (i)

$\mathrm{ran}(D_{\rm B})=\mathrm{ran}(D_{{\rm B}+\mathcal{C}_{D}})$ and it is
closed if and only if $B+\mathcal{C}_{D}$ is a boundary condition (i.e. closed
in $\check{\mathrm{H}}(D)$).

  2. (ii)

$\ker(D_{\rm B})$ is finite-dimensional if and only if $B\cap\mathcal{C}_{D}$
is finite-dimensional.

  3. (iii)

$\mathrm{ran}(D_{\rm B})$ has finite algebraic codimension if and only if
$B+\mathcal{C}_{D}$ has finite algebraic codimension in
$\check{\mathrm{H}}(D)$ if and only if $\mathrm{ran}(D_{\rm B})$ is closed and
$\mathrm{ran}(D_{\rm B})^{\perp}$ is finite-dimensional.

###### Proof.

Item i follows from Theorem B.23, Proposition 2.1, and Lemma B.20. Item ii
follows from Proposition 2.8. Item iii follows from Proposition B.26. ∎

######  Corollary 2.10.

Let $D$ be an elliptic differential operator of order $m>0$ and
$B\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma,E)$ be
closed in $\check{\mathrm{H}}(D)$. Then
$B+\mathcal{C}_{D}\subseteq\check{\mathrm{H}}(D)$ is closed.

###### Proof.

The space $B$ defines a semi-regular boundary condition by Proposition 2.5. By
Proposition 2.6, $D_{\rm B}$ has closed range so
$B+\mathcal{C}_{D}\subseteq\check{\mathrm{H}}(D)$ is closed by Theorem 2.9. ∎

A significant consequence of Theorem 2.9 is that it allows us to relate the
closedness of the range of a realisation of a boundary condition to the
closedness of an associated boundary condition. It may seem that this
implicitly asserts that an operator needs to be closed in order to have closed
range. This is not the case, as highlighted by the following quintessential
example.

######  Example 2.11.

Consider the generalised boundary condition

$B_{\rm Sob}=\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E),$

for an elliptic operator $D$ of order $m>0$. The space $B_{\rm Sob}$ is not a
closed subspace of $\check{\mathrm{H}}(D)$, so the associated operator
$D_{{\rm B}_{\rm Sob}}$ is not a closed operator. One can in fact show that
$\overline{D_{{\rm B}_{\rm Sob}}}=D_{{\rm B}_{\rm Sob}+\mathcal{C}}=D_{\max}$.
We can prove that $D_{{\rm B}_{\rm Sob}}$ has closed range and as such,
$B_{\rm Sob}+\mathcal{C}_{D}$ is closed in $\check{\mathrm{H}}(D)$. We return
to this example below in in Example 3.21 below and prove, amongst other
things, that $B_{\rm Sob}+\mathcal{C}_{D}=\check{\mathrm{H}}(D)$.

Let us prove that $D_{{\rm B}_{\rm Sob}}$ has closed range. By a similar
argument as in Proposition 2.5, we see that

$\mathrm{dom}(D_{{\rm B}_{\rm Sob}})={\rm H}^{\rm m}(M;E).$

We can assume that $M$ is a domain with smooth boundary in a compact manifold
$\hat{M}$, that $E,F$ extend to Hermitian vector bundles
$\hat{E},\hat{F}\to\hat{M}$ and that there exists an elliptic differential
operator $\hat{D}$ of order $m$ extending $D$ to $\hat{M}$. Let
$\hat{T}\in\Psi^{-m}_{\rm cl}(\hat{M};\hat{F},\hat{E})$ denote a parametrix of
$\hat{D}$ such that $\hat{D}\hat{T}-1$ is a projection onto the finite-
dimensional space $\ker(\hat{D}^{\dagger})$ and $\hat{T}\hat{D}-1$ is a
projection onto the finite-dimensional space $\ker(\hat{D})$. We define the
continuous operator $T:=\hat{T}|:{\rm L}^{2}(M;F)\to{\rm H}^{\rm m}(M;E)$
which by construction inverts $D:{\rm H}^{\rm m}(M;E)\to{\rm L}^{2}(M;F)$
(viewed as a continuous operator) from the right up to smoothing operators.
From the compactness of $(1-DT)$ and Fredholm’s theorem, we conclude that

$\mathrm{ran}(D_{{\rm B}_{\rm Sob}})=\mathrm{ran}(D:{\rm H}^{\rm
m}(M;E)\to{\rm L}^{2}(M;F))\supseteq\mathrm{ran}(DT:{\rm L}^{2}(M;F)\to{\rm
L}^{2}(M;F)),$

has finite algebraic codimension and is closed in ${\rm L}^{2}(M;F)$.

By understanding closedness of the ranges of an operator, we are able to
understand which boundary conditions yield Fredholm operators. In applications
to index theory, it is essential to understand which boundary conditions yield
such realisations. This is described in the following theorem.

######  Theorem 2.12.

Let $D$ be an elliptic operator of order $m>0$ and $B$ be a boundary
condition. Then the following are equivalent:

  1. (i)

$(B,\mathcal{C}_{D})$ is a Fredholm pair in $\check{\mathrm{H}}(D)$;

  2. (ii)

$D_{\rm B}$ is a Fredholm operator.

If either of these equivalent conditions hold, we have that

$B^{\ast}\cap\mathcal{C}_{D^{\dagger}}\cong\faktor{\check{\mathrm{H}}(D)}{(B+\mathcal{C}_{D})}.$

Moreover,

$\operatorname{ind}(D_{\rm
B})=\operatorname{ind}(B,\mathcal{C}_{D})+\dim\ker(D_{\min})-\dim\ker(D_{\min}^{\dagger}).$

###### Proof.

Follows from Theorem B.28. ∎

######  Remark 2.13.

A direct corollary of Theorem 2.12 is that whenever $B$ and $B^{\prime}$ are
two Fredholm boundary condition that are norm-continuously homotopic via
Fredholm boundary conditions222Norm-continuously homotopic in the sense that
$B$ and $B^{\prime}$ are images of norm-continuously homotopic projectors
where all projections along the homotopy define Fredholm boundary conditions.
then it holds that

$\operatorname{ind}(D_{\rm B})=\operatorname{ind}(D_{{\rm B}^{\prime}}).$

However, Theorem 8.1 below provides a more general result concerning homotopy
invariance so we omit the details of this argument.

######  Corollary 2.14.

Let $D$ be an elliptic operator of order $m>0$. Assume that
$B\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$ is
closed in $\check{\mathrm{H}}(D)$ and satisfies that
$B^{*}\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma;F)$. Then $(B,\mathcal{C}_{D})$ is a Fredholm pair
in $\check{\mathrm{H}}(D)$ and

$B^{\ast}\cap\mathcal{C}_{D^{\dagger}}\cong\faktor{\check{\mathrm{H}}(D)}{(B+\mathcal{C}_{D})}.$

###### Proof.

Follows from Theorem 2.12 by noting that our assumptions are equivalent to $B$
being regular which implies that it is Fredholm by Proposition 2.6. ∎

###  2.3. The boundary pairing

Consider $\check{\mathscr{H}}_{D}$ with the quotient map
$\mathrm{dom}(D_{\max})\to\faktor{\mathrm{dom}(D_{\max})}{\mathrm{dom}(D_{\min})}=\check{\mathscr{H}}_{D}$
as a Cauchy data space. Then, by abstract nonsense, the boundary pairing

$\omega_{D}:\check{\mathscr{H}}_{D^{\dagger}}\times\check{\mathscr{H}}_{D}\to\mathbb{C},$

defined for $f\in\mathrm{dom}(D^{\dagger}_{\rm max})$,
$g\in\mathrm{dom}(D_{\rm max})$ as

$\quad\omega_{D}([f],[g]):=\langle D^{\dagger}f,g\rangle_{{\rm
L}^{2}(M;E)}-\langle f,Dg\rangle_{{\rm L}^{2}(M;F)},$

is well-defined and continuous. By an abuse of notation, we consider the
boundary pairing as a sesquilinear mapping
$\omega_{D}:\check{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D)\to\mathbb{C}$.

To describe $\omega_{D}$ on $\check{\mathrm{H}}(D)$ and
$\check{\mathrm{H}}(D^{\dagger})$ in greater detail, we introduce the
following $m\times m$-matrix

$\tau=\begin{pmatrix}0&0&\cdots&0&0&1\\\ 0&0&\cdots&0&1&0\\\
0&0&\cdots&1&0&\vdots\\\ \vdots&\vdots&\mathinner{\mkern 1.0mu\raise
1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern
2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&0&\vdots&\\\ 0&1&\cdots&0&0&0\\\
1&0&\cdots&0&0&0\end{pmatrix}$ (1)

We introduce the notation $A_{l}=(A_{l}(x_{n}))_{x_{n}\in[0,1]}$ for the
family of differential operators ${\rm C}^{\infty}(\Sigma,E)\to{\rm
C}^{\infty}(\Sigma,F)$ of order $l$ such that

$D=\sum_{l=0}^{m}A_{l}D_{x_{n}}^{m-l},$

near $\Sigma$. The following proposition follows from the computations on
[seeley65, page 794]. See also [gerdsgreenbook, Proposition 1.3.2].

######  Proposition 2.15.

Let $D$ be an elliptic differential operator of order $m>0$. There exists an
$m\times m$ matrix of differential operators

$\widetilde{\scalebox{1.5}{a}}=\begin{pmatrix}A_{0}&0&\cdots&0&0&0\\\
A_{1,m-1}&A_{0}&\cdots&0&0&0\\\ A_{2,m-2}&A_{1,m-2}&\ddots&0&\vdots&\vdots\\\
\vdots&\vdots&\ddots&A_{0}&0&0\\\
A_{m-2,2}&A_{m-3,2}&\cdots&A_{1,2}&A_{0}&0\\\
A_{m-1,1}&A_{m-2,1}&\cdots&A_{2,1}&A_{1,1}&A_{0}\end{pmatrix},$

where $A_{j,k}$ is a differential operator of order $j$ with the same
principal symbol as $A_{j}|_{x_{n}=0}$, such that

$\scalebox{1.5}{a}:=-i\tau\widetilde{\scalebox{1.5}{a}}=-i\begin{pmatrix}A_{m-1,1}&A_{m-2,1}&\cdots&A_{2,1}&A_{1,1}&A_{0}\\\
A_{m-2,2}&A_{m-3,2}&\cdots&A_{1,2}&A_{0}&0\\\ A_{m-3,3}&\mathinner{\mkern
1.0mu\raise 1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise
4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&&A_{0}&0&\vdots\\\
\vdots&&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern
2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\hbox{.}\mkern
1.0mu}&0&\vdots&\\\ A_{1,m-2}&A_{0}&\mathinner{\mkern 1.0mu\raise
1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern
2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&&&\\\
A_{0}&0&\cdots&0&0&0\end{pmatrix},$

satisfies

$\langle D^{\dagger}u,v\rangle_{{\rm L}^{2}(M;E)}-\langle u,Dv\rangle_{{\rm
L}^{2}(M,F)}=\langle\gamma u,\scalebox{1.5}{a}\gamma v\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})},$

for all $u\in{\rm C}^{\infty}(M;F)$ and $v\in{\rm C}^{\infty}(M;E)$.

The fact that the order of the differential operators appearing in
$\widetilde{\scalebox{1.5}{a}}$ remains constant along diagonals in the matrix
shows that, in fact,
$\widetilde{\scalebox{1.5}{a}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m},F\otimes\mathbb{C}^{m})$ via the Douglis-
Nirenberg calculus we review in Subsection 3.1. The reader should note that we
abuse notation and identify $A_{0}$ with $A_{0}|_{x_{m}=0}$ which is of order
$0$, i.e. $A_{0}\in{\rm C}^{\infty}(\Sigma;\mathrm{Hom}(E,F))$. Since $A_{0}$
is the restriction of the principal symbol to $x_{n}=0$, $\xi^{\prime}=0$ and
$\xi_{n}=1$, ellipticity implies that $A_{0}:E|_{\Sigma}\to F|_{\Sigma}$ is a
vector bundle isomorphism.

######  Lemma 2.16.

Let $D$ be an elliptic differential operator of order $m>0$. Let
$\widetilde{\scalebox{1.5}{a}}$ and $\widetilde{\scalebox{1.5}{a}}_{\dagger}$
denote the matrices of differential operators constructed from $D$ and
$D^{\dagger}$, respectively, as in Proposition 2.15. Recalling that
$\scalebox{1.5}{a}=-i\tau\widetilde{\scalebox{1.5}{a}}$, it holds that

$\scalebox{1.5}{a}^{*}+\scalebox{1.5}{a}_{\dagger}=0.$

###### Proof.

The lemma follows from that  a defines the boundary pairing as in Proposition
2.15 and the symmetry condition on the boundary pairing in Proposition B.4. ∎

######  Lemma 2.17.

The operators $\widetilde{\scalebox{1.5}{a}}$ (from Proposition 2.15) and
$\tau$ satisfy the following.

  1. (i)

For any $s\in\mathbb{R}$, $\tau$ defines a unitary isomorphism

$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm s+j}(\Sigma;F)\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm m-1+s-j}(\Sigma;F)$

  2. (ii)

The matrix of differential operators $\widetilde{\scalebox{1.5}{a}}$ is
invertible as a matrix of differential operators and

$\widetilde{\scalebox{1.5}{a}}^{-1}=\begin{pmatrix}A_{0}^{-1}&0&\cdots&0&0&0\\\
B_{1,m-1}&A_{0}^{-1}&\cdots&0&0&0\\\
B_{2,m-2}&B_{1,m-2}&\ddots&0&\vdots&\vdots\\\
\vdots&\vdots&\ddots&A_{0}^{-1}&0&0\\\
B_{m-2,2}&B_{m-3,2}&\cdots&B_{1,2}&A_{0}^{-1}&0\\\
B_{m-1,1}&B_{m-2,1}&\cdots&B_{2,1}&B_{1,1}&A_{0}^{-1}\end{pmatrix},$

where $B_{j,k}$ is a differential operator of order $j$.

  3. (iii)

For any $s\in\mathbb{R}$, $\widetilde{\scalebox{1.5}{a}}$ defines a Banach
space isomorphism

$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm s-j}(\Sigma;E)\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm s-j}(\Sigma;F),$

and $\scalebox{1.5}{a}=-i\tau\widetilde{\scalebox{1.5}{a}}$ defines a Banach
space isomorphism

$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm s+j}(\Sigma;E)\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm m-1+s-j}(\Sigma;F),$

###### Proof.

Item i is immediate. Item ii follows from that
$A_{0}^{-1}\widetilde{\scalebox{1.5}{a}}$ is lower triangular with the
identity operator on the diagonal, and as such, the inverse of
$A_{0}^{-1}\widetilde{\scalebox{1.5}{a}}$ is computed from polynomial
operations on its entries through Gaussian elimination. The fact that the
differential operator $B_{j,k}$ is of order $j$ follows from a short
inspection of the Gaussian elimination. To prove item iii, we note that any
matrix $(C_{j,k})_{j,k=0}^{m-1}$ of differential operators with the order of
$C_{j,k}$ being $j-k$, acts continuously on $\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
s-j}(\Sigma;F)$. Now item iii follows from item ii. ∎

Following Appendix B, we use the notation

$\omega_{D}:\check{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D)\to\mathbb{C},$

for the sesquilinear boundary pairing

$\omega_{D}(\eta,\xi):=\langle D^{\dagger}u,v\rangle_{{\rm
L}^{2}(M;E)}-\langle u,Dv\rangle_{{\rm L}^{2}(M,F)},$

for $u\in\gamma^{-1}(\\{\eta\\})$ and $v\in\gamma^{-1}(\\{\xi\\})$. This
pairing is well defined and non-degenerate by Proposition B.4. Below in
Theorem 4.12, we shall see that it is perfect. For now we settle with the
pairing being perfect on the Sobolev spaces.

######  Proposition 2.18.

The boundary pairing
$\omega_{D}:\check{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D)\to\mathbb{C}$
extends by continuity to a perfect pairing

$\omega_{D,s}:\bigoplus_{j=0}^{m-1}{\rm H}^{\rm-
s-j}(\Sigma;F)\times\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-1+s-j}(\Sigma;E)\to\mathbb{C},$

for any $s\in\mathbb{R}$ with

$\omega_{D,s}(\eta,\xi)=\langle\eta,\scalebox{1.5}{a}\xi\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})},$

for $\eta\in\bigoplus_{j=0}^{m-1}{\rm H}^{\rm-s-j}(\Sigma;F)$ and
$\xi\in\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-1+s-j}(\Sigma;E)$.

###### Proof.

By Proposition 2.15,
$\omega_{D}([\xi],[\xi^{\prime}])=\langle\xi,\scalebox{1.5}{a}\xi^{\prime}\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}$ for $\xi\in{\rm C}^{\infty}(\Sigma;F)$
and $\xi^{\prime}\in{\rm C}^{\infty}(\Sigma;E)$. The statement now follows
from Lemma 2.17. ∎

###  2.4. Well-posedness in the absence of kernels

In the situation of boundary conditions without kernels, we are able to
understand some properties of associated PDEs.

######  Proposition 2.19.

Let $D$ be an elliptic operator of order $m>0$. Suppose that $B$ is a semi-
regular boundary condition such that $\ker(D_{{\rm B}})=0$. Then, the spectrum
of $D_{\rm B}^{\ast}D_{{\rm B}}$ consists solely of discrete spectrum in
$(0,\infty)$ and letting $\lambda_{1}$ denote its smallest nonzero eigenvalue,
we have the Poincaré inequality

$\sqrt{\lambda_{1}}\|u\|_{{\rm L}^{2}(M;E)}\leq\|D_{{\rm B}}u\|_{{\rm
L}^{2}(M;F)}$

for all $u\in\mathrm{dom}(D_{{\rm B}})$.

Under an additional hypothesis on $B$, we give a precise description of the
spectral asymptotics of $D_{\rm B}^{\ast}D_{{\rm B}}$ in Corollary 7.7 below.

###### Proof.

The operator $D_{\rm B}^{\ast}D_{{\rm B}}$ satisfies $\mathrm{dom}(D_{\rm
B}^{\ast}D_{{\rm B}})\subset{\rm H}^{\rm m}(M;E)$, and moreover, it is easy to
see that it is non-negative self-adjoint. By the Rellich embedding theorem, we
obtain that $\mathrm{spec}(D_{\rm B}^{\ast}D_{{\rm B}})$ is discrete, real,
non-negative. Also, $\ker(D_{\rm B}^{\ast}D_{{\rm B}})=\ker(D_{{\rm B}})=0$
and therefore, $\mathrm{spec}(D_{\rm B}^{\ast}D_{{\rm
B}})=\left\\{0<\lambda_{1}\leq\lambda_{2}\leq\dots\right\\}$. Then, via
numerical range considerations,

$\left\langle D_{\rm B}^{\ast}D_{{\rm
B}}u,u\right\rangle\geq\lambda_{1}\|u\|^{2}.$

for $u\in\mathrm{dom}(D_{\rm B}^{\ast}D_{{\rm B}})$. However,
$\mathrm{dom}(D_{\rm B}^{\ast}D_{\rm B})$ is dense in
$\mathrm{dom}(\sqrt{D_{\rm B}^{\ast}D_{\rm B}})=\mathrm{dom}(D_{\rm B})$ and
therefore, we obtain the desired inequality. ∎

######  Proposition 2.20.

Let $D$ be an elliptic operator of order $m>0$. Suppose that $B$ is a semi-
regular boundary condition and that $\ker(D_{\rm B})=0$. Then,

$\begin{cases}D_{\rm B}u=f\\\ \gamma(u)\in B\end{cases}$

for $f\in\mathrm{ran}(D_{{\rm B}})$ is well-posed.

###### Proof.

By Proposition 2.6, we have that $D_{\rm B}$ and $D_{\rm B}^{\ast}$ has closed
range. But ${\rm L}^{2}(M;E)=\ker(D_{\rm B})\oplus\mathrm{ran}(D_{\rm
B}^{\ast})=\mathrm{ran}(D_{\rm B}^{\ast})$, and so on applying [Yosida,
Corollary 1, Chapter VII, Section 5], we obtain that

$D_{{\rm B}}:\mathrm{dom}(D_{\rm B})\subset{\rm
L}^{2}(\Sigma;E)\to\mathrm{ran}(D_{\rm B})$

has a bounded inverse. Therefore,

$\|u\|_{D_{\rm B}}\simeq\|u\|+\|Du\|=\|{D_{{\rm
B}}^{-1}f}\|+\|f\|\lesssim\|f\|.$

This is exactly that the problem in the statement of the corollary is well-
posed. ∎

We apply this to understand the well-posedness of the Dirichlet problem. An
important and well known notion is _weak UCP (unique continuation property)_
for an operator $D$. This means that for smooth $u$, when $Du=0$ and $u=0$ on
a nonempty open subset $\Omega\subset M$, then $u=0$. Similarly, _weak inner
UCP_ is satisfied by $D$ if for smooth $u$, when $Du=0$ and
$u{{\lvert}}_{\Sigma}=0$, this implies $u=0$.

######  Corollary 2.21.

Let $D$ be an elliptic differential operator satisfying weak inner UCP. Then,

$\begin{cases}Du=f\\\ \gamma(u)=0\end{cases}$

is well-posed on $\mathrm{ran}(D_{\mathcal{C}_{D}})=\mathrm{ran}(D_{\min})$.

###### Proof.

We obtain well-posedness on $\mathrm{ran}(D_{\min})$ from Proposition 2.20
with the semi-regular boundary condition $B=0$ upon showing that
$\ker(D_{\min})=0$. By elliptic regularity, $\ker(D_{\min})\subset{\rm
C}^{\infty}(M;E)$ so the weak inner UCP property yields that
$\ker(D_{\min})=0$. Lastly, we have that
$\mathrm{ran}(D_{\min})=\mathrm{ran}(D_{0+\mathcal{C}_{D}})=\mathrm{ran}(D_{\mathcal{C}_{D}})$
from Theorem 2.9 item i. ∎

Now, let us focus on the first order case. In the first order case, $D_{\rm
min}$ corresponds to the Dirichlet problem for $D$. As discussed in [BBL2009,
Section 1.2], weak UCP implies weak inner UCP for elliptic first order
differential operators. However, focusing on Dirac-type operators, the
following holds.

######  Corollary 2.22.

Let $D$ be first order and Dirac-type. Then,

$\begin{cases}Du=f\\\ \gamma(u)=0\end{cases}$

is well-posed on $\mathrm{ran}(D_{\mathcal{C}_{D}})=\mathrm{ran}(D_{\min})$.

###### Proof.

As mentioned in [BBL2009, Remark 2.2], it is well known that all first order
Dirac-type operators satisfy weak UCP. This, in turn, implies weak inner UCP,
and therefore, the conclusion follows. ∎

######  Remark 2.23.

It is a well known fact that well-posedness fails for the Dirichlet problem
for first order Dirac-type operators on ${\rm L}^{2}$. However, the corollary
shows that on a very large space, well-posedness actually holds. In fact, by
Proposition 2.6, it is easily seen that the obstruction to well-posedness is
indeed large. It is precisely $\ker(D^{\dagger}_{max})=\ker(D_{\min}^{*})$. On
a manifold of dimension exceeding 1, we see from Theorem 3.11 that this is,
indeed, an infinite dimensional space. That is, in general, the obstruction to
well-posedness for the Dirichlet problem is infinite dimensional.

###  2.5. Local, pseudo-local and bundle-like boundary conditions

Let us give some examples of boundary conditions.

######  Definition 2.24.

Let $D$ be an elliptic operator of order $m>0$ and $B$ a boundary condition.

  * •

We say that $B$ is pseudo-local if there exists an $m\times m$ matrix $P$ of
pseudo-differential operators on $E|_{\Sigma}$ with $P^{2}=P$ and

$B=\check{\mathrm{H}}(D)\cap\ker(P:\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})).$

We say that the pseudo-local boundary condition $B$ is defined from $P$, and
to indicate the dependence on $P$ we sometimes use the notation $B_{P}$ for
$B$.

  * •

We say that $B$ is local if there exists a differential operator $b$ on
$E|_{\Sigma}\otimes\mathbb{C}^{m}\to E^{\prime}$ (for some vector bundle
$E^{\prime}\to\Sigma$) such that

$B=\check{\mathrm{H}}(D)\cap\ker(b:\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E^{\prime})).$

We say that the local boundary condition $B$ is defined from $b$.

  * •

If $B$ is a local boundary condition defined from $b$ on
$E|_{\Sigma}\otimes\mathbb{C}^{m}\to E^{\prime}$ where $E^{\prime}\subseteq
E|_{\Sigma}\otimes\mathbb{C}^{m}$ is a subbundle and $b$ is a projection onto
a complement of $E^{\prime}$, we say that the associated boundary condition is
bundle-like and defined from $E^{\prime}$.

In [BBan], the bundle-like boundary conditions were called local. However, in
the higher order case more general local boundary conditions naturally arise
and are well studied, e.g. in [lionsmagenes, grubb68, vishik].

The reader should note that a bundle-like boundary condition is both local and
pseudo-local. If $B$ is a local boundary condition defined from $b$, then

$\mathrm{dom}(D_{\rm B})=\\{u\in\mathrm{dom}(D_{\max}):b\gamma u=0\\}.$

Here $b\gamma u$ is interpreted in a distributional sense. In particular, $B$
is indeed a boundary condition because the topology on $\check{\mathrm{H}}(D)$
induced from the space of distributions is weaker than the norm topology of
$\check{\mathrm{H}}(D)$. A similar argument shows that a pseudo-local boundary
condition indeed is a boundary condition.

######  Example 2.25.

For $0\leq k\leq m$, we consider the order zero operator $b$ that projects
onto the first $k$ coordinates of $E|_{\Sigma}\otimes\mathbb{C}^{m}$. Let
$B_{k}$ be the associated bundle-like boundary condition. We have that

$\displaystyle B_{k}=$
$\displaystyle\\{\xi\in\check{\mathrm{H}}(D):\xi_{0}=\xi_{1}=\ldots=\xi_{k-1}=0\\},\quad\mbox{and}$
$\displaystyle\mathrm{dom}(D_{{\rm B}_{k}})=$
$\displaystyle\\{u\in\mathrm{dom}(D_{\rm
max}):\gamma_{0}u=\gamma_{1}u=\ldots\gamma_{k-1}u=0\\}=\mathrm{dom}(D_{\rm
max})\cap{\rm H}^{\rm k}_{\rm 0}(M).$

######  Proposition 2.26.

Let $D$ be an elliptic operator of order $m>0$ and $B_{P}$ a pseudo-local
boundary condition defined from $P$. Then $(B_{P})^{*}$ is the pseudo-local
boundary condition $B_{P_{\dagger}}$ for $D^{\dagger}$ defined from
$P_{\dagger}:=(\scalebox{1.5}{a}^{*})^{-1}(1-P^{*})\scalebox{1.5}{a}^{*}$
where  a is the differential operator from Proposition 2.15.

###### Proof.

We remark that $(\scalebox{1.5}{a}^{*})^{-1}(1-P^{*})\scalebox{1.5}{a}^{*}$ is
again an idempotent matrix of pseudodifferential operators because
$(\scalebox{1.5}{a}^{*})^{-1}$ is a matrix of differential operators by Lemma
2.17. The space $B^{*}$ is characterised by the property that $\eta\in B^{*}$
if and only if $\omega_{D}(\eta,\xi)=0$ for all $\xi\in B$. We need to prove
that

$B^{*}=\check{\mathrm{H}}(D^{\dagger})\cap\ker((1-P^{*})\scalebox{1.5}{a}^{*}:\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})).$

Since
$B=\check{\mathrm{H}}(D)\cap\ker(P:\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m}))$,
any $\xi\in B$ satisfies $\xi=(1-P)\xi$. In particular, if
$\eta\in\check{\mathrm{H}}(D^{\dagger})\cap\ker((1-P^{*})\scalebox{1.5}{a}^{*}:\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m}))$
and $\xi\in B$ then

$\omega_{D}(\eta,\xi)=\langle\eta,\scalebox{1.5}{a}(1-P)\xi\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}=\langle(1-P^{*})\scalebox{1.5}{a}^{*}\eta,\xi\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}=0.$

We conclude that
$\check{\mathrm{H}}(D^{\dagger})\cap\ker((1-P^{*})\scalebox{1.5}{a}^{*}:\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m}))\subseteq
B^{*}$.

To prove the converse inclusion, if $\eta\in B^{*}$ then for any
$\xi_{0}\in{\rm C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m})$ we have that

$\omega_{D}(\eta,(1-P)\xi_{0})=\langle(1-P^{*})\scalebox{1.5}{a}^{*}\eta,\xi_{0}\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}.$

Since $\xi:=(1-P)\xi_{0}\in B$, we have that
$\langle(1-P^{*})\scalebox{1.5}{a}^{*}\eta,\xi_{0}\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}=0$ for any $\xi_{0}\in{\rm
C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m})$. In particular,
$(1-P^{*})\scalebox{1.5}{a}^{*}\eta=0$ in distributional sense if $\eta\in
B^{*}$. Therefore
$\check{\mathrm{H}}(D^{\dagger})\cap\ker((1-P^{*})\scalebox{1.5}{a}^{*}:\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m}))\supseteq
B^{*}$.

∎

######  Example 2.27.

Returning to Example 2.25, assume for simplicity that $E=F$ and that the
elliptic operator $D$ takes the form $D=D_{x_{n}}^{m}+A$ near the boundary for
a differential operator $A$ on $\Sigma$ (with no dependence on $x_{n}$). A
short computation with integration by parts show that
$\scalebox{1.5}{a}=0^{*}\sigma(A)\otimes\tau$ where $0:\Sigma\to T^{*}\Sigma$
denotes the zero section and $\tau$ is as in Equation (1). Using this it is
readily verified via Proposition 2.26 that $B_{k}^{*}=B_{m-k-1}$ for
$k=0,\ldots,m$ with the convention that $B_{-1}=0$. If the order of $D$ is
even, then $B_{m/2}$ should be interpreted as a Dirichlet condition. We return
to this condition below in Subsection 2.7 for scalar properly elliptic
operators.

###  2.6. Elliptic first order operators

The situation on which we have modelled our description of boundary conditions
is that of first order operators described in [BB, BBan]. For simplicity, we
continue to focus on compact manifolds with boundary, while [BB, BBan] allowed
for non-compact manifolds with compact boundary. The theory of boundary value
problems for first order elliptic operators is further described in [G99] and
the special case of Dirac operators is studied in [bosswojc].

We follow the setup of Subsection 2.1 and, throughout this subsection,
consider an elliptic differential operator $D$ order $m=1$. Associated to such
an operator $D$, there are adapted boundary operators $A$ on the bundle
$E|_{\Sigma}$ which are first order elliptic differential operators on
$\Sigma$ whose principal symbols satisfy

$\upsigma(A)(x,\xi^{\prime})=\upsigma(D)(x,\xi^{\prime}=0,\xi_{n}=1)^{-1}\,\circ\,\upsigma(D)(x,\xi^{\prime}).$

Alternatively, an adapted boundary operator $A$ is an operator on $\Sigma$
such that near $\Sigma$ we can write

$D=\sigma\cdot(\partial_{x_{n}}+A+R_{0}),$

for an $x_{n}$-dependent first order differential operator $R_{0}$ on $\Sigma$
such that $R_{0}|_{x_{n}=0}$ is of order zero. Here we have shortened the
notation $\sigma=\upsigma(D)(x,\xi^{\prime}=0,\xi_{n}=1)$. The reader should
note that in the notation of Proposition 2.15,

$\sigma=A_{0}=\scalebox{1.5}{a}\quad\mbox{and}\quad
A_{1}=\sigma\cdot(A+R_{0}).$

In [BBan] by Bär and Bandara, the authors show that there exists an
_admissible cut_ , which is an $r\in\mathbb{R}$ so that $A_{r}:=A-r$ is
invertible $\omega_{r}$-bisectorial. This means that $\mathrm{spec}(A_{r})$ is
contained in a closed bisector
$S_{\omega_{r}}=\left\\{\zeta\in\mathbb{C}:\pm\arg\zeta\leq\omega_{r}\right\\}$
of angle $\omega_{r}<\pi/2$ in the complex plane, and that for
$\mu\in(\omega_{r},\pi/2)$, there exists a constant $C_{\mu}>0$ so that
whenever $\zeta$ is outside of the closed bisector $S_{\mu}$ of angle $\mu$,
we have that $|\zeta|\|(\zeta-A_{r})^{-1}\|\leq C_{\mu}$. The framework in
[BBan] extends the work of Bär and Ballmann in [BB] where they make the
additional assumption that $A$ can be chosen self-adjoint.

Given that the spectrum of $A_{r}$ is contained in the open left and right
half-planes, a consequence of the bisectoriality of $A_{r}$ is that we can
consider spectral projectors $\chi^{\pm}(A_{r})$, where $\chi^{\pm}(\zeta)=1$
for $\pm\mathrm{Re}\ \zeta>0$ and $0$ otherwise. Not only do these spectral
projectors exist, they are pseudo-differential operators of order zero (see
[grubbsec]). This means that they act boundedly across all Sobolev scales and
we define the space

$\check{\mathrm{H}}_{A}(D):=\chi^{-}(A_{r}){\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)\oplus\chi^{+}(A_{r}){\rm
H}^{\rm-\frac{1}{2}}(\Sigma;E).$

The following is an important property of this space that we will use later.
We refer its proof to Appendix A.

######  Lemma 2.28.

Let $A$ be an adapted boundary operator for a first order elliptic operator
acting on a manifold with boundary of dimension $>1$. For any admissible cut
$r\in\mathbb{R}$, the spaces $\chi^{\pm}(A_{r}){\rm L}^{2}(\Sigma;E)$ are
infinite dimensional. Moreover, the space
$\check{\mathrm{H}}_{A}(D)\neq\left\\{0\right\\}$ is an infinite-dimensional
space with

${\rm
H}^{\rm{\frac{1}{2}}}(\Sigma;E)\subseteq\check{\mathrm{H}}_{A}(D)\subseteq{\rm
H}^{\rm-{\frac{1}{2}}}(\Sigma;E),$

and each inclusion is dense.

In [BBan, Theorem 2.3], the trace map $\gamma:u\mapsto u{{\lvert}}_{\Sigma}$,
initially defined on ${\rm C}^{\infty}(\Sigma;E)$, is extended to a bounded
surjection

$u\mapsto
u{{\lvert}}_{\Sigma}:\mathrm{dom}(D_{\max})\to\check{\mathrm{H}}_{A}(D)$

with kernel $\ker(u\mapsto u{{\lvert}}_{\Sigma})=\mathrm{dom}(D_{\min})$. That
is, $\check{\mathrm{H}}(D)=\check{\mathrm{H}}_{A}(D)$ with $\gamma$ is a
Cauchy data space. The topology of $\check{\mathrm{H}}(D)$ is given purely in
terms of an associated differential operator on the boundary, namely, the
adapted boundary operator $A$. We give a direct argument for the equality
$\check{\mathrm{H}}(D)=\check{\mathrm{H}}_{A}(D)$ below in Example 3.33.

A pseudo-differential projection $P$ of order zero, for which
$1-P-\chi^{+}(A_{r})$ is a Fredholm operator induces a regular boundary
condition via $B_{P}=(1-P){\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$. In particular,
for any admissible cut $r\in\mathbb{R}$, $P_{r}=\chi^{+}(A_{r})$ is such a
projection. That is, $B_{r}=\chi^{-}(A_{r}){\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$
is a regular boundary condition. In particular, the (generalised) APS-
realisation $D_{\rm APS}$, defined from

$\mathrm{dom}(D_{\rm APS}):=\\{u\in H^{1}(M;E):\chi^{+}(A_{r})\gamma(u)=0\\},$

is regular. For notational convenience, we assume $r=0$ and drop it from the
notation for the remainder of the paper. The reader can find further examples
of boundary conditions for first order elliptic operators in [BB, BBan,
leschgor] and [G99, Section 4].

###  2.7. Scalar properly elliptic operators

Let us consider an example that historically has played an important role and
has long been well understood, see for instance [lionsmagenes, grubb68,
vishik]. We consider an elliptic differential operator $D$ of order $2m$
acting between sections of $E$ and $F$. We say that the differential operator
is scalar if $E$ and $F$ are line bundles; the terminology is justified by the
fact that $\mathrm{Hom}(E,F)$ is a line bundle (trivialisable at $\Sigma$ due
to the existence of $A_{0}\in C^{\infty}(\Sigma;\mathrm{Aut}(E,F))$), and as
such the symbol calculus is completely scalar. We recall the notion of a
properly elliptic scalar operator, see [schechter59].

######  Definition 2.29.

Let $D$ be a scalar elliptic differential operator of order $2m$. We say that
$D$ is properly elliptic if for each $(x^{\prime},\xi^{\prime})\in
S^{*}\Sigma$, the polynomial equation

$\sigma(D)|_{\Sigma}(x^{\prime},\xi^{\prime},z)=0,$ (2)

has exactly $m$ complex solutions with positive imaginary part.

######  Remark 2.30.

Let us consider some special cases where scalar elliptic differential
operators are automatically properly elliptic. If $\sigma(D)|_{\Sigma}$ is
real-valued, then $z$ is a solution of Equation (2) if and only if $\bar{z}$
is, so real valued principal symbol ensures that a scalar elliptic
differential operator is properly elliptic.

If $\dim(M)>2$, the fibres of the cosphere bundle of $\Sigma$ are connected.
Since $D$ is elliptic, Equation (2) has no real solutions and as such the
number of solutions with positive imaginary part is locally constant. We note
that if $z$ is a solution to Equation (2) at $(x^{\prime},\xi^{\prime})$, then
$-z$ is a solution to Equation (2) at $(x^{\prime},-\xi^{\prime})$ because of
the symmetry condition
$\sigma(D)|_{\Sigma}(x^{\prime},\xi^{\prime},z)=\sigma(D)|_{\Sigma}(x^{\prime},-\xi^{\prime},-z)$.
This proves that all scalar elliptic differential operators are properly
elliptic if $\dim(M)>2$.

For a scalar properly elliptic differential operator $D$ of order $2m$, we
define the bundle-like boundary condition $B_{\rm Dir}$ from the subbundle
$E^{\prime}:=E\otimes\mathbb{C}^{m}$, viewed as a subbundle of
$E\otimes\mathbb{C}^{2m}$ by embedding it in the last $m$ coordinates. Compare
to Example 2.25. We write $D_{\rm Dir}:=D_{{\rm B}_{\rm Dir}}$ and call it the
Dirichlet realisation of $D$. More explicitly, we have that

$\mathrm{dom}(D_{\rm Dir})=\\{u\in\mathrm{dom}(D_{\rm
max}):\gamma_{0}u=\gamma_{1}u=\cdots=\gamma_{m-1}u=0\\}.$

The following theorem reformulates several results from [grubb68] to our
setting, the reader is referred to [grubb68] for proofs.

######  Theorem 2.31.

Let $D$ be a scalar properly elliptic differential operator of order $2m$.

  1. (i)

$B_{\rm Dir}$ is a regular boundary condition, and so the realisation $D_{\rm
Dir}$ of $D$ with domain

$\displaystyle\mathrm{dom}(D_{\rm Dir})=$ $\displaystyle{\rm H}^{\rm
2m}(M;E)\cap{\rm H}^{\rm m}_{\rm 0}(M;E)=$ $\displaystyle=$
$\displaystyle\\{u\in{\rm H}^{\rm
2m}(M;E):\gamma_{0}u=\gamma_{1}u=\cdots=\gamma_{m-1}u=0\\},$

is regular.

  2. (ii)

Also $D^{\dagger}$ is properly elliptic and

$D_{\rm Dir}^{*}=D^{\dagger}_{\rm Dir},$

is the realisation of $D^{\dagger}$ with domain ${\rm H}^{\rm 2m}(M;F)\cap{\rm
H}^{\rm m}_{\rm 0}(M;F)$.

  3. (iii)

The map

$\mathcal{C}_{D}\to\bigoplus_{j=0}^{m-1}{\rm H}^{\rm-j-{\frac{1}{2}}}(M;E),$

induced from the projection $\bigoplus_{j=0}^{2m-1}{\rm
H}^{\rm-j-{\frac{1}{2}}}(M;E)\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-j-{\frac{1}{2}}}(M;E)$, is a Fredholm operator.

  4. (iv)

We can write

$\mathrm{dom}(D_{\rm max})={\rm H}^{\rm 2m}(M;E)\cap{\rm H}^{\rm m}_{\rm
0}(M;E)\oplus_{\ker D_{\rm Dir}}\ker(D_{\rm max}),$

and

$\check{\mathrm{H}}(D)=\bigoplus_{j=m}^{2m-1}{\rm H}^{\rm
2m-j-{\frac{1}{2}}}(M;E)\oplus_{\gamma\ker(D_{\rm Dir})}\mathcal{C}_{D}.$

The reader should note that the theorem above holds for more general local
(regular) boundary conditions than Dirichlet conditions, and are studied using
different methods in [grubb68]. We remark that from these type of results,
several Fredholm type results follow readily for properly elliptic boundary
value problems with regular boundary conditions.

##  3\. Calderón projectors and the Douglis-Nirenberg calculus

In this section we will review results of Hörmander and Seeley concerning
Calderón projectors. See more in [horIII, Chapter XX], [seeley65] and also in
[grubbdistop, Chapter 11]. Calderón projections are projections on function
spaces on the boundary $\Sigma$ onto the space of boundary values of
homogeneous solutions $Du=0$ in the interior, i.e. the Hardy space
$\mathcal{C}_{D}$. For higher order operators, it is necessary to keep track
of different order traces, so even though the Calderón projection is a matrix
of pseudodifferential operators, its entries will have different order.
However, the orders remain constant along the diagonals in the Calderón
projection which makes it amenable for study via the calculus of Douglis-
Nirenberg.

###  3.1. Approximate Calderón projectors

First, we will in this subsection recall some constructions found in
Hörmander’s book [horIII]. It will provide a way of constructing the full
symbol of the Calderón projection.

As above, we write $x_{n}$ for the coordinate near the boundary defined from
the transversal vector field $\vec{T}$. The coordinate $x_{n}$ identifies a
neighbourhood of the boundary with a collar $\Sigma\times[0,1]$ with the
boundary defined by the equation $x_{n}=0$. Consider the differential operator
$D_{x_{n}}=-i\partial_{x_{n}}$ defined in the collar neighbourhood of the
boundary. Near the boundary, we can write

$D=\sum_{j=0}^{m}A_{j}D_{x_{n}}^{m-j}.$

For each $j$, $A_{j}=(A_{j}(x_{n}))_{x_{n}\in[0,1]}$ is a family of
differential operators ${\rm C}^{\infty}(\Sigma,E)\to{\rm
C}^{\infty}(\Sigma,F)$ of order $j$. Let $a$ denote the principal symbol of
$D$ and $a_{j}$ the principal symbol of $A_{j}$. In coordinates
$x=(x^{\prime},x_{n})$ on $\Sigma\times[0,1]$ with associated cotangent
coordinates $\xi=(\xi^{\prime},\xi_{n})$ on $T^{*}M|_{\Sigma}$, near $\Sigma$
we can write

$a(x^{\prime},x_{n},\xi^{\prime},\xi_{n})=\sum_{j=0}^{m}a_{j}(x^{\prime},x_{n},\xi^{\prime})\xi_{n}^{m-j}.$

We define the boundary symbol $\sigma_{\partial}(A)$ as the ordinary
differential operator valued function on $T^{*}\Sigma$ given by

$\sigma_{\partial}(D)(x^{\prime},\xi^{\prime}):=\sum_{j=0}^{m}a_{j}(x^{\prime},0,\xi^{\prime})D_{t}^{m-j}.$
(3)

We consider $\sigma_{\partial}(D)$ as an element of ${\rm
C}^{\infty}(T^{*}\Sigma,\mathrm{Hom}(E|_{\Sigma},F|_{\Sigma})\otimes\mathcal{L}({\rm
C}^{\infty}[0,\infty)))$. The conormal symbol of $D$ is the symbol of the
boundary symbol

$\sigma_{\rm
cn}(D)(x^{\prime},\xi^{\prime},\xi_{n}):=\sum_{j=0}^{m}a_{j}(x^{\prime},0,\xi^{\prime})\xi_{n}^{m-j},$
(4)

so that $\sigma_{\partial}(D)(x^{\prime},\xi^{\prime})=\sigma_{\rm
cn}(D)(x^{\prime},\xi^{\prime},D_{t})$ and $\sigma_{\rm
cn}(D)=\sigma(D)|_{\Sigma}$. For more details on the yoga of boundary symbols,
see [gerdsgreenbook, rempelschulze, elmarbdmover] or [melroseAPS] for related
notions.

The following result follows from [horIII, Theorem XX.1.3], which we state
below as Theorem 3.8.

######  Proposition 3.1.

Let $D$ be elliptic of order $m>0$. The following construction produces a
well-defined vector bundle

$E_{+}(D)\to S^{*}\Sigma.$

Define $E_{+}(D)$ as the subset
$E_{+}(D)\subseteq\pi^{*}(E)|_{S^{*}\Sigma}\otimes\mathbb{C}^{m}$ whose fibre
over $(x^{\prime},\xi^{\prime})\in S^{*}\Sigma$ consists of
$(v_{0},v_{1},\ldots,v_{m-1})\in E_{x}\otimes\mathbb{C}^{m}$ such that the
solution $v\in{\rm C}^{\infty}([0,\infty),E_{x})$ of the ordinary differential
equation

$\begin{cases}\sigma_{\partial}(D)(x^{\prime},\xi^{\prime})v(t)=0,\;t>0,\\\
D^{j}_{t}v(0)=v_{j},\;j=0,\ldots,m-1,\end{cases}$ (5)

is exponentially decaying as $t\to+\infty$.

The subset $E_{-}(D)\subseteq\pi^{*}(E)|_{S^{*}\Sigma}$ given as the set of
vectors whose solution to (5) is exponentially decaying as $t\to-\infty$ is a
well-defined vector bundle defining a complement to $E_{+}(D)$ in
$\pi^{*}(E)|_{S^{*}\Sigma}\otimes\mathbb{C}^{m}$.

######  Definition 3.2.

Let $p_{+}(D)\in{\rm
C}^{\infty}(S^{*}\Sigma,\mathrm{Hom}(\pi^{*}(E)|_{S^{*}\Sigma}\otimes\mathbb{C}^{m}))$
denote the projection onto $E_{+}(D)$ along $E_{-}(D)$ and
$p_{-}(D):=1-p_{+}(D)$ its complementary projection.

######  Proposition 3.3.

Let $D$ be elliptic of order $m>0$. It holds that

$p_{+}(D)=\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}^{-1})p_{-}(D^{\dagger})^{*}\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}),$

where  a is the matrix of differential operators from Proposition 2.15 and
$\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})$ is its principal symbol (in the
Douglis-Nirenberg calculus we review below)

$\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})=-i\begin{pmatrix}\sigma_{m-1}(A_{m-1})&\sigma_{m-2}(A_{m-2})&\cdots&\sigma_{2}(A_{2})&\sigma_{1}(A_{1})&A_{0}\\\
\sigma_{m-2}(A_{m-2})&\sigma_{m-3}(A_{m-3})&\cdots&\sigma_{1}(A_{1})&A_{0}&0\\\
\sigma_{m-3}(A_{m-3})&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern
7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}&&A_{0}&0&\vdots\\\ \vdots&&\mathinner{\mkern
1.0mu\raise 1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise
4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&0&\vdots&\\\
\sigma_{1}(A_{1})&A_{0}&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern
7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}&&&\\\ A_{0}&0&\cdots&0&0&0\end{pmatrix}.$

In particular, $\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})$ induces
isomorphisms

$E_{\pm}(D)\cong E_{\mp}(D^{\dagger}).$

We postpone the proof of Proposition 3.3 until after the statement of Theorem
3.8 below.

######  Proposition 3.4.

Let $D$ be an elliptic differential operator of order $m>0$ acting on a
manifold of $\dim(M)>1$. The vector bundles $E_{+}(D),E_{-}(D)\to S^{*}\Sigma$
are non-trivial over each component of $S^{\ast}\Sigma$.

###### Proof.

We note that $\sigma_{\rm
cn}(D)(x^{\prime},\xi^{\prime},z)=(-1)^{m}\sigma_{\rm
cn}(D)(x^{\prime},-\xi^{\prime},-z)$, so $z\in\mathbb{C}$ solves
$\det\sigma_{\rm cn}(D)(x^{\prime},\xi^{\prime},z)=0$ if and only if
$\det\sigma_{\rm cn}(D)(x^{\prime},-\xi^{\prime},-z)=0$. Since
$E\otimes\mathbb{C}^{m}=E_{+}(D)\oplus E_{-}(D)$ by Proposition 3.1, we
conclude that given $(x^{\prime},\xi^{\prime})\in S^{*}\Sigma$, either
$E_{+}(D)_{(x^{\prime},\xi^{\prime})}\neq 0$ or
$E_{-}(D)_{(x^{\prime},\xi^{\prime})}\neq 0$. Therefore, given
$(x^{\prime},\xi^{\prime})\in S^{*}\Sigma$ then at least one of the vector
spaces $E_{+}(D)_{(x^{\prime},\xi^{\prime})}$ and
$E_{+}(D)_{(x^{\prime},-\xi^{\prime})}$ is non-zero and at least one of the
vector spaces $E_{-}(D)_{(x^{\prime},\xi^{\prime})}$ and
$E_{-}(D)_{(x^{\prime},-\xi^{\prime})}$ is non-zero. ∎

######  Remark 3.5.

We remark that if $\dim(M)>2$, then $\pi_{0}(S^{*}\Sigma)=\pi_{0}(\Sigma)$ and
in this case $E_{+}(D),E_{-}(D)\to S^{*}\Sigma$ are non-trivial over each
component of $S^{*}\Sigma$. If $\dim(M)=2$, the “odd-to-even” part of a spinc
Dirac operator twisted by a line bundle provides an example where $E_{+}(D)$
and $E_{-}(D)$ satisfy $\mathrm{rk}(E_{+}(D))+\mathrm{rk}(E_{-}(D))=1$ but are
non-vanishing over each connected component of the boundary; their fibrewise
supports are on complementary components of
$S^{*}\Sigma=S^{1}\times\Sigma\dot{\cup}S^{1}\times\Sigma$.

To construct the Calderón projection, we use the Douglis-Nirenberg calculus.
The reader is referred to [agmon, grubb77], see also [horIII, Chapter XIX.5],
for details. We introduce some notation. If $\boldsymbol{E}\to\Sigma$ is a
Hermitian vector bundle orthogonally graded by $\mathbb{R}$ (viewed as a
discrete group), for $s\in\mathbb{R}$ we define the graded Sobolev space

${\mathbb{H}}^{{s}}(\Sigma,\boldsymbol{E}):=\oplus_{s^{\prime}+t=s}{\rm
H}^{\rm s^{\prime}}(\Sigma,\boldsymbol{E}[t]),$

where $\boldsymbol{E}=\oplus_{t}\boldsymbol{E}[t]$ is the grading of
$\boldsymbol{E}$. If $\boldsymbol{E},\boldsymbol{F}\to\Sigma$ are two
Hermitian vector bundles orthogonally graded by $\mathbb{R}$ (viewed as a
discrete group), we can define the space $\Psi^{\boldsymbol{m}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ of Douglis-Nirenberg operators from
$\boldsymbol{E}$ to $\boldsymbol{F}$ of order $m\in\mathbb{R}$ as the subspace
of classical pseudodifferential operators $\Psi_{\rm
cl}^{*}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ that act homogeneously jointly
in the Sobolev degree and the gradings of the vector bundles. More precisely,
writing $\boldsymbol{E}=\oplus_{k=1}^{n_{E}}E[s_{k}]$ and
$\boldsymbol{F}=\oplus_{j=1}^{n_{F}}F[r_{j}]$ for some
$s_{1},\ldots,s_{n_{E}},r_{1},\ldots,r_{n_{F}}\in\mathbb{R}$, a matrix
$(A_{jk})_{j=1,\ldots,n_{E},k=1,\ldots,n_{F}}\in\Psi_{\rm
cl}^{*}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ belongs to
$\Psi^{\boldsymbol{\alpha}}_{\rm cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ if
and only if the order $\alpha_{j,k}$ of $A_{j,k}$ satisfies
$\alpha_{j,k}=\alpha-r_{k}+s_{j}$ for all $j$ and $k$. By construction,
$\Psi^{-\boldsymbol{\infty}}(\Sigma;\boldsymbol{E},\boldsymbol{F}):=\cap_{s\in\mathbb{R}}\Psi^{-\boldsymbol{s}}(\Sigma;\boldsymbol{E},\boldsymbol{F})=\Psi^{-\infty}(\Sigma;\boldsymbol{E},\boldsymbol{F})$
– the ordinary smoothing operators. We write $\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E}):=\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{E})$.

We define the principal symbol $\sigma_{\boldsymbol{\alpha}}(A)$ to be the
matrix of principal symbols $(\sigma_{\alpha_{j,k}}(A_{j,k}))_{j,k}$ which
defines an element of ${\rm
C}^{\infty}(S^{*}\Sigma,\mathrm{Hom}(\pi^{*}\boldsymbol{E},\pi^{*}\boldsymbol{F}))$.
We say that $A\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ is elliptic if
$\sigma_{\boldsymbol{\alpha}}(A)$ is invertible in all points, i.e. that
$\sigma_{\boldsymbol{\alpha}}(A)\in{\rm
C}^{\infty}(S^{*}\Sigma,\mathrm{Iso}(\pi^{*}\boldsymbol{E},\pi^{*}\boldsymbol{F}))$.
By the standard construction, $A\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ is elliptic if and only if there
exists a parametrix $B\in\Psi^{-\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{F},\boldsymbol{E})$.

By construction, any $A\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ defines a continuous operator

$A:{\mathbb{H}}^{{s}}(\Sigma;\boldsymbol{E})\to{\mathbb{H}}^{{s^{\prime}}}(\Sigma;\boldsymbol{F}),$

for any $s,s^{\prime}\in\mathbb{R}$ with $s^{\prime}\leq s-\alpha$. If
$\sigma_{\boldsymbol{\alpha}}(A)\neq 0$, this operator is compact if and only
if $s^{\prime}<s-\alpha$. For $\sigma_{\boldsymbol{\alpha}}(A)=0$, this
operator is compact whenever $s^{\prime}\leq s-\alpha$. This operator is
Fredholm if and only if $A$ is elliptic (cf. [horIII, Theorem XIX.5.1 and
XIX.5.2]).

The graded vector bundles we shall be concerned with take the form
$E\otimes\mathbb{C}^{m}$ for a trivially graded Hermitian vector bundle $E$.
We shall tacitly grade this vector bundle as follows. We view $\mathbb{C}^{m}$
as a graded bundle by

$\mathbb{C}^{m}[s]=\begin{cases}\mathbb{C},\;&\mbox{if $s=0,1,\ldots,m-1$},\\\
0,\;&\mbox{otherwise}.\end{cases}.$

We grade $E\otimes\mathbb{C}^{m}$ as a tensor product, i.e.

$(E\otimes\mathbb{C}^{m})[s]=\begin{cases}E,\;&\mbox{if
$s=0,1,\ldots,m-1$},\\\ 0,\;&\mbox{otherwise}.\end{cases}$

In particular,

${\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})=\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm s-j}(\Sigma;E).$

We also introduce the graded bundle $\mathbb{C}^{m}_{\rm op}$ which is the
trivial vector bundle of rank $m$, graded by

$\mathbb{C}^{m}_{\rm
op}[s]=\mathbb{C}^{m}[-s]=\begin{cases}\mathbb{C},\;&\mbox{if
$s=0,-1,\ldots,-m+1$},\\\ 0,\;&\mbox{otherwise}.\end{cases}.$

The reader should note that $\mathbb{C}^{m}_{\rm op}$ can be identified with
the graded dual bundle of $\mathbb{C}^{m}$, and the Hermitian structure of $E$
allow us to identify $E\otimes\mathbb{C}^{m}_{\rm op}$ with the graded dual
bundle of $E\otimes\mathbb{C}^{m}$. In particular,

${\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})=\bigoplus_{j=0}^{m-1}{\rm H}^{\rm s+j}(\Sigma;E).$

We can conclude the following proposition.

######  Proposition 3.6.

The ${\rm L}^{2}$-pairing induces a perfect pairing

${\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})\times{\mathbb{H}}^{-{s}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\to\mathbb{C},$

for any $s\in\mathbb{R}$.

######  Definition 3.7.

Let $D$ be an elliptic differential operator of order $m>0$ on a manifold $M$
with boundary $\Sigma$ acting between sections of the Hermitian vector bundles
$E$ and $F$. An approximate Calderón projection for $D$ is a classical pseudo-
differential operator $Q\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ which is order zero in the Douglis-
Nirenberg sense and satisfies the following

  * •

$Q^{2}-Q$ is smoothing.

  * •

$Q$ is an approximate projection onto the Hardy space $\mathcal{C}_{D}$ in the
following sense:

    1. (1)

If $u\in\ker(D_{\rm max})$ then $v:=\gamma
u\in{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ satisfies
that $v-Qv\in{\rm C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m})$.

    2. (2)

For any $v\in{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
there is a $u\in\ker(D_{\rm max})+{\rm C}^{\infty}(M,E)$ (smooth up to the
boundary) such that $Qv=\gamma u$.

######  Theorem 3.8 ([horIII], Theorem XX.1.3).

Let $D$ be an elliptic differential operator of order $m>0$. There exists an
approximate Calderón projection $Q\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ for $D$ with principal symbol given by

$\sigma_{\boldsymbol{0}}(Q)=p_{+}(D),$

(as defined in Defintion 3.2) and the full symbol of
$Q=(Q_{j,k})_{j,k=0}^{m-1}$ computed from the formula [horIII, Equation
XX.1.7]:

$Q_{j,k}f=\sum_{l=0}^{m-1-k}(-i)^{j+l+1}\gamma_{j}\circ
T[A_{m-l-k-1}f\otimes\delta_{t=0}^{(l)}],$ (6)

where $T$ is a pseudodifferential parametrix to $D$ computed in a
neighbourhood of $\Sigma$.

######  Remark 3.9.

We remark that the formula for $Q_{j,k}$ might seem difficult to parse at
first. But the fact that the trace mappings are from inside of $M$ makes it
possible to compute the full symbol by means of residue calculus and the
Paley-Wiener theorem. For instance, [horIII, Equation XX.1.8] computes the
principal symbol of $Q$ by residue calculus as

$\displaystyle\sigma_{\boldsymbol{0}}(Q)_{j,k}(x^{\prime},\xi^{\prime})\equiv$
$\displaystyle\sigma_{k-j}(Q_{j,k})(x^{\prime},\xi^{\prime})=$ (7)
$\displaystyle=$
$\displaystyle\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\sum_{l=0}^{m-k-1}\xi_{n}^{j+l}a(x^{\prime},0,\xi,\xi_{n})^{-1}a_{m-k-l-1}(x^{\prime},0,\xi^{\prime})\right]$

where $(x^{\prime},\xi^{\prime})$ are coordinates on $T^{*}\Sigma$ and
$((x^{\prime},t),(\xi^{\prime},\xi_{n}))$ are coordinates on $T^{*}M$ near
$\Sigma$. The proof of [horIII, Theorem XX.1.3] identifies this expression
with $p_{+}(D)$. The lower order terms in $Q$ are, in general, hard to
compute. We shall give some examples of such computations below for order 1
operators. We can condense the formula (7) into

$\displaystyle\sigma_{\boldsymbol{0}}(Q)(x^{\prime},\xi^{\prime})=\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})\right],$
(8)

where $e(\xi_{n}):=v(\xi_{n})\otimes v(\xi_{n})^{*}$ and
$v(\xi_{n})=(\xi_{n}^{j})_{j=0}^{m-1}$. Here, we abuse notation and view
$a^{-1}|_{T^{*}M|_{\Sigma}\setminus\Sigma}$ and $e$ as endomorphisms of
$E|_{\Sigma}\otimes\mathbb{C}^{m}$. Note that
$a^{-1}|_{T^{*}M|_{\Sigma}\setminus\Sigma}$ and $e$ commute because they act
on different factors in the tensor product $E|_{\Sigma}\otimes\mathbb{C}^{m}$.

###### Proof of Proposition 3.3.

In light of Theorem 3.8 and Remark 3.9, the proof consists of an exercise with
boundary symbols and residue calculus. We compute that

$\displaystyle p_{-}(D^{\dagger})^{*}(x^{\prime},\xi^{\prime})=$
$\displaystyle\left(-\sum_{\mathrm{Im}(\xi_{n})<0}\mathrm{Res}_{\xi_{n}}\left[a^{*}(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}_{\dagger})(x^{\prime},\xi^{\prime})\right]\right)^{*}=$
$\displaystyle=$
$\displaystyle-\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\left(a^{*}(x^{\prime},0,\xi^{\prime},\overline{\xi_{n}})^{-1}e(\overline{\xi_{n}})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}_{\dagger})(x^{\prime},\xi^{\prime})\right)^{*}\right]=$
$\displaystyle=$
$\displaystyle-\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}^{*}_{\dagger})(x^{\prime},\xi^{\prime})a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\right]=$
$\displaystyle=$
$\displaystyle\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\right].$

In the first equality we used that $p_{-}(D^{\dagger})$ is the complementary
projection to $p_{+}(D^{\dagger})$ which is the sum of all residues in the
lower half plane, with a minus sign due to the orientation. In the second
equality we used that the adjoint is antilinear, and interchanges the upper
and lower half plane. In the last equality we used Lemma 2.16. We arrive at

$\displaystyle\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})$
$\displaystyle(x^{\prime},\xi^{\prime})^{-1}p_{-}(D^{\dagger})^{*}(x^{\prime},\xi^{\prime})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})=$
$\displaystyle=$
$\displaystyle\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})^{-1}\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\right]\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})=$
$\displaystyle=$
$\displaystyle\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})\right]=p_{+}(D).\qed$

######  Corollary 3.10.

Let $D$ be an elliptic differential operator of order $m>0$ acting on a
manifold of $\dim(M)>1$ and let $Q$ be the approximate Calderón projection
from Theorem 3.8. For any $s\in\mathbb{R}$, the operator

$Q:{\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m}),$
(9)

is continuous but neither $Q$ nor $1-Q$ is compact.

This result should be compared to Lemma A.2 in the appendix.

###### Proof.

Continuity of (9) follows from that $Q$ is order zero in the Douglis-Nirenberg
sense. By standard arguments in pseudodifferential calculus, (9) is compact
for some $s$ if and only if it is compact for all $s$ if and only if the
principal symbol vanishes. The principal symbol of $Q$ is $p_{+}(D)$ which is
non-trivial by Proposition 3.4, so $Q$ is never compact on
${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})$. By the same argument, the
principal symbol of $1-Q$ is $p_{-}(D)$ which is non-trivial by Proposition
3.4, so $1-Q$ is never compact on
${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})$. ∎

######  Theorem 3.11.

Let $D$ be an elliptic differential operator of order $m>0$ on a manifold of
dimension $>1$. Then $\ker(D_{\rm max})$ is infinite-dimensional.

Despite this result being well known and an immediate consequence of results
of Seeley that we recall in the next subsection, let us provide a short proof
that remains at a symbolic level for an approximate Calderón projection.

###### Proof.

By Proposition 2.1, $\ker(D_{\rm max})$ is infinite-dimensional if and only if
the Hardy space $\mathcal{C}_{D}$ is infinite-dimensional. We will argue by
contradiction.

Assume that $\mathcal{C}_{D}$ is finite-dimensional. Since the space

$\mathcal{C}_{D}\subseteq{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

is finite-dimensional it is closed. Consider the operator
$Q:{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
as in Equation (9). By Theorem 3.8, for any
$v\in{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ it holds
that

$Qv\in\mathcal{C}_{D}+{\rm C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m}).$

We conclude that
$Q:{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
factors over the inclusion

$\mathcal{C}_{D}+{\rm
C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m})\hookrightarrow{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

which is compact if $\mathcal{C}_{D}$ is finite-dimensional. Therefore, if
$\mathcal{C}_{D}$ is finite-dimensional then $Q$ is compact on
${\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ which
contradicts Corollary 3.10. ∎

######  Corollary 3.12.

Assume that $(F,h^{F})=(E,h^{E})$ and let $D:{\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(M;E)$ be an elliptic differential operator of order $m>0$ acting
on a manifold of $\dim(M)>1$. We then have that
$\mathrm{spec}(D_{\max})=\mathrm{spec}_{\mathrm{pt}}(D_{\max})=\mathbb{C}$.
Consequently, $\mathrm{spec}(D_{\min})=\mathbb{C}$.

###### Proof.

Fix $\lambda\in\mathbb{C}$, and note that $D-\lambda$ is again an elliptic
differential operator of order $m>0$. Now, Theorem 3.11 implies that
$\dim\ker(D_{\max}-\lambda)=\infty$. But that is exactly that
$\lambda\in\mathrm{spec}_{\mathrm{pt}}(D_{\max})$ and so we conclude that
$\mathrm{spec}(D_{\max})=\mathrm{spec}_{\mathrm{pt}}(D_{\max})=\mathbb{C}$.
For the statement concerning $D_{\min}$, we note that by mirroring this
argument, we obtain that $\mathrm{spec}(D_{\max}^{\dagger})=\mathbb{C}$ and
since $D_{\min}=(D_{\max}^{\dagger})^{\ast}$, we obtain that
$\mathrm{spec}(D_{\min})=\mathrm{spec}(D_{\max}^{\dagger})^{\mathrm{conj}}=\mathbb{C}$.
∎

######  Remark 3.13.

For $D$ with $\ker(D_{\min})=0$ (i.e., first order Dirac-type operators), we
have that $0\not\in\mathrm{spec}_{\mathrm{pt}}(D_{\min})$. However, from
Corollary 3.12, we know that $0\in\mathrm{spec}(D_{\min})$. In fact,
$0\in\mathrm{spec}_{\mathrm{res}}(D_{\min})$, i.e., it is in the residue
spectrum. This is because we have that ${\rm
L}^{2}(M;E)=\ker(D_{\max}^{\dagger})\oplus\mathrm{ran}(D_{\min})$ from
Proposition 2.6 which shows that $D_{\min}:\mathrm{dom}(D_{\min})\to{\rm
L}^{2}(M;E)$ does not have dense range. In fact, the range fails to be dense
on the infinite dimensional subspace $\ker(D_{\max}^{\dagger})$. However,
despite these shortcomings, and a complete lack of spectral theory,
Proposition 2.20 shows that the Dirichlet problem is indeed well-posed when
restricted to $\mathrm{ran}(D_{\min})=\mathrm{ran}(D_{\mathcal{C}_{D}})$ (see
more in Subsection 2.4).

######  Corollary 3.14.

Assume that $(F,h^{F})=(E,h^{E})$ and that $D:{\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(M;E)$ is first order elliptic and Dirac-type. Then,
$\mathrm{spec}(D_{\min})=\mathrm{spec}_{\mathrm{res}}(D_{\min})=\mathbb{C}$.

###### Proof.

For $\lambda\in\mathbb{C}$, the operator $D_{\lambda}=D-\lambda$ is again
first order Dirac-type. By Remark 3.13, we conclude that
$0\in\mathrm{spec}_{\mathrm{res}}((D_{\lambda})_{\min})$. But
$(D_{\lambda})_{\min}=D_{\min}-\lambda$ and so the conclusion holds. ∎

###  3.2. Seeley’s results on Calderón projectors

######  Definition 3.15.

Let $D$ be an elliptic differential operator of order $m>0$ on a manifold $M$
with boundary $\Sigma$ acting between sections of the Hermitian vector bundles
$E$ and $F$. A Calderón projection for $D$ is an approximate Calderón
projection $P_{\mathcal{C}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ such that it is a projection onto

$\mathcal{C}_{D}\subseteq{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$

######  Remark 3.16.

Calderón projectors are not uniquely determined. For instance, if
$P_{\mathcal{C}}$ is a Calderón projection, then for any self-adjoint
smoothing operator $R$ which is “small”333How small can be made precise by
pointwise estimates on the kernel, but it is irrelevant for the sake of this
argument. and $R|_{\mathcal{C}_{D}}=0$ we have that
$P_{\mathcal{C}}(1+R)^{-1}$ also satisfies the conditions of the theorem. _We
shall tacitly use only the Calderón projection constructed in the next theorem
of Seeley._

Let us recall the following theorem of Seeley. We remark that Seeley proved
this result in the larger generality of higher regularity (cf. Theorem 6.1
below) on ${\rm L}^{p}$-spaces, $1<p<\infty$. For $p\neq 2$, the spaces on the
boundary are Besov spaces and not Sobolev spaces.

######  Theorem 3.17.

[Seeley [seeley65]] Let $D$ be an elliptic differential operator of order
$m>0$ on a manifold $M$ with boundary $\Sigma$ acting between sections of the
Hermitian vector bundles $E$ and $F$. Then there exists a Calderón projection
$P_{\mathcal{C}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ whose full symbol is uniquely determined
as that of the approximate Calderón projection in Theorem 3.8. In particular,

$\sigma_{\boldsymbol{0}}(P_{\mathcal{C}})=p_{+}(D).$

In fact, Seeley obtained even stronger results in [seeley65] that we partly
summarise in the following theorem.

######  Theorem 3.18.

Let $D$ be an elliptic differential operator of order $m>0$ on a manifold $M$
with boundary $\Sigma$ acting between sections of the Hermitian vector bundles
$E$ and $F$. There exists a Calderón projection $P_{\mathcal{C}}$ such that

  1. (i)

There exists a continuous Poisson operator

$\mathcal{K}:{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\to\ker(D_{\rm
max}),$

with $P_{\mathcal{C}}=\gamma\circ\mathcal{K}$ (and in particular,
$\mathcal{K}=\mathcal{K}\circ P_{\mathcal{C}}$).

  2. (ii)

The image of

$\gamma:\mathrm{dom}(D_{\rm
max})\to{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

is precisely the subspace

$P_{\mathcal{C}}{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus(1-P_{\mathcal{C}}){\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$

In particular, the identity map induces an isomorphism of Banach spaces

$\check{\mathrm{H}}(D)\cong
P_{\mathcal{C}}{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus(1-P_{\mathcal{C}}){\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$

###### Proof.

The first item is proven in [seeley65]. The second item is a reformulation of
a statement on [seeley65, top of page 782] which was stated but not proven
there. We include its proof for the convenience of the reader.

To prove the second item, it suffices to prove that

$(1-P_{\mathcal{C}})\gamma\mathrm{dom}(D_{\rm
max})\subseteq{\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$

We can assume that $M$ is a domain with smooth boundary in a compact manifold
$\hat{M}$, that $E$ and $F$ extend to Hermitian vector bundles
$\hat{E},\hat{F}\to\hat{M}$ and that there exists an elliptic differential
operator $\hat{D}$ of order $m$ extending $D$ to $\hat{M}$. Write $D^{\prime}$
for the elliptic differential operator of order $m$ defined from restricting
$\hat{D}$ to $M^{c}$. Take a $g\in(1-P_{\mathcal{C}})\gamma\mathrm{dom}(D_{\rm
max})\subseteq\gamma\mathrm{dom}(D_{\rm max})$ and pick
$f\in\mathrm{dom}(D_{\rm max})$ lifting $g$. Since
$g\in(1-P_{\mathcal{C}})\gamma\mathrm{dom}(D_{\rm max})$, [seeley65, Lemma 5]
implies that there exists an $f^{\prime}\in\ker(D^{\prime})\subseteq{\rm
L}^{2}(M^{c},\hat{E}|_{M^{c}})$ lifting $g$. We define $F\in{\rm
L}^{2}(\hat{M},E)$ as $f$ on $M$ and as $f^{\prime}$ on $M^{c}$. Since
$\gamma(f)=g=\gamma(f^{\prime})$, we have that $F\in\mathrm{dom}(\hat{D}_{\rm
max})$. By elliptic regularity on the closed manifold $\hat{M}$, it holds that
$\mathrm{dom}(\hat{D}_{\rm max})={\rm H}^{\rm m}(\hat{M},E)$. We conclude that
$f\in{\rm H}^{\rm m}(M,E)$ and therefore
$g=\gamma(f)\in{\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
∎

######  Remark 3.19.

It is also proven in [seeley65] that the complementary projection
$(1-P_{\mathcal{C}})$ is (up to a finite rank projection) a projection onto
the Hardy space of the complementary manifold with boundary
$\hat{M}\setminus\mathring{M}$. Note that Seeley’s construction depends on the
choice of the closed manifold $\hat{M}$ and the extension of $D$ to an
elliptic operator $\hat{D}$ on $\hat{M}$. In other words, the Calderon
projection is a projection onto the Hardy space along the Hardy space of the
complement. For this reason, the Calderon projection was denoted by $P^{+}$ in
[seeley65] and $C^{+}$ in [grubbdistop]. Our notation $P_{\mathcal{C}}$ is
chosen to avoid confusion with the notation for spectral projectors from [BB,
BBan].

We remark that a Calderon projection $P_{\mathcal{C}}$ such that
$(1-P_{\mathcal{C}})$ is (up to a finite rank projection) a projection onto
the Hardy space of the complementary manifold is uniquely determined modulo
smoothing operator. We therefore implicitly will use the Calderon projection
$P_{\mathcal{C}}$ from Theorem 3.18 throughout the paper.

######  Corollary 3.20.

Let $D$ be an elliptic differential operator of order $m>0$. The short exact
sequence of Hilbert spaces

$0\to\mathrm{dom}(D_{\rm min})\to\mathrm{dom}(D_{\rm
max})\xrightarrow{\gamma}\check{\mathrm{H}}(D)\to 0,$

is split by the continuous mapping

$E:\check{\mathrm{H}}(D)\to\mathrm{dom}(D_{\rm max}),\quad
E=\mathcal{K}+E_{0}\circ(1-P_{\mathcal{C}}),$

where
$E_{0}:{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\rm
H}^{\rm m}(M;E)$ is any continuous splitting of the trace mapping ${\rm
H}^{\rm
m}(M;E)\to{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

###### Proof.

We have that $E=EP_{\mathcal{C}}+E(1-P_{\mathcal{C}})$. The operators
$EP_{\mathcal{C}}=\mathcal{K}$ and
$E(1-P_{\mathcal{C}})=E_{0}(1-P_{\mathcal{C}})$ are continuous by Theorem
3.17, so $E$ is continuous. Moreover, we have that $\gamma
E=\gamma\mathcal{K}+\gamma
E_{0}(1-P_{\mathcal{C}})=P_{\mathcal{C}}+(1-P_{\mathcal{C}})=1$ so $E$ is a
splitting. ∎

######  Example 3.21.

Let us provide some further context for Example 2.11. We again consider the
generalised boundary condition $B_{\rm Sob}=\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma;E)$ for an elliptic differential operator $D$ of
order $m>0$. Letting
$\mathrm{P}_{\mathcal{C}^{\mathrm{c}}}=(I-\mathrm{P}_{\mathcal{C}})$, the
complementary projection to $\mathrm{P}_{\mathcal{C}}$,

$B_{\rm
Sob}+\mathcal{C}_{D}=\mathrm{P}_{\mathcal{C}^{\mathrm{c}}}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus\mathcal{C}_{D}=\mathcal{C}^{\mathrm{c}}\oplus\mathcal{C}=\check{\mathrm{H}}(D),$

by Theorem 3.18. We can now use Theorem B.23 to give an alternate proof of the
fact that $\mathrm{ran}(D_{{\rm B}_{\rm Sob}})$ is closed. We can also use
Lemma B.20 to conclude $\mathrm{ran}(D_{{\rm B}_{\rm
Sob}})=\mathrm{ran}(D_{\max})$ which we formulate in the following corollary.

######  Corollary 3.22.

Let $D$ be an elliptic differential operator of order $m>0$ on a manifold $M$
with boundary $\Sigma$ acting between sections of the Hermitian vector bundles
$E$ and $F$. Then it holds that

$\mathrm{ran}(D_{\max})=D{\rm H}^{\rm m}(M;E),$

as closed subspaces of ${\rm L}^{2}(M;F)$ whose codimension is
$\dim\ker(D^{\dagger}_{\rm min})$.

The next result is another corollary of Theorem 3.18.

######  Corollary 3.23.

Let $D$ be an elliptic differential operator of order $m>0$. The boundary
condition
$\mathcal{C}^{c}:=(1-P_{\mathcal{C}}){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
is regular and satisfies:

  * •

$\mathcal{C}^{c}$ is closed in
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and in
$\check{\mathrm{H}}(D)$ and for $x\in\mathcal{C}^{c}$,

$\|x\|_{{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})}\simeq\|x\|_{\check{\mathrm{H}}(D)}.$

  * •

$\mathcal{C}^{c}\cap\mathcal{C}_{D}=0$ and
$\mathcal{C}^{c}+\mathcal{C}_{D}=\check{\mathrm{H}}(D)$.

  * •

The inclusion maps defines a Banach space isomorphism

$\mathcal{C}^{c}\oplus\mathcal{C}_{D}\cong\check{\mathrm{H}}(D),$

where $\mathcal{C}_{D}$ is equipped with either the norm from
$\check{\mathrm{H}}(D)$ or the norm from
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ (the two are
equivalent on $\mathcal{C}_{D}$).

######  Remark 3.24.

It follows from the proof Theorem 2.9 and Corollary 3.23 that an elliptic
differential operator $D$ admits an invertible realisation if and only if
$\ker(D_{\min})=0$ and $\ker(D_{\min}^{\dagger})=0$. Indeed, for any
realisation $D_{\rm B}$ we have the inclusions
$\ker(D_{\min})\subseteq\ker(D_{\rm B})$ and
$\ker(D_{\min}^{\dagger})=\ker(D_{\rm B}^{*})$ so $\ker(D_{\min})=0$ and
$\ker(D_{\min}^{\dagger})=0$ is a necessary condition. The converse follows
from that $\ker(D_{\min})=0$ and $\ker(D_{\min}^{\dagger})=0$ if and only if
$D_{\mathcal{C}^{c}}$ is invertible by the proof Theorem 2.9 and Corollary
3.23.

For general regular boundary conditions, we have a statement similar to
Corollary 3.23:

######  Proposition 3.25.

Let $D$ be an elliptic differential operator of order $m>0$ and $B$ be a
regular boundary condition. Then the following holds:

  1. (i)

$(B,\mathcal{C}_{D})$ is a Fredholm pair in $\check{\mathrm{H}}(D)$ and

$B^{\ast}\cap\mathcal{C}_{D^{\dagger}}\cong\faktor{\check{\mathrm{H}}(D)}{(B+\mathcal{C}_{D})}.$

  2. (ii)

There is a finite dimensional space
$F\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and a subspace $B_{0}\subseteq B+F$ of finite codimension defining a regular
boundary condition with $\ker D_{\rm B_{0}}=\ker(D_{\min})$ for which we can
write

$\mathrm{dom}(D_{\rm max})=\mathrm{dom}(D_{{\rm B}_{0}})\oplus_{\ker
D_{\min}}\ker D_{\rm max},$

and

$\check{\mathrm{H}}(D)=B_{0}\oplus\mathcal{C}_{D}.$

The contents of item ii) in Theorem 3.25 can be interpreted as the statement
that any regular boundary condition is up to finite dimensional perturbations
a complement to the Hardy space in the Cauchy data space. In light of Theorem
2.12, or rather Corollary B.29, there are constraints on the dimension of $F$
and the codimension of $B_{0}\subseteq B+F$ given by the identity:

$\dim(F)-\dim\left(\faktor{B+F}{B_{0}}\right)=\dim\ker(D_{\min})-\dim\ker(D_{\min}^{\dagger})-\operatorname{ind}(D_{\rm
B}).$

###### Proof.

Item i follows from Theorem 2.12. If $B$ satisfies that $\ker(D_{\rm
B}^{*})=\ker(D_{\rm min}^{\dagger})$ and $\ker D_{\rm B}=\ker(D_{\min})$, then
with $B_{0}=B$ and $F=0$, item ii follows from Corollary B.27 applied to the
FAP with closed range $\mathcal{T}_{\rm B}:=(D_{\rm B},D^{\dagger}_{\rm
min})$.

To prove item ii, we first prove that for any regular boundary condition $B$
there is a finite dimensional space
$F\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
with $(B+F)\cap\mathcal{C}_{D}=B\cap\mathcal{C}_{D}$ and
$B+F+\mathcal{C}_{D}=\check{\mathrm{H}}(D)$. Indeed, $B+F$ is clearly semi-
regular and will be regular since $(B+F)^{*}\subseteq B^{*}$. To construct
such an $F$, we note that item i implies that the inclusion
$\faktor{B}{B\cap\mathcal{C}_{D}}\to\faktor{\check{\mathrm{H}}(D)}{\mathcal{C}_{D}}$
has finite codimension. Indeed, we have that

$\faktor{(\check{\mathrm{H}}(D)/\mathcal{C}_{D})}{(B/B\cap\mathcal{C}_{D})}=\faktor{\check{\mathrm{H}}(D)}{(B+\mathcal{C}_{D})}.$

As such, there are plenty of finite-dimensional subspaces
$F\subseteq\check{\mathrm{H}}(D)$ with
$(B+F)\cap\mathcal{C}_{D}=B\cap\mathcal{C}_{D}$ and
$B+F+\mathcal{C}_{D}=\check{\mathrm{H}}(D)$. That we can take
$F\subseteq\check{\mathrm{H}}(D)\cap{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
follows from Example 3.21 which implies that the inclusion
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\hookrightarrow\faktor{\check{\mathrm{H}}(D)}{\mathcal{C}_{D}}$
induces an isomorphism

$\faktor{{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})}{\left({\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap\mathcal{C}_{D}\right)}\cong\faktor{\check{\mathrm{H}}(D)}{\mathcal{C}_{D}}.$

To finish the proof, we need to construct $B_{0}$. Upon replacing $B$ with
$B+F$, the argument in the preceding paragraph allow us to assume that the
regular boundary condition $B$ satisfies $\ker(D_{\rm B}^{*})=\ker(D_{\rm
min}^{\dagger})$. We need to construct a regular boundary condition
$B_{0}\subseteq B$ of finite codimension such that
$B_{0}+\mathcal{C}_{D}=B+\mathcal{C}_{D}$ and $B_{0}\cap\mathcal{C}_{D}=0$.
Consider any
$B_{0}\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
which forms a complement of $B\cap\mathcal{C}_{D}$ in $B$. Since $B$ is
regular,
$B\cap\mathcal{C}_{D}\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
is finite-dimensional so $B_{0}\subseteq B$ has finite codimension. Moreover,
$B_{0}$ is closed in $\check{\mathrm{H}}(D)$ and by construction
$B_{0}+\mathcal{C}_{D}=B+\mathcal{C}_{D}$ and $B_{0}\cap\mathcal{C}_{D}=0$. It
remains to prove that we can take the complement $B_{0}$ to satisfy that
$B_{0}^{*}\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;F\otimes\mathbb{C}^{m})$.
We have that $B^{*}\subseteq B_{0}^{*}$ has finite codimension and using
Proposition 2.18 we deduce that we can choose $B_{0}$ such that
$\faktor{B_{0}^{*}}{B^{*}}\subseteq\faktor{{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;F\otimes\mathbb{C}^{m})}{B^{*}}$
which proves the theorem. ∎

###  3.3. Boundary decomposing projectors

With Theorem 3.17, we are able to prove results of use when studying graphical
decompositions of regular boundary conditions, see more in [BBan] and
Subsection 5 below. We make the following definition that we shall make use of
in the first order case below in Subsection 5

######  Definition 3.26.

Let $D$ be an elliptic differential operator of order $m>0$. We say that a
projection $\mathcal{P}_{+}$ is boundary decomposing for $D$ if the following
conditions are satisfied

  1. (P1)

$\mathcal{P}_{+}:{\mathbb{H}}^{\alpha}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{\alpha}(\Sigma;E\otimes\mathbb{C}^{m})$
is a bounded projection for
$\alpha\in\left\\{-\frac{1}{2},m-\frac{1}{2}\right\\}$,

  2. (P2)

$\mathcal{P}_{+}:\check{\mathrm{H}}(D)\to\check{\mathrm{H}}(D)$, and
$\mathcal{P}_{-}:=(I-\mathcal{P}_{+}):\check{\mathrm{H}}(D)\to{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
and

  3. (P3)

$\|u\|_{\check{\mathrm{H}}(D)}\simeq\|\mathcal{P}_{-}u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|\mathcal{P}_{+}u\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$

The next proposition follows from the open mapping theorem.

######  Proposition 3.27.

Let $D$ be an elliptic differential operator of order $m>0$ and
$\mathcal{P}_{+}$ a boundary decomposing projection for $D$. Then

$\mathcal{P}_{+}\check{\mathrm{H}}(D)=\mathcal{P}_{+}{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E)\quad\mbox{and}\quad\mathcal{P}_{-}\check{\mathrm{H}}(D)=\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E),$

with equivalent norms.

Let us state yet another corollary of Theorem 3.17.

######  Corollary 3.28.

The Calderón projection $P_{\mathcal{C}}$ of an elliptic differential operator
$D$ is boundary decomposing for $D$. Moreover, the complementary subspace
$\mathcal{C}^{\mathrm{c}}:=(I-\mathrm{P}_{\mathcal{C}})\check{\mathrm{H}}(D)=(1-P_{\mathcal{C}}){\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
defines a regular boundary condition with
$\ker(D_{\mathcal{C}^{\mathrm{c}}})=\ker(D_{\min})$ and characterises
$\mathrm{dom}(D_{\rm max})$ via the following continuously split short exact
sequence of Hilbert spaces

$0\to\mathrm{dom}(D_{\mathcal{C}^{\mathrm{c}}})\to\mathrm{dom}(D_{\rm
max})\xrightarrow{P_{\mathcal{C}}\circ\gamma}\mathcal{C}\to 0.$

###### Proof.

By the fact that $\mathrm{P}_{\mathcal{C}}$ is a pseudo-differential operator
of order zero in the Douglis-Nirenberg calculus, it automatically satisfies 1.
The properties 2 and 3 follow from Theorem 3.18. It follows that
$\mathcal{C}^{\mathrm{c}}$ is a regular boundary condition. Since the
restricted trace map is a surjection
$\ker(D_{\mathcal{C}^{\mathrm{c}}})\to\mathcal{C}^{\mathrm{c}}\cap\mathcal{C}=0$
with kernel $\ker(D_{\min})$, we conclude that
$\ker(D_{\mathcal{C}^{\mathrm{c}}})=\ker(D_{\min})$. ∎

######  Example 3.29.

The Hardy space itself is a boundary condition, so we may consider the closed
operator $D_{\mathcal{C}_{D}}$. The realisation $D_{\mathcal{C}_{D}}$ is
called the soft extension, or the Krein extension, and is self-adjoint if $D$
is formally self-adjoint. Its properties were further studied in [grubbsoft].
It is clear that $\ker(D_{\mathcal{C}_{D}})=\ker(D_{\max})$, which we have
noted is infinite dimensional if $\dim(M)>1$. By Proposition B.10, setting
$B_{2}=\check{\mathrm{H}}(D)$ and $B_{1}=\mathcal{C}_{D}$, we obtain that

$\faktor{\mathrm{dom}(D_{\max})}{\mathrm{dom}(D_{\mathcal{C}_{D}})}\cong\faktor{\check{\mathrm{H}}(D)}{\mathcal{C}_{D}}\cong\mathcal{C}^{\mathrm{c}},$

where $\mathcal{C}^{\mathrm{c}}$ is the kernel of $\mathrm{P}_{\mathcal{C}}$.
By Corollary 3.28, we have that $\mathcal{C}^{\mathrm{c}}$ is infinite
dimensional. That is, $D_{\mathcal{C}}$ is an operator which differs from
$D_{\max}$ on an infinite dimensional subspace but still has infinite
dimensional kernel.

This can be repeated on replacing $\mathcal{C}$ by any infinite dimensional
closed subspace of $\mathcal{C}$. Therefore, there’s an abundance of boundary
conditions with infinite dimensional kernel which is different from $D_{\max}$
on an infinite dimensional subspace.

We now state a sufficient condition for a projection to be boundary
decomposing.

######  Theorem 3.30.

Let $D$ be an elliptic differential operator of order $m>0$ and $P$ a
continuous projection on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. Assume that
$P$ satisfies the following:

  * •

$P$ is continuous also in the $\check{\mathrm{H}}(D)$-norm and the
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$-norm.

  * •

The operator $A:=P_{\mathcal{C}}-(1-P)$ defines a Fredholm operator on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and by
continuity extends to a Fredholm operator on $\check{\mathrm{H}}(D)$.

Then $P$ is boundary decomposing, and in particular

$\|u\|_{\check{\mathrm{H}}(D)}\simeq\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|Pu\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$

###### Proof.

By the open mapping theorem and density it suffices to prove that the
assumptions on $P$ ensure that

$\|(1-P)u\|_{\check{\mathrm{H}}(D)}\simeq\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\quad\mbox{and}\quad\|Pu\|_{\check{\mathrm{H}}(D)}\simeq\|Pu\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})},$

for $u\in{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.