Since we have a continuous embedding
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\hookrightarrow\check{\mathrm{H}}(D)$,
we conclude that there is an estimate
$\|(1-P)u\|_{\check{\mathrm{H}}(D)}\lesssim\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
for $u\in{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. We
now prove the converse estimate. Since $A$ is Fredholm on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$, we conclude
that there is a finite rank operator $F$ on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ such that

$\|u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\simeq\|Au\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|Fu\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$

By density, we can assume that $F$ extends to a finite rank operator on
$\check{\mathrm{H}}(D)$. We then have that

$\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\lesssim\|A(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|F(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$
(10)

But $A(1-P)=(P_{\mathcal{C}}-(1-P))(1-P)=(P_{\mathcal{C}}-1)(1-P)$ so

$\displaystyle\|A(1-P)u\|$
${}_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|F(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}=$
(11) $\displaystyle=$
$\displaystyle\|(1-P_{\mathcal{C}})(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|F(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}=$
$\displaystyle=$
$\displaystyle\|(1-P_{\mathcal{C}})(1-P)u\|_{\check{\mathrm{H}}(D)}+\|F(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\lesssim$
$\displaystyle\lesssim$
$\displaystyle\|(1-P)u\|_{\check{\mathrm{H}}(D)}+\|F(1-P)u\|_{\check{\mathrm{H}}(D)}\lesssim\|(1-P)u\|_{\check{\mathrm{H}}(D)}.$

In the second last estimate we used that all norms on finite-dimensional
spaces are equivalent and in the third estimate we used that $F$ extends to a
finite rank operator on $\check{\mathrm{H}}(D)$. Combining the estimates (10)
and (11) we arrive at the estimate
$\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\lesssim\|(1-P)u\|_{\check{\mathrm{H}}(D)}$.

The proof that
$\|Pu\|_{\check{\mathrm{H}}(D)}\simeq\|Pu\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
goes ad verbatim with $\check{\mathrm{H}}(D)$ instead of
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ instead of
$\check{\mathrm{H}}(D)$ and using that $A$ is Fredholm on
$\check{\mathrm{H}}(D)$. ∎

######  Lemma 3.31.

Let $D$ be an elliptic differential operator of order $m>0$ and
$T\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$. Then $T$
defines a bounded operator on $\check{\mathrm{H}}(D)$ if and only if
$(1-P_{\mathcal{C}})TP_{\mathcal{C}}\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$.

###### Proof.

Assume first that
$(1-P_{\mathcal{C}})TP_{\mathcal{C}}\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ holds. The property
$(1-P_{\mathcal{C}})TP_{\mathcal{C}}\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ implies that

$\displaystyle T=$ $\displaystyle
P_{\mathcal{C}}TP_{\mathcal{C}}+(1-P_{\mathcal{C}})T(1-P_{\mathcal{C}})+P_{\mathcal{C}}T(1-P_{\mathcal{C}})+(1-P_{\mathcal{C}})TP_{\mathcal{C}}$
$\displaystyle=$ $\displaystyle
P_{\mathcal{C}}TP_{\mathcal{C}}+(1-P_{\mathcal{C}})T(1-P_{\mathcal{C}})+P_{\mathcal{C}}T(1-P_{\mathcal{C}})+\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m}).$

It now follows that $T$ acts boundedly on $\check{\mathrm{H}}(D)$ from the
facts that $\Psi^{\boldsymbol{-m}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$
acts boundedly on $\check{\mathrm{H}}(D)$ and $\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ acts boundedly
${\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{{t}}(\Sigma;E\otimes\mathbb{C}^{m})$
for all $s,t\in\mathbb{R}$ with $t\leq s$.

Conversely, assume that $T$ acts boundedly on $\check{\mathrm{H}}(D)$. Since
$T\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$, the
argument above implies that the pseudodifferential operator
$(1-P_{\mathcal{C}})TP_{\mathcal{C}}$ act boundedly on
$\check{\mathrm{H}}(D)$. In particular, $(1-P_{\mathcal{C}})TP_{\mathcal{C}}$
will have to extend to a continuous operator
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and so it must be an operator of order $-m$, i.e.
$(1-P_{\mathcal{C}})TP_{\mathcal{C}}\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$. ∎

######  Corollary 3.32.

Let $D$ be an elliptic differential operator of order $m>0$ and
$P\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$ a
projection. Assume that $P$ satisfies the following:

  * •

$[P_{\mathcal{C}},P]\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$.

  * •

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

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.

Since
$(1-P_{\mathcal{C}})PP_{\mathcal{C}}=-[P_{\mathcal{C}},P]P_{\mathcal{C}}\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$, the projection $P$ extends to a
continuous projection on $\check{\mathrm{H}}(D)$ by Lemma 3.31. Therefore $P$
defines a continuous projection on
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and
$\check{\mathrm{H}}(D)$. Since $A:=P_{\mathcal{C}}-(1-P)$ is elliptic, it
defines a Fredholm operator on
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

Let us now prove that $A$ extends to a Fredholm operator on
$\check{\mathrm{H}}(D)$. Let $R$ denote a parametrix of $A$. We proceed by
computing in the formal symbol algebra $\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})/\Psi^{-\infty}(\Sigma;E\otimes\mathbb{C}^{m})$.
For a zeroth order pseudodifferential operator $T$ we write $\lfloor T\rfloor$
for its associated full symbol. In this notation, $\lfloor R\rfloor=\lfloor
A\rfloor^{-1}$. We compute that

$\displaystyle\lfloor[P_{\mathcal{C}},R]\rfloor=$ $\displaystyle[\lfloor
P_{\mathcal{C}}\rfloor,\lfloor A\rfloor^{-1}]=-\lfloor A\rfloor^{-1}[\lfloor
P_{\mathcal{C}}\rfloor,\lfloor A\rfloor]\lfloor A\rfloor^{-1}=$
$\displaystyle=$ $\displaystyle-\lfloor A\rfloor^{-1}[\lfloor
P_{\mathcal{C}}\rfloor,\lfloor P\rfloor]\lfloor A\rfloor^{-1}=-\lfloor
R[P_{\mathcal{C}},P]R\rfloor.$

Therefore it holds that
$[P_{\mathcal{C}},R]=-R[P_{\mathcal{C}},P]R+\Psi^{-\infty}(\Sigma;E\otimes\mathbb{C}^{m})$.
Since $[P_{\mathcal{C}},P]\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$, we conclude that
$(1-P_{\mathcal{C}})RP_{\mathcal{C}}=-[P_{\mathcal{C}},R]P_{\mathcal{C}}\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ so $R$ acts boundedly on
$\check{\mathrm{H}}(D)$. Therefore, density and the fact that smoothing
operators act compactly on $\check{\mathrm{H}}(D)$ implies that $A$ extends to
a Fredholm operator on $\check{\mathrm{H}}(D)$.

The corollary now follows from Theorem 3.30. ∎

Below in Theorem 4.15 we shall see that under assumptions on $P$ similar to
those in Corollary 3.32,
$B_{P}:=(1-P){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
is a regular boundary condition. For the precise relations between the
assumptions in Theorem 4.15 and Corollary 3.32, see the formulation of Theorem
4.

######  Example 3.33.

Let us briefly comment on the case of a first order elliptic differential
operator $D$. In this case, Corollary 3.32 shows that any
$P\in\Psi^{0}(\Sigma;E)$ with $P$ and $P_{\mathcal{C}}$ commuting at principal
symbol level (as elements of $C^{\infty}(S^{*}M;\mathrm{End}(\pi^{*}E))$) and
with $p_{+}(D)-(1-\sigma_{0}(P))\in
C^{\infty}(S^{*}M;\mathrm{Aut}(\pi^{*}E))$, defines a boundary decomposing
projection. Below in Lemma 9.1, we prove that $p_{+}(D)$ coincides with the
principal symbol of the positive spectral projection $\chi^{+}(A)$ for an
adapted boundary operator. In particular, $P:=\chi^{+}(A)$ defines a boundary
decomposing projection by Corollary 3.32. This proves the claimed equality
$\check{\mathrm{H}}(D)=\check{\mathrm{H}}_{A}(D)$ from Subsection 2.6, thus
reconciling the topological considerations of $\check{\mathrm{H}}(D)$ from
Subsection 3.2 with that of [BBan] in the first order case.

######  Example 3.34.

Consider the case of a second order elliptic differential operator $D$ acting
on a vector bundle $E$ with scalar positive principal symbol (e.g. a Laplace
type operator on a vector bundle). In this case, it is readily verified from
solving Equation (5) that

$\sigma_{\boldsymbol{0}}(P_{\mathcal{C}})(x^{\prime},\xi^{\prime})=\frac{1}{2}\begin{pmatrix}1_{E}&|\xi^{\prime}|^{-1}1_{E}\\\
|\xi^{\prime}|1_{E}&1_{E}\end{pmatrix},$

for the Riemannian metric on $\Sigma$ induced from the Riemannian metric that
$\sigma_{2}(D)$ defines on $M$. We can consider the projectors defining
Dirichlet or Neumann conditions (from the rule
$B_{P}:=(1-P){\mathbb{H}}^{m-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{2})$ or
the explicit condition $P\gamma u=0$) are given by

$P_{D}:=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}\quad\mbox{and}\quad
P_{N}:=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}.$

We then have that

$[P_{D},P_{\mathcal{C}}]=\frac{1}{2}\begin{pmatrix}0&|\xi^{\prime}|^{-1}1_{E}\\\
-|\xi^{\prime}|1_{E}&0\end{pmatrix}\quad\mbox{and}\quad[P_{N},P_{\mathcal{C}}]=\frac{1}{2}\begin{pmatrix}0&-|\xi^{\prime}|^{-1}1_{E}\\\
|\xi^{\prime}|1_{E}&0\end{pmatrix}.$

Therefore,
$(1-P_{\mathcal{C}})P_{N}P_{\mathcal{C}},(1-P_{\mathcal{C}})P_{D}P_{\mathcal{C}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{2})$ with non-vanishing principal symbol. By
Lemma 3.31, we conclude that neither $P_{D}$ nor $P_{N}$ act boundedly on
$\check{\mathrm{H}}(D)$. In particular, these projectors are not boundary
decomposing. A more appropriate projector is considered in Subsection 6.3.

##  4\. The Cauchy data space of an elliptic differential operator

In this section we will refine our analysis of the Cauchy data space and
boundary conditions further.

###  4.1. Constructing regular pseudodifferential boundary conditions

Theorem 3.18 implies that we can describe the Cauchy data space
$\check{\mathrm{H}}(D)$ in terms of the Calderón projection. Unfortunately,
the construction of the Calderón projection requires knowledge from the
interior of $M$ and not just symbolic information of $D$ at the boundary. In
this subsection, we give a construction of projectors from the Douglis-
Nirenberg calculus that rely only on the germ of $D$ at the boundary yet still
describes the Cauchy data space as in Corollary 3.23. The considerations of
this subsection implies that many structural properties of
$\check{\mathrm{H}}(D)$ extends to the case of noncompact $M$ with compact
boundary. Recall the definition of a boundary decomposing projection from
Definition 3.26.

######  Lemma 4.1.

Let $D$ be an elliptic differential operator of order $m>0$. Then the
following scheme produces a boundary decomposing pseudodifferential projection
$P\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$ with
$P-P_{\mathcal{C}}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$.

  1. (1)

Choose $P_{1}\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$
with $\sigma_{\boldsymbol{0}}(P_{1})=p_{+}(D)$.

  2. (2)

Construct $P_{2}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ by choosing a
$T_{2}\in\Psi^{-\boldsymbol{1}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$ with
$\sigma_{-\boldsymbol{1}}(T_{2})=\sigma_{-\boldsymbol{1}}(P_{1}-P_{\mathcal{C}})$
and set $P_{2}:=P_{1}-T_{2}$.

  3. (3)

Proceed iteratively and define $P_{k+1}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ from $P_{k}$ by choosing a
$T_{k+1}\in\Psi^{-\boldsymbol{k}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$
with
$\sigma_{-\boldsymbol{k}}(T_{k+1})=\sigma_{-\boldsymbol{k}}(P_{k}-P_{\mathcal{C}})$
and set $P_{k+1}:=P_{k}-T_{k+1}$. Stop at $k=m$.

  4. (4)

Choose a compact complex contour $\Gamma$ in the open right half-plane
containing $1$ in its interior domain and not intersecting the spectrum of
$P_{m}:{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
and define $P$ as the Riesz projection

$P:=\frac{1}{2\pi i}\int_{\Gamma}(\lambda-P_{m})^{-1}\mathrm{d}\lambda.$

Moreover,
$B_{P}:=(1-P){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ is
a regular boundary condition for $D$ that characterises $\mathrm{dom}(D_{\rm
max})$ via the following continuously split short exact sequence of Hilbert
spaces

$0\to\mathrm{dom}(D_{{\rm B}})\to\mathrm{dom}(D_{\rm
max})\xrightarrow{P\circ\gamma}P{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\to
0.$

###### Proof.

We need to verify that the stated scheme is possible to carry out and that it
produces a boundary decomposing pseudodifferential projection
$P\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$ that
defines a regular boundary condition. For the proof, it is beneficial to bear
in mind that if $e$ is an idempotent in a unital algebra, then the spectrum of
$e$ is $\\{0,1\\}$ and for $\lambda\neq 0,1$ we have that

$(\lambda-e)^{-1}=\lambda^{-1}(1-e)+(\lambda-1)^{-1}e.$

In particular, if $e$ is an idempotent in a unital algebra its Riesz
projectors are well defined and that onto the component $\\{1\\}$ coincides
with $e$.

Surjectivity of the symbol mappings ensures that the construction of
$P_{m}\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is
possible. Indeed, $P_{m}$ will in local charts have a full symbol that up to
degree $-m$ is determined by the formula (6). We note that the spectrum of
$P_{m}:{\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})$
is by elliptic regularity independent of $s\in\mathbb{R}$. Since
$\sigma_{\boldsymbol{0}}(P_{m})$ by construction is a projection, the
essential spectrum of
$P_{m}:{\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})$
is precisely the set $\\{0,1\\}\subseteq\mathbb{C}$. This ensures the
existence of the complex contour $\Gamma$. By construction, $P$ will be a
projection and belong to $\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ (since $\Gamma$ is compact).

To prove that $P-P_{\mathcal{C}}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ it suffices to prove that $(\lambda-
P_{m})^{-1}-(\lambda-P_{\mathcal{C}})^{-1}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ for $\lambda\in\Gamma$. By construction,
$\sigma_{-\boldsymbol{m-1}}(P_{m}-P_{\mathcal{C}})=0$, so
$P_{m}-P_{\mathcal{C}}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ and $(\lambda-P_{m})^{-1}-(\lambda-
P_{\mathcal{C}})^{-1}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ follows from the identity

$(\lambda-P_{m})^{-1}-(\lambda-P_{\mathcal{C}})^{-1}=(\lambda-
P_{m})^{-1}(P_{\mathcal{C}}-P_{m})(\lambda-P_{\mathcal{C}})^{-1}.$

Since $P-P_{\mathcal{C}}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$, $P$ is boundary decomposing because of
Corollary 3.32. The proof that $P$ defines a regular boundary condition is
postponed until Theorem 4.14 below. ∎

######  Remark 4.2.

The reader should be aware that the condition
$P-P_{\mathcal{C}}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is far from sufficient to guarantee that
$P-P_{\mathcal{C}}$ is compact on $\check{\mathrm{H}}(D)$. This happens
despite $P-P_{\mathcal{C}}$ being compact on
${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})$ for all $s\in\mathbb{R}$ (it
even has singular values behaving like $O(k^{-m/\dim(M)})$ as $k\to\infty$).
We give an example thereof in Proposition 9.12 below in order $m=1$.

We also note that the condition
$P-P_{\mathcal{C}}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is far from necessary for $P$ to be
boundary decomposing (cf. Corollary 3.32) or even for
$B_{P}:=(1-P){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ to
be a regular boundary condition for $D$ (cf. Theorem 4.14 below).

######  Proposition 4.3 (Localizability of $\check{\mathrm{H}}(D)$).

Let $D$ be an elliptic differential operator of order $m>0$ on a manifold with
boundary $M$. Pick a point $x_{0}\in\Sigma$ and coordinates $\kappa:U\to
U_{0}\subseteq M$ around $x_{0}$. Let $D_{0}$ be an elliptic differential
operator of order $m$ on $U$ such that $D_{0}$ and $\kappa^{*}D$ coincide on
$\kappa^{-1}(U_{0}\cap\Sigma)=U\cap\partial\mathbb{R}^{n}_{+}$ to order $m$.
Let $Q_{U}\in\Psi^{\boldsymbol{0}}_{\rm cl}(U;\kappa^{*}E)$ differ from an
approximate Calderón projection for $D_{0}$ by a term in
$\Psi^{\boldsymbol{-m}}_{\rm cl}(U;\kappa^{*}E)$.

Then for any $u\in\mathcal{D}^{\prime}(\Sigma;E)$, it holds that
$\chi_{0}u\in\check{\mathrm{H}}(D)$ for all $\chi_{0}\in{\rm C}^{\infty}_{\rm
c}(U_{0})$ if and only if
$(1-Q_{U})[\chi\kappa^{*}u]\in{\mathbb{H}}^{{m-{\frac{1}{2}}}}_{\rm\rm
loc}(U;\kappa^{*}E\otimes\mathbb{C}^{m})$ and
$Q_{U}[\chi\kappa^{*}u]\in{\mathbb{H}}^{{-{\frac{1}{2}}}}_{\rm\rm
loc}(U;\kappa^{*}E\otimes\mathbb{C}^{m})$ for all $\chi\in{\rm
C}^{\infty}_{\rm c}(U)$.

###### Proof.

It follows from Theorem 3.8 that for any $\chi,\chi^{\prime}\in{\rm
C}^{\infty}_{\rm c}(U)$,
$\chi(\kappa^{*}P_{\mathcal{C}}-Q_{U})\chi^{\prime}\in\Psi^{-\boldsymbol{m}}_{\rm
cl}(U;\kappa^{*}E)$. Therefore,
$(1-P_{\mathcal{C}})\chi_{0}u\in{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
for all $\chi_{0}\in{\rm C}^{\infty}_{\rm c}(U_{0})$ if and only if
$(1-Q_{U})[\chi\kappa^{*}u]\in{\mathbb{H}}^{{m-{\frac{1}{2}}}}_{\rm\rm
loc}(U;\kappa^{*}E\otimes\mathbb{C}^{m})$ for all $\chi\in{\rm
C}^{\infty}_{\rm c}(U)$ and
$P_{\mathcal{C}}\chi_{0}u\in{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
for all $\chi_{0}\in{\rm C}^{\infty}_{\rm c}(U_{0})$ if and only if
$Q_{U}[\chi\kappa^{*}u]\in{\mathbb{H}}^{{-{\frac{1}{2}}}}_{\rm\rm
loc}(U;\kappa^{*}E\otimes\mathbb{C}^{m})$ for all $\chi\in{\rm
C}^{\infty}_{\rm c}(U)$. The proposition follows. ∎

###  4.2. The boundary pairing and Calderón projectors of the formal adjoint

In this subsection, we shall study the relation between a Calderón projection
of $D$ and a Calderón projection of its formal adjoint $D^{\dagger}$. We apply
this to the boundary pairing. First, we shall revisit the operators $\tau$ and
a from Subsection 2.3. The next proposition follows from Lemma 2.17 and the
following observations.

For a matrix $T=(T_{i,j})_{i,j=0}^{m-1}$ of pseudo-differential operators on a
closed manifold, we write $T^{*}=(T_{j,i}^{*})_{i,j}$ for its ${\rm
L}^{2}$-adjoint and

$T^{\\#}:=\tau T^{*}\tau=(T_{m-i-1,m-j-1}^{*})_{i,j},$

where $\tau$ is defined in (1). A short computation shows that if
$T\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m},F\otimes\mathbb{C}^{m})$ then
$T^{\\#}\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m},E\otimes\mathbb{C}^{m})$ and
$T^{*}\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m}_{\rm op},E\otimes\mathbb{C}^{m}_{\rm op})$.

######  Proposition 4.4.

The operators  a , $\widetilde{\scalebox{1.5}{a}}$, and $\tau$ from Subsection
2.3 satisfy the following.

  1. (i)

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

${\mathbb{H}}^{s}(\Sigma;F\otimes\mathbb{C}^{m}_{\rm
op})\to{\mathbb{H}}^{{s+m-1}}(\Sigma;F\otimes\mathbb{C}^{m}),$

and for any $\alpha\in\mathbb{R}$, conjugation by $\tau$ defines a product
preserving isomorphism

$\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m},F\otimes\mathbb{C}^{m})\to\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm op},F\otimes\mathbb{C}^{m}_{\rm op}).$

  2. (ii)

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

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

and  a defines a Banach space isomorphism

${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{{s-m+1}}(\Sigma;F\otimes\mathbb{C}^{m}_{\rm
op}),$

and for any $\alpha\in\mathbb{R}$, conjugation by  a (i.e. the mapping
$T\mapsto\scalebox{1.5}{a}T\scalebox{1.5}{a}^{-1}$) defines a product
preserving isomorphism

$\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})\to\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m}_{\rm op}).$

######  Proposition 4.5.

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. It holds that
$\widetilde{\scalebox{1.5}{a}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m},F\otimes\mathbb{C}^{m})$ and
$\widetilde{\scalebox{1.5}{a}}_{\dagger}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m},E\otimes\mathbb{C}^{m})$ are elliptic and
invertible in the Douglis-Nirenberg calculus and satisfy

$\widetilde{\scalebox{1.5}{a}}^{\\#}=\widetilde{\scalebox{1.5}{a}}_{\dagger}.$

###### Proof.

The proposition is a consequence of Proposition 2.15 and Lemma 2.16. ∎

The following theorem refines Proposition 3.3 to the operator level.

######  Theorem 4.6.

Let $D$ be an elliptic differential operator of order $m>0$. We assume that
$P_{\mathcal{C}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is a Calderón projection for $D$ and
$P_{\mathcal{C}^{\dagger}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m})$ is a Calderón projection for
$D^{\dagger}$. Then it holds that

$\displaystyle P_{\mathcal{C}^{\dagger}}=$
$\displaystyle\scalebox{1.5}{a}_{\dagger}^{-1}(1-P_{\mathcal{C}}^{*})\scalebox{1.5}{a}_{\dagger}+\Psi^{-\infty}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m})=$ $\displaystyle=$
$\displaystyle[\scalebox{1.5}{a}(1-P_{\mathcal{C}})\scalebox{1.5}{a}^{-1}]^{*}+\Psi^{-\infty}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m}).$

###### Proof.

Before proving the main identity of the theorem, we remark that
$\scalebox{1.5}{a}_{\dagger}^{-1}(1-P_{\mathcal{C}}^{*})\scalebox{1.5}{a}_{\dagger}=[\scalebox{1.5}{a}(1-P_{\mathcal{C}})\scalebox{1.5}{a}^{-1}]^{*}$
by Lemma 2.16 and
$\scalebox{1.5}{a}(1-P_{\mathcal{C}})\scalebox{1.5}{a}^{-1}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m}_{\rm op})$ by Proposition 4.4.

Let us proceed with the proof. It follows from [grubb68, Chapter II, Lemma
1.2] that, up to a smoothing operator, we have

$\omega_{D}(\xi,\eta)=\omega_{D}(P_{\mathcal{C}^{\dagger}}\xi,\eta)-\omega_{D}(\xi,P_{\mathcal{C}}\eta),$

for $\xi\in\check{\mathrm{H}}(D^{\dagger})$ and
$\eta\in\check{\mathrm{H}}(D)$. By Proposition 2.15 and Lemma 2.16, this is
equivalent to the equality (valid up to a smoothing operator)

$\langle\scalebox{1.5}{a}_{\dagger}\xi,\eta\rangle_{{\rm
L}^{2}(\Sigma;E\otimes\mathbb{C}^{m})}=\langle\scalebox{1.5}{a}_{\dagger}\xi,P_{\mathcal{C}}\eta\rangle_{{\rm
L}^{2}(\Sigma;E\otimes\mathbb{C}^{m})}-\langle
P_{\mathcal{C}^{\dagger}}\xi,\scalebox{1.5}{a}\eta\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}.$

In particular, we have that

$\langle\scalebox{1.5}{a}_{\dagger}\xi,(1-P_{\mathcal{C}})\eta\rangle_{{\rm
L}^{2}(\Sigma;E\otimes\mathbb{C}^{m})}=-\langle
P_{\mathcal{C}^{\dagger}}\xi,\scalebox{1.5}{a}\eta\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})},$

up to a smoothing operator. We deduce from Lemma 2.16 that
$(1-P_{\mathcal{C}}^{*})\scalebox{1.5}{a}_{\dagger}=\scalebox{1.5}{a}_{\dagger}P_{\mathcal{C}^{\dagger}}$
up to a smoothing operator. ∎

######  Remark 4.7.

The reader should compare Theorem 4.6 to Proposition 2.26 describing adjoints
of boundary conditions defined from projectors.

######  Corollary 4.8.

Let $D$ be an elliptic differential operator of order $m>0$. We assume that
$P_{\mathcal{C}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is a Calderón projection for $D$ and
$P_{\mathcal{C}^{\dagger}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;F\otimes\mathbb{C}^{m})$ is a Calderón projection for
$D^{\dagger}$. Then  a extends to a continuous and invertible operator

$\scalebox{1.5}{a}:\begin{matrix}(1-P_{\mathcal{C}}){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\\\
\bigoplus\\\
P_{\mathcal{C}}{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\end{matrix}\longrightarrow\begin{matrix}(1-P_{\mathcal{C}^{\dagger}}^{*}){\mathbb{H}}^{{{\frac{1}{2}}-m}}(\Sigma;F\otimes\mathbb{C}^{m}_{\rm
op})\\\ \bigoplus\\\
P_{\mathcal{C}^{\dagger}}^{*}{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;F\otimes\mathbb{C}^{m}_{\rm
op})\end{matrix}.$

Moreover, in this decomposition

$\scalebox{1.5}{a}=\begin{pmatrix}0&\scalebox{1.5}{a}^{+}\\\
\scalebox{1.5}{a}^{-}&0\end{pmatrix}+\Psi^{-\infty}(\Sigma;E\otimes\mathbb{C}^{m},F\otimes\mathbb{C}^{m}),$

where

$\scalebox{1.5}{a}^{+}=(1-P_{\mathcal{C}^{\dagger}}^{*})\scalebox{1.5}{a}P_{\mathcal{C}}\quad\mbox{and}\quad\scalebox{1.5}{a}^{-}=P_{\mathcal{C}^{\dagger}}^{*}\scalebox{1.5}{a}(1-P_{\mathcal{C}}).$

###### Proof.

This follows directly from Proposition 4.4 and the relation between
$P_{\mathcal{C}}$ and $P_{\mathcal{C}^{\dagger}}$ from Theorem 4.6. ∎

######  Definition 4.9.

Let $D$ be an elliptic differential operator of order $m>0$. We define the
space

$\hat{\mathrm{H}}(D)=\begin{matrix}(1-P_{\mathcal{C}}^{*}){\mathbb{H}}^{{{\frac{1}{2}}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\\\ \bigoplus\\\
P_{\mathcal{C}}^{*}{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\end{matrix}$

######  Lemma 4.10.

The ${\rm L}^{2}$-pairing induces a perfect pairing
$\left\langle\cdot,\cdot\right\rangle:\hat{\mathrm{H}}(D)\times\check{\mathrm{H}}(D)^{\ast}$
and hence a Banach space isomorphism

$\hat{\mathrm{H}}(D)\cong\check{\mathrm{H}}(D)^{*}.$

Moreover, the operator  a defines an isomorphism

$\scalebox{1.5}{a}:\check{\mathrm{H}}(D)\to\hat{\mathrm{H}}(D^{\dagger}).$

###### Proof.

By Theorem 3.18 we have that

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

Proposition 3.6 implies that the ${\rm L}^{2}$-pairing induces a Banach space
isomorphism $\hat{\mathrm{H}}(D)\cong\check{\mathrm{H}}(D)^{*}$. The fact that
$\scalebox{1.5}{a}:\check{\mathrm{H}}(D)\to\hat{\mathrm{H}}(D^{\dagger})$ is
an isomorphism follows from Corollary 4.8. ∎

######  Remark 4.11.

If $D$ is an elliptic differential operator of order $m=1$ with adapted
boundary operator $A$, the results of [BBan] shows that

$\hat{\mathrm{H}}(D)=\chi^{-}(A^{\ast}){\rm
H}^{\rm-{\frac{1}{2}}}(\Sigma;E)\oplus\chi^{+}(A^{\ast}){\rm
H}^{\rm{\frac{1}{2}}}(\Sigma;E).$

We use the notation $\hat{\mathrm{H}}_{A}(D)$ for the right hand side of this
equation. The reader should be aware that in [BBan], this space was denoted by
$\hat{\mathrm{H}}(A^{\ast})$.

######  Theorem 4.12.

Let $D$ be an elliptic differential operator of order $m>0$. The boundary
pairing

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

takes the form

$\omega_{D}(\xi,\xi^{\prime})=\langle\xi,\scalebox{1.5}{a}\xi^{\prime}\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})},\quad\xi\in\check{\mathrm{H}}(D^{\dagger}),\;\xi^{\prime}\in\check{\mathrm{H}}(D),$

and is a perfect pairing.

###### Proof.

The formula for the boundary pairing for smooth sections follows from
Proposition 2.15 and that it can be written as the stated ${\rm
L}^{2}$-pairing follows from Lemma 4.10. ∎

###  4.3. Adjoint boundary conditions and a sufficient condition for
regularity

In light of the results of last subsection, we can further describe adjoint
boundary conditions. In particular, we can derive a sufficient condition for
certain semi-regular boundary conditions to be regular.

######  Proposition 4.13.

Let $D$ be an elliptic differential operator of order $m>0$ and
$B\subseteq\check{\mathrm{H}}(D)$ a generalised boundary condition. The
adjoint boundary condition $B^{*}\subseteq\check{\mathrm{H}}(D^{\dagger})$
takes the form

$B^{*}=(\scalebox{1.5}{a}B)^{\perp}=\scalebox{1.5}{a}_{\dagger}^{-1}B^{\perp},$

where $\perp$ denotes the induced annihilators with respect to ${\rm
L}^{2}$-pairing.

###### Proof.

Using the description in Lemma 4.10, the space
$(\scalebox{1.5}{a}B)^{\perp}\subseteq\check{\mathrm{H}}(D^{\dagger})$ is the
annihilator of $\scalebox{1.5}{a}B\subseteq\hat{\mathrm{H}}(D^{\dagger})$ in
the perfect ${\rm L}^{2}$-pairing
$\hat{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D^{\dagger})\to\mathbb{C}$.
By Theorem 4.12, $(\scalebox{1.5}{a}B)^{\perp}$ is the annihilator of $B$ in
the boundary pairing $\omega_{D}$, so $B^{*}=(\scalebox{1.5}{a}B)^{\perp}$
follows. By a similar argument,
$\scalebox{1.5}{a}_{\dagger}B^{*}\subseteq\hat{\mathrm{H}}(D)$ is the
annihilator of $B\subseteq\check{\mathrm{H}}(D)$ in the ${\rm L}^{2}$-pairing,
so $\scalebox{1.5}{a}_{\dagger}B^{*}=B^{\perp}$ which implies
$B^{*}=\scalebox{1.5}{a}_{\dagger}^{-1}B^{\perp}$. ∎

The reader should compare Proposition 4.13 to Proposition 2.26 describing the
adjoint boundary condition of a pseudo-local boundary condition. We shall now
state a theorem providing a rather general sufficient condition for
regularity.

######  Theorem 4.14.

Let $D$ be an elliptic differential operator of order $m>0$ and $P$ a
continuous projection on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ that extends
by continuity to a projection on
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. Assume that
the operator $A:=P_{\mathcal{C}}-(1-P)$ defines a Fredholm operator on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ that extends
by continuity to a Fredholm operator on
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. Then $P$
defines an regular boundary condition

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

for $D$. In other words, we have that

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

and this space defines the domain of an regular realisation of $D$.

###### Proof.

By using the same argument as in the proof of Theorem 3.30, we see that if
$A:=P_{\mathcal{C}}-(1-P)$ is Fredholm on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$, then

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

for $u\in{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. In
particular, $B$ is closed in $\check{\mathrm{H}}(D)$. This proves that $B$ is
a semi-regular boundary condition. It remains to prove that $B^{*}$ is a semi-
regular boundary condition.

We claim that
$B^{*}=(1-P_{\dagger}){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;F\otimes\mathbb{C}^{m})$
and that this is a semi-regular boundary condition. Here we use the notation
$P_{\dagger}:=\scalebox{1.5}{a}_{\dagger}^{-1}(1-P^{*})\scalebox{1.5}{a}_{\dagger}$,
where $P^{*}:{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\to{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm op})$
is the continuous projection defined from the ${\rm L}^{2}$-pairing
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\times{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\to\mathbb{C}$. By the arguments in the preceding paragraph and
Proposition 2.26, this claim follows if
$A_{\dagger}:=P_{\mathcal{C}^{\dagger}}-(1-P_{\dagger})$ is Fredholm on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;F\otimes\mathbb{C}^{m})$. However,
using Theorem 4.6 we have that

$A_{\dagger}=\scalebox{1.5}{a}_{\dagger}^{-1}A^{*}\scalebox{1.5}{a}_{\dagger},$

where we by an abuse of notation write $A$ for its Fredholm extension to
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and $A^{*}$ for
its ${\rm L}^{2}$-adjoint on
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm op})$. The
fact that $A_{\dagger}$ is Fredholm now follows from that $A$ is Fredholm on
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. ∎

The next result specialises Theorem 4.14 to pseudolocal boundary conditions
and characterises those that are regular. We encourage the reader to compare
the assumptions on $P$ in the next corollary to those in Corollary 3.32.

######  Theorem 4.15.

Let $D$ be an elliptic differential operator of order $m>0$ and
$P\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$ a
projection. Define the generalised boundary condition

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

Then the following are equivalent:

  1. i)

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

  2. ii)

$B_{P}$ is an regular boundary condition for $D$.

  3. iii)

$B_{P}$ is a Fredholm boundary condition for $D$.

We remark that the space $B_{P}$ defined in the theorem above is a priori
different from the pseudo-local boundary condition defined from $P$ as in
Definition 2.24. A posteriori, they produce the same boundary condition under
any of the assumptions i)-iii) in Theorem 4.15.

###### Proof.

Item i) implies item ii) by Theorem 4.14. Item ii) implies item iii) by
Proposition 2.6. Using [horIII, Theorem XIX.5.1 and XIX.5.2], to prove that
iii) implies i) it suffices to prove that $P_{\mathcal{C}}-(1-P)$ is Fredholm
as soon as $B_{P}$ is Fredholm boundary condition. We note that $B_{P}$ is
Fredholm if and only if $B_{P}^{*}$ is Fredholm if and only if
$(P_{\mathcal{C}}\check{\mathrm{H}}(D),B_{P})$ and
$((1-P_{\mathcal{C}}^{*})\hat{\mathrm{H}}(D),B_{P}^{\perp})$ are Fredholm
pairs. As such, $P_{\mathcal{C}}-(1-P)$ is Fredholm whenever $B_{P}$ is a
Fredholm boundary condition by combining [BBan, Lemma A.9] with Atkinson’s
theorem. ∎

To add some more flavor to the regularity condition on a pseudo-local boundary
condition as in Theorem 4.15, let us compare this to the well studied notion
of Shapiro-Lopatinskii ellipticity.

######  Definition 4.16.

Let $D$ be an elliptic differential operator of order $m>0$ and $B$ a pseudo-
local boundary condition defined from a pseudo-differential projection
$P\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$. We say
that $P$ is Shapiro-Lopatinskii elliptic if for any
$(x^{\prime},\xi^{\prime})\in T^{*}\Sigma$, when the differential equation

$\begin{cases}\sigma_{\partial}(D)(x^{\prime},\xi^{\prime})v(t)=0,\quad
t>0,\\\
\sigma_{\boldsymbol{0}}(P)(x^{\prime},\xi^{\prime})\left(D^{j}_{t}v(0)\right)_{j=0}^{m-1}=h,\end{cases}$
(12)

has a solution it is uniquely determined and exponentially decaying.

Compare the definition of Shapiro-Lopatinskii elliptic pseudo-differential
projectors the the construction of $E_{+}(D)$ in Proposition 3.1. The reader
can find more on Shapiro-Lopatinskii ellipticity of pseudo-local boundary
conditions in [schulzeseiler]. Similarly, for more general boundary
conditions, if for any $(x^{\prime},\xi^{\prime})\in T^{*}\Sigma$, the
differential equation

$\begin{cases}\sigma_{\partial}(D)(x^{\prime},\xi^{\prime})v(t)=0,\quad
t>0,\\\
\sigma(b)(x^{\prime},\xi^{\prime})\left(D^{j}_{t}v(0)\right)_{j=0}^{m-1}=h,\end{cases}$

has a unique exponentially decaying solution (for a suitably defined symbol
$\sigma(b)$) for any right hand side $h$, the problem is said to be Shapiro-
Lopatinskii elliptic. The reader can find more on Shapiro-Lopatinskii
ellipticity of a local boundary condition in [agranovichencyclopedia]. The
constructions in [agranovichencyclopedia] show how local considerations at the
boundary prove to be a highly efficient tool for determining regularity of a
local boundary condition. Indeed for the purpose of verifying regularity of
local boundary conditions, local semiclassical methods are more direct than
the global perspective on a boundary condition as a subspace
$B\subseteq\check{\mathrm{H}}(D)$ that dominates this paper.

######  Example 4.17.

We return to the study of Dirichlet and Neumann conditions on a second order
elliptic differential operator $D$ acting on a vector bundle $E$ with scalar
positive principal symbol as in Example 3.34. We have that

$\displaystyle\sigma_{\boldsymbol{0}}(P_{\mathcal{C}}-(1-P_{D}))(x^{\prime},\xi^{\prime})$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1_{E}&|\xi|^{\prime-1}1_{E}\\\
|\xi^{\prime}|1_{E}&-1_{E}\end{pmatrix}\quad\mbox{and}$
$\displaystyle\sigma_{\boldsymbol{0}}(P_{\mathcal{C}}-(1-P_{N}))(x^{\prime},\xi^{\prime})$
$\displaystyle=\frac{1}{2}\begin{pmatrix}-1_{E}&|\xi|^{\prime-1}1_{E}\\\
|\xi^{\prime}|1_{E}&1_{E}\end{pmatrix}.$

It follows that $P_{\mathcal{C}}-(1-P_{D})$ and $P_{\mathcal{C}}-(1-P_{N})$
are elliptic. Therefore Theorem 4.15 reproves the well known fact that
Dirichlet and Neumann conditions are regular. A short computation (or using
Proposition 4.18 below) reproves the well known fact that Dirichlet and
Neumann conditions are Shapiro-Lopatinskii elliptic. By Example 3.34, the
projectors $P_{D}$ and $P_{N}$ are not boundary decomposing which shows that
regularity of a boundary condition defined from a projection is not equivalent
to the boundary decomposing property of a projector.

We now return to the general case of pseudo-local boundary conditions defined
from a projection in the Douglis-Nirenberg calculus.

######  Proposition 4.18.

Let $D$ be an elliptic differential operator of order $m>0$ and
$P\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$. Then $P$
is Shapiro-Lopatinskii elliptic if and only if
$P_{\mathcal{C}}-(1-P)\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is elliptic.

In particular, a Shapiro-Lopatinskii elliptic pseudo-local boundary condition
is regular in the sense of Subsection 2.2.

###### Proof.

The condition of Shapiro-Lopatinskii ellipticity is equivalent to
$\sigma_{0}(P)$ defining an isomorphism between $E_{+}(D)$ and
$\sigma_{\boldsymbol{0}}(P)(E\otimes\mathbb{C}^{m})$. By [BBan, Proof of
Theorem 2.15], this is equivalent to $p_{+}(D)-\sigma_{\boldsymbol{0}}(1-P)$
being an isomorphism in all points, or in other words that
$P_{\mathcal{C}}-(1-P)$ is elliptic. ∎

##  5\. Characterisations of regularity via graphical decompositions

A graphical characterisation for regular boundary conditions was first given
in [BB], motivated by the desire to capture regular boundary conditions beyond
those that are given by pseudodifferential projectors. The main application in
mind was a remarkably conceptual proof of the relative index theorem,
originally due to Gromov-Lawson [GL]. In [BB], the boundary decomposing
projectors were assumed to be self-adjoint spectral projectors, with the
assumption that the adapted operator on the boundary was self-adjoint. A
particular luxury in this setting was that the obtained decomposition could be
equated with orthogonal complements in ${\rm L}^{2}$. This was subsequently
generalised [BBan, Theorem 2.9] for general first order operators, where
orthogonal complements were no longer available and the decomposition had to
be obtained separately for the boundary condition and its adjoint. In
particular, this meant that there were double the number of spaces in the
decomposition and we consider the most significant step towards our
presentation here for the general order case.

###  5.1. Preliminaries

As above, we assume that $D$ is an elliptic differential operator and
$\mathcal{P}_{+}$ is a boundary decomposing projection for $D$ (in the sense
of Definition 3.26 on page 3.26). We use the notation
$\mathcal{P}_{-}=1-\mathcal{P}_{+}$ and $\mathcal{P}_{+}^{*}$ for the
appropriate dual operator to $\mathcal{P}_{+}$ with respect to the duality
induced by ${\rm L}^{2}$ inner product.

######  Lemma 5.1.

The pairing

$\left\langle\cdot,\cdot\right\rangle:{\mathcal{P}}^{\ast}_{\pm}{\mathbb{H}}^{{-\alpha}}(\Sigma;E\otimes\mathbb{C}^{n}_{\rm
op})\times\mathcal{P}_{\pm}{\mathbb{H}}^{{\alpha}}(\Sigma;E\otimes\mathbb{C}^{m})\to\mathbb{C},$

induced from the ${\rm L}^{2}(\Sigma;E)$ inner product is a perfect pairing
for $\alpha\in\\{-\frac{1}{2},m-\frac{1}{2}\\}$.

###### Proof.

The ${\rm L}^{2}$-inner product $\left\langle\cdot,\cdot\right\rangle$ induces
a perfect pairing between
${\mathbb{H}}^{{-\alpha}}(\Sigma;E\otimes\mathbb{C}^{n}_{\rm op})$ and
${\mathbb{H}}^{{\alpha}}(\Sigma;E\otimes\mathbb{C}^{m})$ by Proposition 3.6.
The desired conclusion follows simply on making the right identifications to
the spaces $\mathcal{B}$, $\mathcal{B}^{\ast}$, $X$, $Y$ in Lemma A.1. ∎

######  Proposition 5.2.

For any boundary decomposing projection $\mathcal{P}_{+}$ for the elliptic
differential operator $D$, we have that:

  1. (i)

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

  2. (ii)

$\hat{\mathrm{H}}(D)=\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\oplus\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$.

###### Proof.

Item i follows from Proposition 3.27. The proof of ii follows from i since
$\check{\mathrm{H}}(D)=\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus\mathcal{P}_{+}{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
Lemma 5.1, and [Kato, Theorem 4.8 (4.12) in Chapter 4, Section 4]. ∎

######  Definition 5.3 (Graphical decomposition).

Let $B$ be a boundary condition and $\mathcal{P}_{+}$ a boundary decomposing
projection for an elliptic differential operator $D$ of order $m>0$. Following
[BBan], we say that a graphical decomposition of $B$ (with respect to
$\mathcal{P}_{+}$) is a decomposition of the following form:

  1. (i)

there exist mutually complementary subspaces $W_{\pm}$ and $V_{\pm}$ of
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ satisfying:

$W_{\pm}\oplus
V_{\pm}=\mathcal{P}_{\pm}{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

  2. (ii)

it holds that
${W_{-}}^{\ast}\subset{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$, and
${W_{\pm}}^{\ast}\subset{\mathbb{H}}^{{{\frac{1}{2}}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$ and the subspaces
$W_{\pm}\subset{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
are finite dimensional,

  3. (iii)

there exists a continuous map $g:V_{-}\to V_{+}$ such that

${g}^{\ast}({V}^{\ast}_{+}\cap{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))\subset{V}^{\ast}_{-}\cap{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}),$

where ${g}^{\ast}$ denotes the adjoint in the induced ${\rm L}^{2}$-pairing
between ${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and
${\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$

$B=\left\\{v+gv:v\in V_{-}\right\\}\oplus W_{+}.$

If a boundary condition admits a graphical decomposition (with respect to
$\mathcal{P}_{+}$), we say it is $\mathcal{P}_{+}$-graphically decomposable.

We remark that spaces $V_{\pm}^{*},W_{\pm}^{\ast}$, denotes subspaces of
${\mathbb{H}}^{{{\frac{1}{2}}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm op})$
obtained via the induced pairing from the ${\rm L}^{2}$-inner product, formed
with respect to the direct sum decomposition
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})=V_{+}\oplus
W_{+}\oplus V_{-}\oplus W_{-}$.

######  Definition 5.4 ($\mathcal{P}_{+}$ Fredholm pair decomposed in
${\mathbb{H}}^{{m-\frac{1}{2}}}$).

Let $B\subset{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
Suppose:

  1. (i)

$B$ is closed subspace of
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$,

  2. (ii)

$(\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}),B)$
and
$(\mathcal{P}_{-}^{*}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))$ are Fredholm pairs in
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm op}))$,
respectively, and

  3. (iii)

$\operatorname{ind}(\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E),B)=-\operatorname{ind}(\mathcal{P}_{-}^{*}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})).$

Then, we say that $B$ is _Fredholm-pair decomposed with respect to
$\mathcal{P}_{+}$ in ${\mathbb{H}}^{{m-\frac{1}{2}}}$_.

######  Definition 5.5 ($\mathcal{P}_{+}$ Fredholm pair decomposed in
$\check{\mathrm{H}}(D)$).

Let $B\subset{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
Suppose:

  1. (i)

$B$ is closed subspace of $\check{\mathrm{H}}(D)$ and
$B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$ is a closed subspace of $\hat{\mathrm{H}}(D)$,

  2. (ii)

$(\mathcal{P}_{+}\check{\mathrm{H}}(D),B)$ and
$(\mathcal{P}_{-}^{*}\hat{\mathrm{H}}(D),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))$ are Fredholm pairs in $\check{\mathrm{H}}(D)$ and $\hat{\mathrm{H}}(D)$
respectively,

  3. (iii)

$\operatorname{ind}(\mathcal{P}_{+}\check{\mathrm{H}}(D),B)=-\operatorname{ind}(\mathcal{P}_{-}^{*}\hat{\mathrm{H}}(D),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})).$

Then, we say that $B$ is _Fredholm-pair decomposed with respect to
$\mathcal{P}_{+}$ in $\check{\mathrm{H}}(D)$_.

Note that the $\perp$ appearing in these definitions taken with respect to the
induced pairing from ${\rm L}^{2}$ between
${\mathbb{H}}^{\frac{1}{2}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm op})$ and
${\mathbb{H}}^{m-\frac{1}{2}}(\Sigma;E\otimes\mathbb{C}^{m})$.

###  5.2. Decomposability

The main theorem we prove in this section is the following.

######  Theorem 5.6.

Let $D$ be an elliptic differential operator of order $m>0$. For
$\mathcal{P}_{+}$ being a boundary decomposing projection for $D$, the
following are equivalent:

  1. (i)

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

  2. (ii)

$B$ is $\mathcal{P}_{+}$ graphically decomposable,

  3. (iii)

$B$ is $\mathcal{P}_{+}$ Fredholm decomposable in
${\mathbb{H}}^{{m-\frac{1}{2}}}$

  4. (iv)

$B$ is $\mathcal{P}_{+}$ Fredholm decomposable in $\check{\mathrm{H}}(D)$.

If one of these equivalent conditions are satisfied, then

$\scalebox{1.5}{a}_{\dagger}{B}^{\ast}=B^{\perp}=\left\\{u-{g}^{\ast}u:u\in{V}^{\ast}_{+}\cap{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\right\\}\oplus{W}^{\ast}_{-}.$ (13)

and

$\displaystyle\operatorname{ind}(\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}),B)$
$\displaystyle=\operatorname{ind}(\mathcal{P}_{+}\check{\mathrm{H}}(D),B),$
(14)
$\displaystyle\operatorname{ind}(\mathcal{P}_{-}^{*}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))$
$\displaystyle=\operatorname{ind}(\mathcal{P}_{-}^{*}\hat{\mathrm{H}}(D),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})).$

In particular, the properties of being graphically decomposable, Fredholm
decomposable in ${\mathbb{H}}^{{m-\frac{1}{2}}}$ and Fredholm decomposable in
$\check{\mathrm{H}}(D)$ are independent of the choice of boundary decomposing
projection.

The proof of this theorem is aided by the structure of [BBan, Theorem 2.9]
which deals with the first order case. In a sense, the essential ingredients
are contained in that proof, although the subtleties of this decomposition
requires significant care. For that purpose, we give full details of the
argument for the general order case. Before entering the proof of Theorem 5.6,
let us give an index theoretical consequence. The following corollary follows
directly from Theorem 2.12 and Equation (14)

######  Corollary 5.7.

Let $D$ be an elliptic differential operator of order $m>0$ and
$B\subseteq{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ a
regular boundary condition for $D$. Write
$\mathcal{C}_{D}^{m-\frac{1}{2}}:=P_{\mathcal{C}}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=\mathcal{C}_{D}\cap{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
The index of $D_{B}$ is given by

$\operatorname{ind}(D_{B})=\operatorname{ind}(B,\mathcal{C}_{D}^{m-\frac{1}{2}})+\dim\ker(D_{\min})-\dim\ker(D_{\min}^{\dagger}).$

in terms of an index of the Fredholm pair
$(B,\mathcal{C}_{D}^{m-\frac{1}{2}})$ in
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

###### Proof that ii$\implies$ i in Theorem 5.6 .

We shall prove that graphical decomposability implies regularity. First, note
that a simple calculation yields

$\left\\{v+gv:v\in
V_{-}\right\\}^{\perp,\hat{\mathrm{H}}(D)}=W_{-}^{\ast}\oplus
W_{+}^{\ast}\oplus\left\\{u-g^{\ast}u:u\in
V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\right\\}.$
(15)

Moreover, we have that
${\mathbb{H}}^{m-\frac{1}{2}}(\Sigma;E\otimes\mathbb{C}^{m})=W_{-}\oplus
V_{-}\oplus W_{+}\oplus V_{+}$, and by Lemma 5.1, and using Lemma A.1, we have
that

$W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}=V_{-}^{\ast}\oplus
W_{-}^{\ast}\oplus V_{+}^{\ast}.$

Therefore,

$W_{+}^{\perp,\hat{\mathrm{H}}(D)}=W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}\cap\hat{\mathrm{H}}(D)=V_{-}^{\ast}\oplus
W_{-}^{\ast}\oplus(V_{+}^{\ast}\cap\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})).$
(16)

By [Kato, Theorem 4.8 (4.12) in Chapter 4, Section 4], we have that

$\displaystyle B^{\perp}$ $\displaystyle=(W_{+}\oplus\left\\{v+gv:v\in
V_{-}\right\\})^{\perp,\hat{\mathrm{H}}(D)}$
$\displaystyle=W_{+}^{\perp,\hat{\mathrm{H}}(D)}\cap\left\\{v+gv:v\in
V_{-}\right\\}^{\perp,\hat{\mathrm{H}}(D)}$
$\displaystyle=W_{-}^{\ast}\oplus\left\\{u-g^{\ast}u:u\in
V_{-}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\right\\},$

where the ultimate equality follows from combining (15) and (16). Note that by
Definition 5.3, we have that
$B^{\perp}\subset{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
Proposition 4.13 yields that

$\scalebox{1.5}{a}_{\dagger}{B}^{\ast}=B^{\perp}=\left\\{u-{g}^{\ast}u:u\in{V}^{\ast}_{+}\cap{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\right\\}\oplus{W}^{\ast}_{-}\subset{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}),$

and therefore, we have that
${B}^{\ast}\subset{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;F\otimes\mathbb{C}^{m})$
so $B$ is regular. ∎

The proof of i$\implies$ ii is considerably more involved. In order to proceed
with the proof i$\implies$ ii, the spaces in the Definition 5.3 need to be
defined. Taking inspiration from [BBan], we define the following spaces:

$\displaystyle
W_{-}^{\ast}:={\mathcal{P}}^{\ast}_{-}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\cap
B^{\perp,{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
(17) $\displaystyle
W_{+}:=\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap
B$ $\displaystyle
V_{-}^{\ast}:={\mathcal{P}}^{\ast}_{-}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
$\displaystyle
V_{+}:=\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$

Note that by definition
$W_{-}^{\ast}\subset{\mathcal{P}}^{\ast}_{-}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$. Therefore, we are able to consider its annihilator
$(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
in ${\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. Recalling
the definition of the hybrid spaces

${\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=\oplus_{j=0}^{m-1}{\rm
H}^{\rm-\frac{1}{2}-j}(\Sigma;E)\supset\oplus_{j=0}^{m-1}{\rm
H}^{\rm\frac{1}{2}-m+j}(\Sigma;E)={\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}).$

Therefore, $V_{-}^{\ast}$ is well-defined. Similarly, it is readily verified
that

${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\subset{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}),$

and therefore, we have that $V_{+}$ is well defined.

This then allows us to define the remaining spaces as follows simply as the
ranges of dual projectors or, as given by Lemma A.1, annihilators with respect
to the corresponding duality.

$\displaystyle
W_{+}^{\ast}:={\mathcal{P}}^{\ast}_{+}{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\cap
V_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
(18) $\displaystyle
W_{-}:=\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap(V_{-}^{\ast})^{\perp,{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
$\displaystyle
V_{+}^{\ast}:={\mathcal{P}}^{\ast}_{+}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
$\displaystyle
V_{-}:=\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$

######  Lemma 5.8.

Assume that $B$ is regular and that $V_{+}$, $V_{-}$, $W_{+}$ and $W_{-}$ are
defined as in the paragraphs above. The following hold:

  1. (i)

$W_{\pm}\subseteq{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and
$W_{\pm}^{\ast}\subset{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$ are finite dimensional. In particular,
$W_{-}^{\ast}={\mathcal{P}}^{\ast}_{-}{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\cap
B^{\perp,{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$.

  2. (ii)

$\mathcal{P}_{-}B\subseteq{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and
$\mathcal{P}_{+}^{\ast}B^{\perp}\subseteq{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$ are closed subspaces.

  3. (iii)

All of the subspaces $V_{+}$, $V_{-}$, $W_{+}$ and $W_{-}$ as well as
$V_{+}^{\ast}$, $V_{-}^{\ast}$, $W_{+}^{\ast}$ and $W_{-}^{\ast}$ listed in
(17) and (18) are closed in
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and
${\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
respectively.

  4. (iv)

$\mathcal{P}_{\pm}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=V_{\pm}\oplus
W_{\pm}$ and
$\mathcal{P}_{\pm}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})=V_{\pm}^{\ast}\oplus W_{\pm}^{\ast}$.

###### Proof.

Consider
$\mathcal{P}_{+}^{\ast}{{\lvert}}_{B^{\perp}}:B^{\perp}\to{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
By definition, we have that $B^{\perp}$ is closed in $\hat{\mathrm{H}}(D)$.
Set $X=(B^{\perp},\|\cdot\|_{\hat{\mathrm{H}}(D)})$,
$Y={\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$,
$Z={\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
and the map $K=\mathcal{P}_{-}^{\ast}{{\lvert}}_{B^{\perp}}:X\to Z$. For for
$b^{\perp}\in B^{\perp}$

$\|b^{\perp}\|_{\hat{\mathrm{H}}(D)}\simeq\|\mathcal{P}_{+}^{\ast}{{\lvert}}_{B^{\perp}}b^{\perp}\|_{{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}+\|\mathcal{P}_{-}^{\ast}{{\lvert}}_{B^{\perp}}b^{\perp}\|_{{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}.$

Note that

$\mathcal{P}_{-}^{\ast}{{\lvert}}_{B^{\perp}}:B^{\perp}\subset{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\to{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\xhookrightarrow{\mathrm{compact}}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$

and so on invoking [BBan, Lemma A.1] with this choice of $X,Y,Z$ and $K$
yields that $\ker\mathcal{P}_{+}^{\ast}{{\lvert}}_{B^{\perp}}$ is finite
dimensional and $P_{+}^{\ast}B^{\perp}$ is closed in
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
Moreover,

$\displaystyle\ker\mathcal{P}_{+}^{\ast}{{\lvert}}_{B^{\perp}}$
$\displaystyle=\left\\{b^{\perp}\in
B^{\perp}:\mathcal{P}_{+}^{\ast}b^{\perp}=0\right\\}=\left\\{b^{\perp}\in
B^{\perp}:b^{\perp}\in\mathcal{P}_{-}^{\ast}\hat{\mathrm{H}}(D)=0\right\\}$
$\displaystyle\hskip
70.0001pt=B^{\perp}\cap\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})=B^{\perp}\cap\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}),$

where the penultimate equality follows from Proposition 5.2 i and the ultimate
equality from the fact that
$B^{\perp}\subset{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.

Similarly, consider the map
$\mathcal{P}_{-}{{\lvert}}_{B}:B\to{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
Since $B\subset{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
we have that whenever $b\in B$,

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

However,

$\mathcal{P}_{+}{{\lvert}}_{B}:B\to{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\xhookrightarrow{\mathrm{compact}}{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}),$

and therefore, on invoking [BBan, Lemma A.1] yields that
$W_{+}=\ker(\mathcal{P}_{+}{{\lvert}}_{B})$ is finite dimensional and
$\mathcal{P}_{-}B$ is closed in
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

Therefore, we have proven the property ii as well as a part of the property i,
namely that $W_{-}^{\ast}$ and $W_{+}$ are finite dimensional and that the
asserted equality for $W_{-}^{\ast}$ holds. We prove that $W_{-}$ and
$W_{-}^{\ast}$ are finite dimensional after proving iii and iv.

For that, we first prove that
$\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})=V_{-}^{\ast}\oplus
W_{-}^{\ast}$. From the formula for $W_{-}^{\ast}$ that we have already
proven, we have that
$W_{-}^{\ast}\subset\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
and it is closed since it is finite dimensional. By construction
$V_{-}^{\ast}\subset\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
Therefore, we have that
$V_{-}^{\ast}+W_{-}^{\ast}\subset\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.

Next, we show that $V_{-}^{\ast}\cap W_{-}^{\ast}=0$. Fix $w\in
V_{-}^{\ast}\cap
W_{-}^{\ast}=\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\cap(W_{-}^{\ast})^{\perp,\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap
W_{-}^{\ast}$. In particular, we have that
$w\in(W_{-}^{\ast})^{\perp,\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap
W_{-}^{\ast}$. Since $w\in W_{-}^{\ast}$ we also have that
$w\in{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
which is in a perfect pairing with
${\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. Therefore, we
have that $\left\langle w,w\right\rangle=0$. Moreover, it is readily verified
that
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\subset\oplus_{j=0}^{m-1}{\rm
L}^{2}(\Sigma;E)$ and therefore,

$\|w\|_{{\rm L}^{2}(\Sigma;E)}^{2}=\left\langle w,w\right\rangle_{{\rm
L}^{2}(\Sigma;E)}=\left\langle
w,w\right\rangle_{{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\times{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}=0,$

from which we conclude that $w=0$. This proves that $V_{-}^{\perp}\oplus
W_{-}^{\perp}\subset\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.

We now prove the reverse containment. For that, let $k=\dim W_{-}^{\ast}$ and
fix a basis $e_{1},\dots,e_{k}$. By considering each
$e_{i}\in\oplus_{j=0}^{m-1}{\rm L}^{2}(\Sigma;E)$, we can run the Gram-Schmidt
process to obtain an ${\rm L}^{2}$-orthonormal basis $\tilde{e}_{i}$.
Therefore, without the loss of generality, assume that
$\left\\{e_{i}\right\\}$ are orthonormal in ${\rm L}^{2}$. For
$u\in{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$, define

$Pu=\sum_{i=1}^{k}\left\langle u,e_{i}\right\rangle e_{i}.$

It is readily checked that this is a projector and in fact,
$P:{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
with
$P{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=W_{-}^{\ast}$
and $\ker
P=(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$.
Therefore, restricting to
$u\in\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\subset{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
we obtain that $u=Pu-(1-P)u$ and $Pu\in W_{-}^{\ast}$. This means that
$(1-P)u\in(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
since
$W_{-}^{\ast}\subset\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
and therefore, $(1-P)u\in V_{-}^{\ast}$. Hence, we conclude that
$V_{-}^{\perp}\oplus
W_{-}^{\perp}=\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
By exploiting the duality yielded in Lemma 5.1, we conclude that $V_{-},W_{-}$
are closed in
$\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
Via Lemma A.1, we obtain
$\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=V_{-}\oplus
W_{-}$.

A similar argument then holds to show that
$\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=V_{+}\oplus
W_{+}$. Showing
$V_{+}+W_{+}\subset\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and that $V_{+}\cap W_{+}=0$ is an easy calculation. The most important point
to show the reverse containment is note that
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\subset{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
This allows for the construction of a projector
$Q:{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\to{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
with
$Q{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})=W_{+}$
and $\ker
Q=W_{+}^{\perp,{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
and consequently, write
$\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\subset
V_{+}\oplus W_{+}$. As before, this yields that $V_{+}^{\ast}$ and
$W_{+}^{\ast}$ are closed subspaces of
$\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
and Lemma A.1 then yields
$\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})=V_{+}^{\ast}\oplus
W_{+}^{\ast}$.

This proves iii and iv. It remains to show that $W_{+}^{\ast}$ and $W_{-}$ are
finite dimensional. For this, note that by using Lemma A.1, we have induced
perfect pairings $\left\langle\cdot,\cdot\right\rangle:W_{+}\times
W_{+}^{\ast}\to\mathbb{C}$ and
$\left\langle\cdot,\cdot\right\rangle:W_{-}\times W_{-}^{\ast}\to\mathbb{C}$.
But that means that $W_{+}^{\prime}\cong W_{+}^{\ast}$ and
$W_{-}^{\prime}\cong W_{-}^{\ast}$, where $W_{+}^{\prime}$ and
$W_{-}^{\prime}$ are dual spaces of $W_{+}$ and $W_{-}$ respectively. By
finite dimensionality we have that $W_{+}^{\prime}\cong W_{+}$ and
$W_{-}^{\prime}\cong W_{-}$ and this finishes the proof. ∎

######  Lemma 5.9.

Under the same assumptions as in Lemma 5.8, the following hold:

  1. (i)

$V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})=\mathcal{P}_{+}^{\ast}B^{\perp}$,
and

  2. (ii)

$V_{-}=\mathcal{P}_{-}B$.

###### Proof.

First, we prove that

$W_{+}=(\mathcal{P}_{+}^{\ast}B^{\perp})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}).$
(19)

Then,

$\displaystyle
x\in(\mathcal{P}_{+}^{\ast}B^{\perp})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
$\displaystyle\iff 0=\left\langle
x,\mathcal{P}_{+}^{\ast}b^{\perp}\right\rangle\ \forall b^{\perp}\in
B^{\perp}$ $\displaystyle\iff
0=\left\langle\mathcal{P}_{+}x,b^{\perp}\right\rangle\ \forall b^{\perp}\in
B^{\perp}$
$\displaystyle\implies\mathcal{P}_{+}x\in(B^{\perp})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$

Therefore,
$x\in(\mathcal{P}_{+}^{\ast}B^{\perp})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
implies that $x=\mathcal{P}_{+}x$. Moreover, since by Proposition 5.2
$\check{\mathrm{H}}(D)\subset{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and
$B^{\perp}=B^{\perp,\hat{\mathrm{H}}(D)}\subset{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$,
we obtain that
$B=(B^{\perp,\hat{\mathrm{H}}(D)})^{\check{\mathrm{H}}(D)}=(B^{\perp})^{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap\check{\mathrm{H}}(D)$.
Therefore,
$x\in(\mathcal{P}_{+}^{\ast}B^{\perp})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
then yields that $x\in\check{\mathrm{H}}(D)$ and hence, $x\in B$. This shows ,

$(\mathcal{P}_{+}^{\ast}B^{\perp})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\subset
B\cap\mathcal{P}_{+}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=W_{+}.$

For the reverse containment, fix $w\in
W_{+}=B\cap\mathcal{P}_{+}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
Therefore, for all $b^{\perp}\in B^{\perp}$, we have that

$\displaystyle 0=\left\langle
w,b^{\perp}\right\rangle_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\times{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
$\displaystyle=\left\langle\mathcal{P}_{+}w,b^{\perp}\right\rangle_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\times{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
$\displaystyle=\left\langle
w,\mathcal{P}_{+}^{\ast}b^{\perp}\right\rangle_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\times{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}.$

Therefore, we have
$w\in(\mathcal{P}_{+}^{\ast}B^{\perp})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
and by assumption
$w\in\mathcal{P}_{+}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
That is precisely that
$w\in(\mathcal{P}_{+}^{\ast}B^{\perp})^{\perp,{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cap\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

Now, by the definition of $V_{+}^{\ast}$ in (18), we have that

$\displaystyle
V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
$\displaystyle=\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
$\displaystyle=\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$

By the description of $W_{+}$ in (19) and on using [Kato, Theorem 4.8 (4.12)
in Chapter 4, Section 4], we obtain

$W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}=\mathcal{P}_{+}^{\ast}B^{\perp}+(\mathcal{P}_{+}{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}))^{\perp,{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}=\mathcal{P}_{+}^{\ast}B^{\perp}+\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}).$

Therefore,

$\displaystyle
V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
$\displaystyle=\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
$\displaystyle=\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\cap(\mathcal{P}_{+}^{\ast}B^{\perp}+\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
$\displaystyle=\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\cap\mathcal{P}_{+}^{\ast}B^{\perp}$
$\displaystyle=\mathcal{P}_{+}^{\ast}B^{\perp}.$

This proves i.

The proof of ii is similar. First, we prove that

$W_{-}^{\ast}=(\mathcal{P}_{-}B)^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}\cap\mathcal{P}_{-}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}).$
(20)

Recall that by Lemma 5.8 i,
$W_{-}^{\ast}={\mathcal{P}}^{\ast}_{-}{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\cap
B^{\perp,{\mathbb{H}}^{{\frac{1}{2}}-m}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$.
and so for $w\in W_{-}^{\ast}$, we have that for all $b\in B$,

$\displaystyle 0$ $\displaystyle=\left\langle
w,b\right\rangle_{{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\times{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
$\displaystyle=\left\langle\mathcal{P}_{-}^{\ast}w,b\right\rangle=\left\langle
w,\mathcal{P}_{-}b\right\rangle_{{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\times{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$

This shows the containment “$\subset$”. For the reverse containment, let
$x\in(\mathcal{P}_{-}B)^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}\cap\mathcal{P}_{-}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
Arguing similarly, for all $b\in B$,

$0=\left\langle
x,\mathcal{P}_{-}b\right\rangle=\left\langle\mathcal{P}_{-}^{\ast}x,b\right\rangle$

which yields that $x=\mathcal{P}_{-}^{\ast}x\in
B^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$.

With this, we now note that

$\displaystyle V_{-}$
$\displaystyle=\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
$\displaystyle=\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap(\mathcal{P}_{-}B+\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}))$
$\displaystyle=\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap\mathcal{P}_{-}B$
$\displaystyle=\mathcal{P}_{-}B.\qed$

###### Proof that i$\implies$ ii in Theorem 5.6 .

We shall prove that regularity implies graphical decomposability. We use the
spaces $V_{+}$, $V_{-}$, $W_{+}$ and $W_{-}$ constructed on the preceding
pages. Since
$B\subset{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\subset{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$,
we have that
$W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}\cap
B\neq 0$. Define

$X_{-}:=\mathcal{P}_{-}{{\lvert}}_{B\cap W_{+}^{\perp}}:B\cap
W_{+}^{\perp}\to\mathcal{P}_{-}B.$

We prove that this map is a bijection and hence, by the open mapping theorem,
it is a Banach space isomorphism between
$(B,\|\cdot\|_{{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})})$
and
$(\mathcal{P}_{-}B,\|\cdot\|_{{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})})$.
First, we establish surjectivity of $X_{-}$. For that, note that

${\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})=W_{+}\oplus
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})},$
(21)

which follows from a construction of a projector as in the proof of iii in
Lemma 5.8. Let $P_{W_{+},W_{+}^{\perp}}$ denote the projection to $W_{+}$
along
$W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
and the complementary projector $P_{W_{+}^{\perp},W_{+}}$. Fix
$x=P_{-}b\in\mathcal{P}_{-}B$ and note

$x=P_{-}(P_{W_{+}^{\perp},W_{+}}b+P_{W_{+},W_{+}^{\perp}}b)=P_{-}P_{W_{+}^{\perp},W_{+}}b.$

Since $W_{+}\subset B$,
$P_{W_{+}^{\perp},W_{+}}b=b-P_{W_{+}^{\perp},W_{+}}b\in B$ and hence
$P_{W_{+}^{\perp},W_{+}}b\in B\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$.
This shows surjectivity. Injectivity follows easily by taking
$0=X_{-}b=P_{-}b$ which allows us to conclude

$b\in\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap
B\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}=W_{+}\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}=0.$

Define projectors
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ as:

$P_{+}:=P_{V_{+},W_{+}\oplus\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\quad\text{and}\quad
Q_{+}:=P_{W_{+},V_{+}\oplus\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$

Furthermore, note that since $V_{-}$ has an induced Hilbert structure and
$\mathcal{P}_{-}B\subset V_{-}$ is closed, we obtain a closed complementary
subspace $V_{-}^{\prime}$ such that $V_{-}=\mathcal{P}_{-}B\oplus
V_{-}^{\prime}$. Using the fact that
$X_{-}:(B,\|\cdot\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})})\to(\mathcal{P}_{-}B,\|\cdot\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})})$
is an isomorphism, and the fact that $V_{-}=\mathcal{P}_{-}B$ from Lemma 5.9
ii, define the
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$-bounded map
$g_{0}:V_{-}\to V_{+}$ by setting

$g_{0}(x)=P_{+}\,\circ\,(X_{-})^{-1}.$

First, we prove that
$W_{+}\oplus\left\\{u+g_{0}u:u\in\mathcal{P}_{-}B\right\\}\subset B$. By
construction, we have that $W_{+}\subset B$, so it remains to show
$\left\\{u+g_{0}u:u\in V_{-}\right\\}\subset B$. For that, we compute

$\displaystyle u+g_{0}u$ $\displaystyle=u+P_{+}\,\circ\,(X_{-})^{-1}u$
$\displaystyle=X_{-}v+P_{+}v$ $\displaystyle=X_{-}v+P_{+}v+Q_{+}v-Q_{+}v$
$\displaystyle=\mathcal{P}_{-}v+\mathcal{P}_{+}v-Q_{+}v$
$\displaystyle=v-Q_{+}v\in B\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}+W_{+}\subset
B,$

where we have set $v:=(X_{-})^{-1}u$.

We now prove the reverse containment that $B\subset
W_{+}\oplus\left\\{u+g_{0}u:u\in V_{-}\right\\}$. Via the decomposition (21),
we have a induced decomposition

${\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})=W_{+}\oplus(W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}\cap{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})).$

This decomposition holds because
$W_{+}\subset{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
and for the same reason, the projectors for this decomposition are the
projectors $P_{W_{+},W_{+}^{\perp}}$ and $P_{W_{+}^{\perp},W_{+}}$ restricted
to
${\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
For $b\in
B\subset{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$,
let $b^{\perp}=P_{W_{+}^{\perp},W_{+}}b$ and $b_{+}=P_{W_{+},W_{+}^{\perp}}b$,
so that $b=b^{\perp}+b_{+}$. It is easy to see that, since $W_{+}\subset B$,
$b^{\perp}\in B\cap
W_{+}^{\perp,{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$.
Let $v:=X_{-}b^{\perp}$. Then,

$\displaystyle b^{\perp}=(X_{-})^{-1}v$
$\displaystyle=\mathcal{P}_{-}(X_{-})^{-1}v+\mathcal{P}_{+}(X_{-})^{-1}v=v+P_{+}(X_{-})^{-1}v+Q_{+}(X_{-})^{-1}v$
$\displaystyle\hskip
100.00015pt=v+g_{0}v+Q_{+}(X_{-})^{-1}v\in\left\\{u+g_{0}u:u\in\mathcal{P}_{-}B\right\\}\oplus
W_{+}.$

Let

$X_{+}^{\ast}:=\mathcal{P}_{+}^{\ast}{{\lvert}}_{B^{\perp}\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}}:B^{\perp}\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\to\mathcal{P}_{+}^{\ast}B^{\perp}.$

Note that, by using a similar projector construction as in the proof of Lemma
5.8 ii, we obtain

${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=W_{-}^{\ast}\oplus(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$
(22)

As before, $w\in
W_{-}^{\ast}\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
yields $w=0$ since, in particular, $w\in\oplus_{j=0}^{m-1}{\rm
L}^{2}(\Sigma;E)$ and therefore, $\left\langle
w,w\right\rangle=\|w\|^{2}_{{\rm L}^{2}}$. For
$x=\mathcal{P}_{+}^{\ast}b^{\perp}\in\mathcal{P}_{+}^{\ast}B^{\perp}$, we
obtain
$x=\mathcal{P}_{+}^{\ast}P_{W_{-}^{\ast},(W_{-}^{\ast})^{\perp}}b^{\perp}$. As
before, since $W_{-}^{\ast}\subset B^{\perp}$, we have that
$P_{W_{-}^{\ast},(W_{-}^{\ast})^{\perp}}b^{\perp}\in
B^{\perp}\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}$
which establishes the surjectivity of $X_{+}^{\ast}$. Injectivity follows
simply on noting that $X_{+}^{\ast}x=0$ implies

$x\in\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap
B^{\perp}\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}=W_{-}^{\ast}\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}=0.$

This shows that $X_{+}^{\ast}$ is a surjection and by the open mapping
theorem, it is an isomorphism between
$(B^{\perp}\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})},\|\cdot\|_{{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})})$
and
$(\mathcal{P}_{+}^{\ast}B^{\perp},\|\cdot\|_{{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})})$,
where the former space was established as a Banach space in Lemma 5.8 ii.

Next, define the following projectors on
${\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$:

$P_{-}^{\ast}:=P_{V_{-}^{\ast},W_{-}^{\ast}\oplus\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}\quad\text{and}\quad
Q_{-}^{\ast}:=P_{W_{-}^{\ast},V_{-}^{\ast}\oplus\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}.$

We have that
$P_{-}^{\ast}{{\lvert}}_{{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
is a bounded map from
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$ to
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
since

$P_{-}^{\ast}u=Q_{-}^{\ast}u-\mathcal{P}_{+}^{\ast}u$

and
$\mathcal{P}_{+}^{\ast}{{\lvert}}_{{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})}$
is a bounded map on
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
and
$Q_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}-m}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})=W_{-}^{\ast}\subset{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
Therefore, using Lemma 5.9 i to equate
$\mathcal{P}_{+}^{\ast}B^{\perp}=V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$,

$h_{0}:=P_{-}^{\ast}\,\circ\,(X_{+}^{\ast})^{-1}:V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})\to
V_{-}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}).$

Since $P_{-}^{\ast}$ is
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
bounded, the map $X_{+}^{\ast}$ was established to be an isomorphism of the
respective Banach spaces it maps between, we have that $h_{0}$ is bounded in
the
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$-norm.

To show
$W_{-}^{\ast}\oplus\left\\{u+h_{0}u:u\in\mathcal{P}_{+}^{\ast}B^{\perp}\right\\}\subset
B^{\perp}$, we note that $W_{-}^{\ast}\subset B^{\perp}$ by construction and
that, setting $w=(X_{+}^{\ast})^{-1}u$ for $u\in P_{+}^{\ast}B^{\perp}$,

$u+h_{0}u=w-Q_{-}\ast w\in B^{\perp}+W_{-}^{\ast}\subset B^{\perp}.$

For the reverse containment, using (22), we write $x=x^{\perp}+x_{-}\in
B^{\perp}\cap(W_{-}^{\ast})^{\perp,{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\oplus
W_{-}^{\ast}$ and letting $w=(X_{+}^{\ast})x^{\perp}$, we deduce that

$x^{\perp}=w+h_{0}w+Q_{-}^{\ast}(X_{+}^{\ast})^{-1}w\in\left\\{u+h_{0}u:v\in
P_{+}^{\ast}B^{\perp}\right\\}\oplus W_{-}^{\ast}.$

Lastly, we show that
$h_{0}=-g_{0}^{\ast}{{\lvert}}_{V_{+}^{\ast}\cap{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})}$. For that, we note that $\left\langle B,B^{\perp}\right\rangle=0$ in
the pairing between $\check{\mathrm{H}}(D)$ and $\hat{\mathrm{H}}(D)$ and so
therefore,

$0=\left\langle u+g_{0}u,v+h_{0}v\right\rangle=\left\langle
u,v\right\rangle+\left\langle u,h_{0}v\right\rangle+\left\langle
g_{0}u,v\right\rangle+\left\langle g_{0}u,h_{0}v\right\rangle.$

However $u\in V_{-}$ and $v\in V_{+}^{\ast}$ and therefore,

$0=\left\langle u,v\right\rangle=\left\langle g_{0}u,h_{0}v\right\rangle.$

From this, we obtain that

$\left\langle u,h_{0}v\right\rangle=\left\langle
u,-g_{0}^{\ast}v\right\rangle$

for all $u\in V_{-}$ and $v\in
V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$.
This shows us that $h_{0}v=-g_{0}^{\ast}v$ for all $v\in
V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$
and moreover, since $h_{0}v\in
V_{-}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}})$,
by construction of $h_{0}$, we obtain that

$g_{0}(V_{+}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}))\subset
V_{-}^{\ast}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\mathrm{op}}).$

This finishes the proof. ∎

Next, we move onto proving the assertions regarding Fredholm pairs
characterisations of regular boundary conditions.

###### Proof that ii$\implies$ iii in Theorem 5.6 .

We shall prove that graphical decomposability implies Fredholm decomposability
in ${\mathbb{H}}^{{m-{\frac{1}{2}}}}$. The argument is exactly as in the proof
of [BBan, Theorem 2.9]. The idea is to use ii to write $B=\left\\{v+gv:v\in
V_{-}\right\\}\oplus W_{+}$ and

$\scalebox{1.5}{a}_{\dagger}{B}^{\ast}=B^{\perp}=\left\\{u-{g}^{\ast}u:u\in{V}^{\ast}_{+}\cap{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\right\\}\oplus{W}^{\ast}_{-}.$

This, along with Proposition 5.2 yields

$\displaystyle
B\cap\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=W_{+},\quad(B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))\cap\hat{\mathrm{H}}(D)=W_{-}^{\ast},$
$\displaystyle\faktor{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}{B+\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\cong\faktor{\check{\mathrm{H}}(D)}{B+\mathcal{P}_{+}\check{\mathrm{H}}(D)}\cong
W_{-},\text{ and}$
$\displaystyle\faktor{{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})}{B^{\perp}+\mathcal{P}_{-}^{\ast}\hat{\mathrm{H}}(D)}\cong\faktor{\hat{\mathrm{H}}(D)}{B^{\perp}+\mathcal{P}_{-}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})}\cong W_{+}^{\ast}.$

Since $W_{\pm}\cong W_{\pm}^{\ast}$, the conclusion follows. ∎

Now, we prove the rest of the theorem in two parts.

###### Proof that iii$\implies$ iv in Theorem 5.6 .

We shall prove that Fredholm decomposability in
${\mathbb{H}}^{{m-\frac{1}{2}}}$ implies Fredholm decomposability in
$\hat{\mathrm{H}}(D)$. The argument is divided into three steps.

  1. (a)

$B$ is closed in $\check{\mathrm{H}}(D)$.

Set
$W_{+}:=B\cap\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=B\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)$,
since $B\subset{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
Therefore, using Proposition 5.2,

$B=W_{+}\oplus(B\cap
W_{+}^{\perp})\quad\text{and}\quad\mathcal{P}_{+}\check{\mathrm{H}}(D)=W_{+}\oplus(\mathcal{P}_{+}\check{\mathrm{H}}(D)\cap
W_{+}^{\perp}).$

Also,
$B\cap\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap
W_{+}^{\perp}=W_{+}\cap W_{+}^{\perp}=0$, and therefore,

$B+\mathcal{P}_{+}\check{\mathrm{H}}(D)=B\oplus(B\cap W_{+}^{\perp}).$

Next, we claim that this space is closed. For that, we use [BBan, Lemma A.4]
to see that
$B+\mathcal{P}_{+}\check{\mathrm{H}}(D)=\mathcal{P}_{-}B\oplus\mathcal{P}_{+}\check{\mathrm{H}}(D)$
and note that
$\mathcal{P}_{-}B\subset\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})=\mathcal{P}_{-}\check{\mathrm{H}}(D)$
is a closed subspace in $\check{\mathrm{H}}(D)$.

Now, using that
$(\mathcal{P}_{-}^{*}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))$ is a Fredholm pair in
${\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm op}))$, we
obtain that
$(\mathcal{P}_{+}{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})),(B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))^{\perp})$ is a Fredholm pair in
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ with opposite
index. Using Proposition 5.2 and [BBan, Lemma A.5], we obtain that
$(\mathcal{P}_{+}\check{\mathrm{H}}(D),\check{B})$ is a Fredholm pair in
$\check{\mathrm{H}}(D)$, where
$\check{B}=(B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))^{\perp}$. This yields
$B\oplus(\check{B}\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)\cap W_{+}^{\perp})$
is closed in $\check{\mathrm{H}}(D)$.

To prove the desired claim, we show that
$\check{B}\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)\cap W_{+}^{\perp}=0$. For
that, it suffices to show that
$\check{B}\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)=B\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)$.
Since $B\subset\check{B}$, and the dimension of
$B\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)$ is finite, it suffices to show
that
$\dim(\check{B}\cap\mathcal{P}_{+}\check{\mathrm{H}}(D))\leq\dim(B\cap\mathcal{P}_{+}\check{\mathrm{H}}(D))$.
For that,

$\displaystyle\dim(\check{B}\cap\mathcal{P}_{+}\check{\mathrm{H}}(D))$
$\displaystyle=\dim\left(\faktor{\hat{\mathrm{H}}(D)}{B^{\perp}+\mathcal{P}_{-}^{\ast}\hat{\mathrm{H}}(D)}\right)$
$\displaystyle=\dim\left(\faktor{\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})}{\mathcal{P}_{+}^{\ast}B^{\perp}}\right)$
$\displaystyle\leq\dim\left(\faktor{\mathcal{P}_{+}^{\ast}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})}{\mathcal{P}_{+}^{\ast}B^{\perp}}\right)$
$\displaystyle=\dim(B\cap\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}))=\dim(B\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)).$

  2. (b)

$(\mathcal{P}_{+}\check{\mathrm{H}}(D),B)$ is a Fredholm pair in
$\check{\mathrm{H}}(D)$, with

$\operatorname{ind}(\mathcal{P}_{+}\check{\mathrm{H}}(D),B)=\operatorname{ind}(\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}),B).$

Contained in the proof of a was that $B+\mathcal{P}_{+}\check{\mathrm{H}}(D)$
is a closed subspace of $\check{\mathrm{H}}(D)$. It is clear that
$B\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)$ is a closed subspace of
$\check{\mathrm{H}}(D)$.

Using [BBan, Lemma A.4], we obtain that

$\faktor{\check{\mathrm{H}}(D)}{B+\mathcal{P}_{+}\check{\mathrm{H}}(D)}\cong\faktor{\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}{\mathcal{P}_{-}B}$

as well as

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

This along with the fact that
$B\cap\mathcal{P}_{+}\check{\mathrm{H}}(D)=B\cap\mathcal{P}_{+}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$
then yields the conclusion.

  3. (c)

$B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$ is closed in $\hat{\mathrm{H}}(D)$ and
$(\mathcal{P}_{-}^{*}\hat{\mathrm{H}}(D),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))$ is a Fredholm pair with

$\operatorname{ind}(\mathcal{P}_{-}^{*}\hat{\mathrm{H}}(D),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))=\operatorname{ind}(\mathcal{P}_{-}^{*}{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})).$

That
$B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$ is closed in $\hat{\mathrm{H}}(D)$ is obtained via a repetition of the
argument in a with the appropriate changes. Similarly, the remaining
conclusion is obtained on mirroring the argument b with the necessary changes.
∎

The last remaining part to establish Theorem 5.6 is given below.

###### Proof that iv$\implies$ i in Theorem 5.6 .

We shall prove that Fredholm decomposability in $\hat{\mathrm{H}}(D)$ implies
regularity. . Since
$B\subset{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and
closed in $\check{\mathrm{H}}(D)$, we only need to show that
$B^{\perp}\subseteq{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$.

By hypothesis, we have that
$\operatorname{ind}(\mathcal{P}_{+}\check{\mathrm{H}}(D),B)=-\operatorname{ind}(\mathcal{P}_{-}^{*}\hat{\mathrm{H}}(D),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))$, and therefore
$\operatorname{ind}(\mathcal{P}_{-}^{*}\hat{\mathrm{H}}(D),B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op}))=\operatorname{ind}(\mathcal{P}_{-}^{*}\hat{\mathrm{H}}(D),B^{\perp})$.
By [BBan, Lemma A.7], we obtain that
$B^{\perp}=B^{\perp}\cap{\mathbb{H}}^{{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})$. ∎

###  5.3. The first order case

For general boundary decomposing projectors $\mathcal{P}_{\pm}$ in the first
order case, a decomposition more in the spirit of [BBan] can be obtained. We
remark that if $\mathcal{P}_{+}$ is boundary decomposing for a first order
elliptic operator, then $\mathcal{P}_{+}$ extends by interpolation to ${\rm
H}^{\rm s}(\Sigma;E)$ for any $s\in[-{\frac{1}{2}},{\frac{1}{2}}]$.

######  Definition 5.10 (Graphical ${\rm L}^{2}$-decomposition).

Let $B$ be a boundary condition and $\mathcal{P}_{+}$ a boundary decomposing
projection for an elliptic differential operator $D$ of order $1$. We say that
a _graphical ${\rm L}^{2}$-decomposition_ (with respect to $\mathcal{P}_{+}$)
of $B$ is the following decomposition:

  1. (i)

there exist mutually complementary subspaces $W_{\pm}$ and $V_{\pm}$ of ${\rm
L}^{2}(\Sigma;E)$ satisfying:

$W_{\pm}\oplus V_{\pm}=\mathcal{P}_{\pm}{\rm L}^{2}(\Sigma;E),$

  2. (ii)

it holds that $W_{\pm}$ are finite dimensional with
$W_{\pm},{W_{\pm}}^{\ast}\subset{\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$, and

  3. (iii)

there exists a bounded linear map $g:V_{-}\to V_{+}$ such that

${g}^{\ast}({V}^{\ast}_{+}\cap{\rm
H}^{\rm\frac{1}{2}}(\Sigma;E))\subset{V}^{\ast}_{-}\cap{\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)\quad\mbox{and}\quad g(V_{-}\cap{\rm
H}^{\rm\frac{1}{2}}(\Sigma;E))\subset V_{+}\cap{\rm
H}^{\rm\frac{1}{2}}(\Sigma;E),$

where ${g}^{\ast}$ denotes the ${\rm L}^{2}$-adjoint and

$B=\left\\{v+gv:v\in V_{-}\cap{\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)\right\\}\oplus W_{+}.$

A boundary condition admitting a graphical ${\rm L}^{2}$-decomposition is said
to be graphically ${\rm L}^{2}$-decomposable.

As before, the spaces $W_{\pm}^{\ast}$ that appear in the definition, as
before, are simply the ranges of the dual projectors. However, our
computations are always in ${\rm L}^{2}$, and therefore, this is simply the
ranges of the adjoint projectors in ${\rm L}^{2}$ with respect to the ${\rm
L}^{2}$ inner product. The special case $\mathcal{P}_{+}=\chi^{+}(A)$, for an
adapted boundary operator $A$, of Definition (5.10) was stated in [BBan,
Definition 2.6].

We emphasise that in Definition 5.10, unlike for the graphical decomposition
in Definition 5.3, the decomposition obtained is in ${\rm L}^{2}$. Indeed, the
following theorem asserts that in the first order case, these decompositions
are equivalent. The precise equivalence is set up as follows: a graphical
${\rm L}^{2}$-decomposition produces a graphical decomposition as in
Definition 5.3 by intersecting with ${\rm H}^{\rm{\frac{1}{2}}}$ and the
converse is constructed from ${\rm L}^{2}$-closures. We present this first
order definition separately as it can be useful to compute in ${\rm L}^{2}$ in
the first order case instead of the more complicated spaces that appear in the
general order decomposition.

######  Theorem 5.11.

For a first order elliptic differential operator $D$, a boundary condition $B$
is regular if and only if it is graphically ${\rm L}^{2}$-decomposable. In
this case, we have that

$B^{\perp}=\left\\{u-{g}^{\ast}u:u\in{V}^{\ast}_{+}\cap{\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)\right\\}\oplus{W}^{\ast}_{-}.$

The proof of this theorem runs as in the proof of [BBan, Theorem 2.9], with
the projectors $\chi^{\pm}(A_{r})$ replaced by $\mathcal{P}_{\pm}$.

##  6\. Higher order boundary regularity

We will now prove results on higher order boundary regularity. These results
rely on the fact that Theorem 3.18 due to Seeley was, in fact, stated in
[seeley65] also for higher order Sobolev spaces.

###  6.1. Preliminaries

For an elliptic differential operator $D$, we let $D_{\rm min,s}$ and $D_{\rm
max,s}$ denote the minimal and maximal, respectively, closures of $D$ acting
densely defined on ${\rm H}^{\rm s}(M;E)\to{\rm H}^{\rm s}(M;F)$. We let
$\gamma_{s}:\mathrm{dom}(D_{\rm
max,s})\to{\mathbb{H}}^{{s-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
denote the trace mapping.

######  Theorem 6.1.

Let $D$ be an elliptic differential operator of order $m>0$. The Calderón
projection $P_{\mathcal{C}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ from Theorem 3.18 satisfies the following.

  1. (i)

The operator $\mathcal{K}$ from Theorem 3.18 defines a continuous Poisson
operator

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

such that $P_{\mathcal{C}}=\gamma_{s}\circ\mathcal{K}_{s}$ on
${\mathbb{H}}^{{s-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

  2. (ii)

The image of

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

is precisely the subspace

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

In particular, the trace map $\gamma_{s}$ induces an isomorphism of Banach
spaces

$\faktor{\mathrm{dom}(D_{\rm max,s})}{\mathrm{dom}(D_{\rm min,s})}\cong
P_{\mathcal{C}}{\mathbb{H}}^{{s-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus(1-P_{\mathcal{C}}){\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$

While the Calderon projection is in general hard to compute, we shall
formulate a condition on a boundary decomposing projection in order for it to
implement higher semiregularity properties. For $s\geq 0$, we introduce the
notation

$\check{\mathrm{H}}^{s}(D):=P_{\mathcal{C}}{\mathbb{H}}^{{s-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus(1-P_{\mathcal{C}}){\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\cong\faktor{\mathrm{dom}(D_{\rm
max,s})}{\mathrm{dom}(D_{\rm min,s})},$

where the last isomorphism is implemented by $\gamma$ by Theorem 6.1.

######  Definition 6.2.

Let $D$ be an elliptic differential operator of order $m>0$ and $s\geq 0$ a
positive number. We say that a projection $\mathcal{P}_{+}$ is $s$-boundary
decomposing for $D$ if $\mathcal{P}_{+}$ is boundary decomposing for $D$ and
additionally satisfies the following conditions

  1. (P1)

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

  2. (P2)

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

  3. (P3)

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

Similarly to Theorem 3.30 and Corollary 3.32, we have the following result
providing a sufficient condition for a projection being $s$-boundary
decomposing.

######  Theorem 6.3.

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

  * •

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

  * •

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

Then $P$ is $s$-boundary decomposing, and in particular

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

In the special case that $P\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is a projection such that

  * •

$[P_{\mathcal{C}},P]\in\Psi^{\boldsymbol{-m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$; and

  * •

the operator $A:=P_{\mathcal{C}}-(1-P)\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is elliptic,

then $P$ is $s$-boundary decomposing.

Let us consider a corollary of Theorem 6.1.

######  Corollary 6.4.

Let $D$ be an elliptic differential operator of order $m>0$ and $s\geq 0$.
Then it holds that

$\displaystyle\mathrm{dom}(D_{\rm max})\cap$ $\displaystyle{\rm H}^{\rm
s+m}(M;E)=$ $\displaystyle=$ $\displaystyle\\{u\in\mathrm{dom}(D_{\rm
max}):D_{\rm max}u\in{\rm H}^{\rm s}(M;F),\;P_{\mathcal{C}}\gamma
u\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\\}.$

###### Proof.

The inclusion $\subseteq$ is clear, we need to prove that if
$u\in\mathrm{dom}(D_{\rm max})$ satisfies that $D_{\rm max}u\in{\rm H}^{\rm
s}(M;F)$ and $P_{\mathcal{C}}\gamma
u\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ then
$u\in{\rm H}^{\rm s+m}(M;E)$. After replacing $u$ with
$u-\mathcal{K}_{s+m}P_{\mathcal{C}}\gamma u$ we can assume that
$P_{\mathcal{C}}\gamma u=0$. In other words, we need to prove that if
$u\in\mathrm{dom}(D_{\mathcal{C}^{c}})$ and $D_{\mathcal{C}^{c}}u\in{\rm
H}^{\rm s}(M;F)$ then $u\in{\rm H}^{\rm s+m}(M;E)$.

Define
$\mathcal{C}^{c}_{s}:=(1-P_{\mathcal{C}}){\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and consider the associated realisation $D_{\mathcal{C}^{c}_{s},s}$ of $D$ on
${\rm H}^{\rm s}(M;E)$. We note that $D_{\mathcal{C}^{c}_{s},s}$ is semi-
regular in the sense that
$\mathrm{dom}(D_{\mathcal{C}^{c}_{s},s})\subseteq{\rm H}^{\rm s+m}(M;E)$.

Theorem 6.1 implies that ${\rm
C}^{\infty}(M;E)\cap\mathrm{dom}(D_{\mathcal{C}^{c}})$ is a core for
$D_{\mathcal{C}^{c}_{s},s}$ for any $s$. Therefore, we have dense inclusions

$\displaystyle D_{\mathcal{C}^{c}}({\rm
C}^{\infty}(M;E)\cap\mathrm{dom}(D_{\mathcal{C}^{c}}))$
$\displaystyle\subset\mathrm{ran}(D_{\mathcal{C}^{c}_{s},s})\quad\mbox{and}$
$\displaystyle D_{\mathcal{C}^{c}}({\rm
C}^{\infty}(M;E)\cap\mathrm{dom}(D_{\mathcal{C}^{c}}))$
$\displaystyle\subset\mathrm{ran}(D_{\mathcal{C}^{c}})\cap{\rm H}^{\rm
s}(M;E).$

However, the spaces
$\mathrm{ran}(D_{\mathcal{C}^{c}_{s},s})=\mathrm{ran}(D_{\max,s})=D{\rm
H}^{\rm s+m}(M;E)$ and $\mathrm{ran}(D_{\mathcal{C}^{c}})\cap{\rm H}^{\rm
s}(M;E)$ are closed in ${\rm H}^{\rm s}(M;E)$, and therefore,
$\mathrm{ran}(D_{\mathcal{C}^{c}_{s},s})=\mathrm{ran}(D_{\mathcal{C}^{c}})\cap{\rm
H}^{\rm s}(M;E)$. The condition that $u\in\mathrm{dom}(D_{\mathcal{C}^{c}})$
with $D_{\mathcal{C}^{c}}u\in{\rm H}^{\rm s}(M;F)$ and $u\in{\rm H}^{\rm
s}(M;E)$ therefore implies that there exists
$v\in\mathrm{dom}(D_{\mathcal{C}^{c}_{s},s})\subseteq{\rm H}^{\rm s+m}(M;E)$
such that $Dv=D_{\mathcal{C}^{c}}u$. We then have that $D(v-u)=0$, so
$\gamma(v-u)\in\mathcal{C}\cap\mathcal{C}^{c}=0$, and therefore
$v-u\in\ker(D_{\min})\subset{\rm C}^{\infty}(M;E)$. Hence, $u=v+(u-v)\in{\rm
H}^{\rm s+m}(M;E)+{\rm C}^{\infty}(M;E)={\rm H}^{\rm s+m}(M;E)$ which proves
the required claim. ∎

###  6.2. Higher order regularity of boundary conditions

In [BB, BBan], higher order regularity of boundary conditions was introduced
under the name of regular boundary conditions. The regularity was stated in
terms of graphical decompositions of regular boundary conditions. We shall
formulate a slightly more general condition.

######  Definition 6.5.

Let $D$ be an elliptic differential operator,
$B\subseteq\check{\mathrm{H}}(D)$ a semi-regular boundary condition (in ${\rm
L}^{2}$-sense), $s\geq 0$ and $P$ an $s$-boundary decomposing projection.

  * •

We say that $B$ is $s$-semiregular (w.r.t $P$) if for any $\xi\in B$ it holds
that $\xi\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
if and only if
$(1-P)\xi\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

  * •

We say that the realisation $D_{B}$ is $s$-semiregular if whenever
$u\in\mathrm{dom}(D_{\rm B})$ satisfies that $D_{\rm B}u\in{\rm H}^{\rm
s}(M;F)$ then $u\in{\rm H}^{\rm s+m}(M;E)$.

The reader should note that the proof of Corollary 6.4 essentially boils down
to proving that $\mathcal{C}^{c}$ is $s$-semiregular for any $s\geq 0$ (w.r.t
$P_{\mathcal{C}}$). Therefore, Corollary 6.4 partially proves the main result
of this section (Theorem 6.6 below) in the special case of the boundary
condition $\mathcal{C}^{c}$.

######  Theorem 6.6 (Higher order boundary regularity).

Let $D$ be an elliptic differential operator of order $m>0$, $s\geq 0$ and
$B\subseteq\check{\mathrm{H}}(D)$ a boundary condition. Then the following are
equivalent:

  1. (1)

$D_{B}$ is $s$-semiregular.

  2. (2)

$B$ is $s$-semiregular with respect to $P_{\mathcal{C}}$

  3. (3)

$B$ is $s$-semiregular with respect to any $s$-boundary decomposing
projection.

###### Proof.

We start by showing that if $P$ is an $s$-boundary decomposing projection and
there is a $\xi\in B$ with
$(1-P)\xi\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$ but
$\xi\notin{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$, then
there is a $u\in\mathrm{dom}(D_{\rm B})$ satisfying that $D_{\rm B}u\in{\rm
H}^{\rm s}(M;F)$ and $u\notin{\rm H}^{\rm s+m}(M;E)$. Indeed, we take a
preimage $u\in\gamma^{-1}(\xi)$ written as
$u=u_{0}+u_{1}\in\mathrm{dom}(D_{\max,s})$ where $\gamma(u_{0})=(1-P)\xi$ and
$\gamma(u_{1})=P\xi$. Since
$(1-P)\xi\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$ we
can assume that $u_{0}\in{\rm H}^{\rm s+m}(M;E)$. However, if
$(1-P)\xi\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$ but
$\xi\notin{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$ then
$P\xi\notin{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$ so
$u_{1}\notin{\rm H}^{\rm s+m}(M;E)$. This proves that 1) implies 3).

It is clear that 3) implies 2). It remains to prove that 2) implies 1), i.e
that if $B$ is $s$-semiregular with respect to $P_{\mathcal{C}}$ then whenever
$u\in\mathrm{dom}(D_{\rm B})$ satisfies that $D_{\rm B}u\in{\rm H}^{\rm
s}(M;F)$ then $u\in{\rm H}^{\rm s+m}(M;E)$. Take a $u\in\mathrm{dom}(D_{\rm
B})$ satisfying that $D_{\rm B}u\in{\rm H}^{\rm s}(M;F)$. By applying the same
argument as in the proof of Corollary 6.4, we have that
$w:=u-\mathcal{K}\gamma
u\in\mathrm{dom}(D_{\mathcal{C}^{c}_{s},s})\subseteq{\rm H}^{\rm s+m}(M;E)$.
On the other hand
$\gamma(w)=\gamma(u)-P_{\mathcal{C}}\gamma(u)=(1-P_{\mathcal{C}})\gamma(u)\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$
so by weak $s$-semiregularity
$\gamma(u)\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$.
Therefore
$P_{\mathcal{C}}\gamma(u)\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(M;E\otimes\mathbb{C}^{m})$
and can conclude that $\mathcal{K}\gamma u=\mathcal{K}_{s+m}\gamma u\in{\rm
H}^{\rm s+m}(M;E)$ by Theorem 6.1. We conclude that $u=w+\mathcal{K}\gamma
u\in{\rm H}^{\rm s+m}(M;E)$. We conclude that 1) and 2) are equivalent. ∎

Based on the graphical decompositions of Section 5, let us provide a method of
producing semiregular boundary conditions.

######  Proposition 6.7.

Let $B$ be a boundary condition for an elliptic differential operator $D$ of
order $m>0$, $s\geq 0$ and $P$ an $s$-boundary decomposing projection. Assume
that $B$ is graphically decomposable with respect to $P$ as in Definition 5.3.
Then $B$ is $s$-semiregular if and only if

$\displaystyle W_{+}$
$\displaystyle\subseteq{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\quad\mbox{and}$
$\displaystyle g(V_{+}$
$\displaystyle\cap{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}))\subseteq
V_{-}\cap{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$

The proof goes ad verbatim to [BBan, Lemma 8.9].

######  Corollary 6.8.

Let $D$ be an elliptic differential operator of order $m>0$ and
$B\subseteq\check{\mathrm{H}}(D)$ a boundary condition. Then the realisation
$D_{\rm B}^{*}D_{\rm B}$ of $D^{\dagger}D$ is regular if $B$ is regular and
$m$-semiregular.

###### Proof.

Before getting down to brass tacks, we note that the operator $D_{\rm
B}^{*}D_{\rm B}$ is self-adjoint, so $D_{\rm B}^{*}D_{\rm B}$ is a regular
realisation if and only if it is a semi-regular realisation. If $B$ is regular
and $m$-semiregular then any $u\in\mathrm{dom}(D_{\rm B}^{*}D_{\rm B})$
satisfies $D_{\rm B}u\in\mathrm{dom}(D_{\rm B}^{*})\subseteq{\rm H}^{\rm
m}(M;F)$, because $B$ is regular. Since $B$ is $m$-semiregular, Theorem 6.6
implies that $u\in{\rm H}^{\rm 2m}(M;E)$ when $D_{\rm B}u\in{\rm H}^{\rm
m}(M;F)$. We conclude that $\mathrm{dom}(D_{\rm B}^{*}D_{\rm B})\subseteq{\rm
H}^{\rm 2m}(M;E)$ if $B$ is regular and $m$-semiregular. ∎

######  Remark 6.9.

We note that an interpolation argument shows that if $D$ is an elliptic
differential operator of order $m>0$ and $B\subseteq\check{\mathrm{H}}(D)$ a
boundary condition such that $D_{\rm B}^{*}D_{\rm B}$ is regular, then $B$ is
semi-regular. Since the proof of Corollary 6.8 only used semi-regularity of
$B^{*}$ and $m$-semiregularity of $B$, we conclude that any $m$-semiregular
boundary condition $B$ with $B^{*}$ semi-regular is regular.

###  6.3. Higher regularity of the Dirichlet problem

As an example of our theory, we consider elliptic operators with a uniquely
solvable Dirichlet problem. Such differential operators play a prominent role
in [grubb68, grubb71, grubb74]. The results of this subsection are well known.
Along the way, we shall exemplify the results of this paper for operators with
positive principal symbols.

For an elliptic differential operator $D$ of even order $m$, we define the
Dirichlet condition as the space

$B_{\rm
Dir}:=\\{\xi=(\xi_{j})_{j=0}^{m-1}\in\check{\mathrm{H}}(D)\subset{\mathbb{H}}^{\frac{1}{2}-m}(\Sigma;E\otimes
C^{m}):\xi_{j}=0,\;j=0,\ldots,\frac{m}{2}-1\\}.$

We say that $D$ has a uniquely solvable Dirichlet problem if $B_{\rm Dir}$ is
a regular boundary condition and $D_{\rm Dir}:=D_{B_{\rm Dir}}$ is an
invertible realisation. We note the following important example.

######  Lemma 6.10.

Let $D=D^{\dagger}$ be a formally selfadjoint elliptic operator of order $m$
acting on the sections of a hermitian vector bundle $E$ on an $n$-dimensional
manifold with boundary. Assume that $D$ has positive interior principal symbol
$\sigma_{D}$. Then $D_{\rm Dir}$ is the realisation with domain
$\mathrm{dom}(D_{\rm Dir}):={\rm H}^{\rm m}(M;E)\cap{\rm H}^{\rm m/2}_{\rm
0}(M;E)$. Moreover, $D_{\rm Dir}$ is regular, self-adjoint and lower
semibounded. In particular, for large enough $\lambda\in\mathbb{R}$,
$D+\lambda$ has a uniquely solvable Dirichlet problem.

We remark that if $D$ has positive interior symbol, the symmetry property
$\sigma_{D}(x,-\xi)=(-1)^{m}\sigma_{D}(x,\xi)$ implies that $m$ is even.

###### Proof.

By extending $D$ to a self-adjoint elliptic differential operator with
positive principal symbol on a closed manifold containing $M$ as a smooth
subdomain, we see that the Gårding inequality implies that for $\lambda>0$
large enough it holds that

$\langle(D+\lambda)\varphi,\varphi\rangle_{{\rm
L}^{2}(M;E)}\sim\|\varphi\|_{{\rm H}^{\rm m/2}_{\rm 0}(M;E)}^{2},$

for $\varphi\in{\rm C}^{\infty}_{\rm c}(M;E)$. Therefore $D_{c}$ is lower
semi-bounded and symmetric, and $D_{\rm Dir}$ is indeed the Friedrichs
extension of $D_{c}$. Therefore, $D_{\rm Dir}$ is self-adjoint and lower
semibounded. Since $D_{\rm Dir}$ is semi-regular by construction, self-
adjointness implies regularity. ∎

We now describe a certain projection used in [grubb68]. By Lemma 6.10 unique
solvability of the Dirichlet problem is no serious restriction for operators
with positive principal symbol. Under this assumption, $\gamma$ restricts to
an isomorphism $\ker(D_{\rm max})\cong\mathcal{C}_{D}$. Since $D_{\rm Dir}$ is
invertible and regular, the operator

$P_{\zeta}:=1-D_{\rm Dir}^{-1}D_{\rm max},$

is a continuous projection on $\mathrm{dom}(D_{\max})$. The projection
$P_{\zeta}$ is the projection onto $\ker(D_{\rm max})$ along
$\mathrm{dom}(D_{\rm Dir})$. As such, $P_{\zeta}$ defines a Banach space
isomorphism

$(1-P_{\zeta})\oplus
P_{\zeta}:\mathrm{dom}(D_{\max})\xrightarrow{\sim}\mathrm{dom}(D_{\rm
Dir})\oplus\ker(D_{\rm max}).$

For more details, see [grubb68, Chapter II]. Under $\gamma$, we arrive at a
decomposition

$\check{\mathrm{H}}(D)=B_{\rm Dir}\oplus\mathcal{C}_{D},$

as Banach spaces when topologising $B_{\rm Dir}$ as a subspace of
${\mathbb{H}}^{m-1/2}(\Sigma;E\otimes\mathbb{C}^{m})$. We let
$P_{\zeta,\partial}$ denote the continuous projection in
$\check{\mathrm{H}}(D)$ onto $\mathcal{C}_{D}$ along $B_{\rm Dir}$; note that
$P_{\zeta,\partial}$ is induced from $P_{\zeta}$.

######  Lemma 6.11.

Let $D$ be an elliptic differential operator of even order with uniquely
solvable Dirichlet problem. Let $s\geq 0$. The projection $P_{\zeta,\partial}$
is $s$-boundary decomposing for $D$ and is of order zero in the Douglis-
Nirenberg calculus $P_{\zeta,\partial}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$.

###### Proof.

We write $\gamma=(\gamma_{D},\gamma_{N})$ where

$\displaystyle\gamma_{D}:$
$\displaystyle\mathrm{dom}(D_{\max})\to{\mathbb{H}}^{-1/2}(\Sigma;E\otimes\mathbb{C}^{m/2}),\quad
u\mapsto(\partial_{x_{n}}^{k}u|_{\Sigma})_{k=0}^{m/2-1},\mbox{ and}$
$\displaystyle\gamma_{N}:$
$\displaystyle\mathrm{dom}(D_{\max})\to{\mathbb{H}}^{-m/2-1/2}(\Sigma;E\otimes\mathbb{C}^{m/2}),\quad
u\mapsto(\partial_{x_{n}}^{k}u|_{\Sigma})_{k=m/2}^{m-1}.$

By [grubb71], there is a Dirichlet-to-Neumann operator
$\Lambda_{DN}\in\Psi^{\boldsymbol{m}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m/2})$ defined from
$\Lambda_{DN}\xi:=\gamma_{N}u$ where $u\in\ker(D_{\max})$ satisfies
$\gamma_{D}u=\xi$. A short computation shows that

$P_{\zeta,\partial}=\begin{pmatrix}1&0\\\ \Lambda_{DN}&0\end{pmatrix},$ (23)

in the decomposition

$\check{\mathrm{H}}(D)\subseteq{\mathbb{H}}^{-1/2}(\Sigma;E\otimes\mathbb{C}^{m})={\mathbb{H}}^{-1/2}(\Sigma;E\otimes\mathbb{C}^{m/2})\oplus{\mathbb{H}}^{-m/2-1/2}(\Sigma;E\otimes\mathbb{C}^{m/2}).$

It follows that $P_{\zeta,\partial}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$.

By construction, $1-P_{\zeta,\partial}$ maps $\check{\mathrm{H}}(D)$ into
$B_{\rm Dir}\subseteq{\mathbb{H}}^{m-1/2}(\Sigma;E\otimes\mathbb{C}^{m})$ and
defines a decomposition $\check{\mathrm{H}}(D)=B_{\rm
Dir}\oplus\mathcal{C}_{D+\lambda}$ of Banach spaces. We conclude that
$P_{\zeta,\partial}$ is boundary decomposing for $D$. The fact that
$P_{\zeta,\partial}$ is $s$-boundary decomposing for all $s\geq 0$ follows
from Theorem 6.3. ∎

######  Example 6.12.

An immediate consequence of Lemma 6.11 and Theorem 6.6 is the well known fact
that the Dirichlet problem is $s$-semiregular for all $s\geq 0$ if it is
uniquely solvable in ${\rm L}^{2}$. With respect to $P_{\zeta,\partial}$, we
can graphically decompose the Dirichlet condition as

$B_{\rm Dir}=\left\\{v+gv:v\in V_{-}\right\\}\oplus W_{+},$

for

$\displaystyle V_{-}=$
$\displaystyle(1-P_{\zeta,\partial})\check{\mathrm{H}}(D)=(1-P_{\zeta,\partial}){\mathbb{H}}^{m-1/2}(\Sigma;E\otimes\mathbb{C}^{m}),$
$\displaystyle V_{+}=$ $\displaystyle
P_{\zeta,\partial}\check{\mathrm{H}}(D)=P_{\zeta,\partial}{\mathbb{H}}^{m-1/2}(\Sigma;E\otimes\mathbb{C}^{m}),$
$\displaystyle g=$ $\displaystyle 0\quad\mbox{and}\quad W_{+}=W_{-}=0.$

######  Example 6.13.

Another consequence of Lemma 6.11 and Theorem 6.6 concerns Robin-type problems
with claims concerning $s$-semiregularity for all $s\geq 0$ if the Dirichlet
problem is uniquely solvable in ${\rm L}^{2}$. This is studied in more detail
and generality in [grubb74].

We give an example in the case that $D$ is second order with positive
principal symbol, e.g. a Laplace type operator. For an element $a\in
C^{\infty}(\Sigma;\mathrm{End}(E))$, consider the boundary condition

$B_{a}:=\\{(\xi_{0},\xi_{1})\in\check{\mathrm{H}}(D):a\xi_{0}+\xi_{1}=0\\}.$

This boundary condition corresponds to the Robin problem for $D$, i.e. for
$f\in{\rm L}^{2}(M;E)$, the equation $D_{B_{a}}u=f$ is equivalent to the Robin
problem

$\begin{cases}Du=f,\quad&\mbox{in $M$}\\\
au|_{\Sigma}+\partial_{n}u|_{\Sigma}=0,\quad&\mbox{on $\Sigma$}.\end{cases}$

The boundary condition $B_{a}$ is a pseudolocal boundary condition
$B_{a}=B_{P_{a}}$ where

$P_{a}=\begin{pmatrix}0&0\\\ a&1\end{pmatrix}.$

A computation with Proposition 2.26 (cf. [grubb74]) shows that $B_{a}$ is
regular.

We compute from Equation (23) that

$(1-P_{\zeta,\partial}(\lambda))\xi=\begin{pmatrix}0\\\
-\Lambda_{DN}\xi_{0}+\xi_{1}\end{pmatrix}.$ (24)

Take an $s\geq 0$, and assume that $\xi\in B_{a}$ satisfies
$(1-P_{\zeta,\partial}(\lambda))\xi\in{\mathbb{H}}^{s+m-1/2}(\Sigma;E\otimes\mathbb{C}^{2})$,
or equivalently (by Equation (24)) that $-\Lambda_{DN}\xi_{0}+\xi_{1}\in{\rm
H}^{\rm s+m-3/2}(\Sigma;E)$. If $(\xi_{0},\xi_{1})\in B_{a}$, we therefore
have that $(\Lambda_{DN}+a)\xi_{0}\in{\rm H}^{\rm s+m-3/2}(\Sigma;E)$. Since
$\Lambda_{DN}\in\Psi^{1}_{\rm cl}(\Sigma;E)$ is elliptic, we conclude from
elliptic regularity that if $\xi=(\xi_{0},\xi_{1})\in B_{a}$ satisfies
$(1-P_{\zeta,\partial}(\lambda))\xi\in{\mathbb{H}}^{s+m-1/2}(\Sigma;E\otimes\mathbb{C}^{2})$
then $\xi_{0}\in{\rm H}^{\rm s+m-1/2}(\Sigma;E)$ so
$\xi\in{\mathbb{H}}^{s+m-1/2}(\Sigma;E\otimes\mathbb{C}^{2})$. We conclude
that $B_{a}$ is $s$-semiregular for any $s\geq 0$ when $a\in
C^{\infty}(\Sigma;\mathrm{End}(E))$. In particular, if $f\in{\rm H}^{\rm
s}(M;E)$, any solution $u\in{\rm L}^{2}(M;E)$ to the Robin problem

$\begin{cases}Du=f,\quad&\mbox{in $M$}\\\
au|_{\Sigma}+\partial_{n}u|_{\Sigma}=0,\quad&\mbox{on $\Sigma$},\end{cases}$

satisfies $u\in{\rm H}^{\rm s+m}(M;E)$.

Let us graphically decompose the Robin condition $B_{a}$ with respect to
$P_{\zeta,\partial}$. For $\xi=(\xi_{0},\xi_{1})=(\xi_{0},-a\xi_{0})\in
B_{a}$, we can write

$\xi=\underbrace{\begin{pmatrix}\xi_{0}\\\
\Lambda_{DN}\xi_{0}\end{pmatrix}}_{\in
P_{\zeta,\partial}\check{\mathrm{H}}(D)}+\underbrace{\begin{pmatrix}0\\\
-\Lambda_{DN}\xi_{0}-a\xi_{0}\end{pmatrix}}_{\in(1-P_{\zeta,\partial})\check{\mathrm{H}}(D)}.$

The operator $\Lambda_{DN}$ is first order elliptic, and so is
$\Lambda_{DN}+a$. So there exists an elliptic $T\in\Psi^{-1}_{\rm
cl}(\Sigma;E)$ such that $1-(\Lambda_{DN}+a)T$ and $1-T(\Lambda_{DN}+a)$ are
smoothing finite rank projectors,
$\mathrm{ran}((\Lambda_{DN}+a)T)=\mathrm{ran}(\Lambda_{DN}+a)$ and
$\ker(T(\Lambda_{DN}+a))=\ker(\Lambda_{DN}+a)$. Define the space

$V_{-}:=\left\\{\begin{pmatrix}0\\\
\eta_{1}\end{pmatrix}\in(1-P_{\zeta,\partial}){\mathbb{H}}^{3/2}(\Sigma;E\otimes\mathbb{C}^{2}):(\Lambda_{DN}+a)T\eta_{1}=\eta_{1}\right\\}.$

The space

$W_{-}:=\left\\{\begin{pmatrix}0\\\
\eta_{1}\end{pmatrix}\in(1-P_{\zeta,\partial}){\mathbb{H}}^{3/2}(\Sigma;E\otimes\mathbb{C}^{2}):(\Lambda_{DN}+a)T\eta_{1}=0\right\\},$

is by the construction of $T$ a finite-dimensional subspace of ${\rm
C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{2})$ such that

$V_{-}\oplus
W_{-}=(1-P_{\zeta,\partial}){\mathbb{H}}^{3/2}(\Sigma;E\otimes\mathbb{C}^{2}).$

Define the space

$V_{+}:=\left\\{\begin{pmatrix}\eta_{0}\\\
\Lambda_{DN}\eta_{0}\end{pmatrix}\in
P_{\zeta,\partial}{\mathbb{H}}^{3/2}(\Sigma;E\otimes\mathbb{C}^{2}):T(\Lambda_{DN}+a)\eta_{0}=\eta_{0}\right\\}.$

The space

$W_{+}:=\left\\{\begin{pmatrix}\eta_{0}\\\
\Lambda_{DN}\eta_{0}\end{pmatrix}\in
P_{\zeta,\partial}{\mathbb{H}}^{3/2}(\Sigma;E\otimes\mathbb{C}^{2}):T(\Lambda_{DN}+a)\eta_{0}=0\right\\},$

is by the construction of $T$ a finite-dimensional subspace of ${\rm
C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{2})$ such that

$V_{+}\oplus
W_{+}=P_{\zeta,\partial}{\mathbb{H}}^{3/2}(\Sigma;E\otimes\mathbb{C}^{2}).$

Define the map $g:V_{-}\to V_{+}$ by

$g\begin{pmatrix}0\\\ \eta_{1}\end{pmatrix}:=\begin{pmatrix}-T\eta_{1}\\\
-\Lambda_{DN}T\eta_{1}\end{pmatrix}.$

A short computation shows that

${g}^{\ast}({V}^{\ast}_{+}\cap{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{2}_{\rm
op}))\subset{V}^{\ast}_{-}\cap{\mathbb{H}}^{{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{2}_{\rm
op}),$

and that

$B_{a}=\left\\{v+gv:v\in V_{-}\right\\}\oplus W_{+}.$

This gives a graphical decomposition of $B_{a}$. The existence of a graphical
decomposition will by Theorem 5.6 give an alternative proof of the fact that
$B_{a}$ is elliptic. The construction of $g$ in terms of a pseudodifferential
operator, combined with Proposition 6.7, gives an alternative explanation for
the fact that $B_{a}$ is $s$-semiregular for any $s\geq 0$.

##  7\. Differential operators with positive principal symbols and their Weyl
laws

In this section we shall prove a Weyl law for general lower semi-bounded self-
adjoint regular realisations of an elliptic differential operator $D$ of order
$m>0$ with positive interior principal symbol. Weyl laws with error estimates
are known under mild assumptions from [agmonasy, horasy], see also [grubb77a,
Theorem 1]. The authors are grateful to Gerd Grubb who showed us how to prove
the Weyl law for general boundary conditions by combining the spectral
asymptotics of the Dirichlet realisation with [grubb68].

The Weyl formula for the asymptotic behaviour of the eigenvalues of Dirichlet
realisations of elliptic operators dates back to Weyl [weyl12], where his
focus was on the Laplacian. Weyl’s result was extended to more general
operators in many contributions to the literature, as summed up e.g. in
Agmon’s paper [agmonasy] (including the treatment by Gårding [garding53] of
variable-coefficient operators). In [agmonasy] and subsequent treatments by
Hörmander [horasy], Duistermaat-Guillemin [MR405514], Seeley [MR506893] and
Ivrii [MR575202] the focus has been on the remainder term. See more in the
survey of Ivrii [ivrii16]. We shall here just study the principal asymptotics,
however for realisations that are more general than those defined by
pseudodifferential boundary conditions.

We now use results of [grubb68, gerdsgreenbook] to describe the resolvent of
$D_{\rm B}$ under the assumptions stated above. The method of proof will be to
reduce the problem to a well behaved boundary condition by studying the
resolvent difference

$(D_{\rm B}+\lambda)^{-1}-(D_{\rm Dir}+\lambda)^{-1},$

for the Dirichlet extension $D_{\rm Dir}$. The idea of studying this resolvent
difference and its localisation to the boundary is due to Birman [MR0142896,
MR0177311]. The precise description that we use follows Grubb [grubb68].
Recall the results of Subsection 6.3 for the Dirichlet realisation. We write
$P_{\zeta,\partial}(\lambda)$ for the boundary decomposing projection on
$\check{\mathrm{H}}(D)$ induced by the projection
$P_{\zeta}(\lambda):=1-(D_{\rm Dir}+\lambda)^{-1}(D_{\rm max}+\lambda)$ on
$\mathrm{dom}(D_{\rm max})$ (cf. Lemma 6.11).

######  Theorem 7.1.

Let $D=D^{\dagger}$ be formally selfadjoint elliptic operator of order $m$
acting on the sections of a hermitian vector bundle $E$ on an $n$-dimensional
manifold with boundary. Assume that $D$ has positive interior principal symbol
$\sigma_{D}$ and that $B$ is a regular and self-adjoint boundary condition.
Then $D_{\rm B}$ has discrete spectrum and for some $\lambda>0$ large enough
then we can write

$(D_{\rm B}+\lambda)^{-1}=Q_{+}+G+\mathfrak{G}_{\rm B},$

where

  1. (i)

$Q_{+}$ is the truncation of a pseudo-differential operator
$\hat{Q}\in\Psi^{-m}_{\rm cl}(\hat{M};\hat{E})$ that acts on a hermitian
vector bundle $\hat{E}$ extending $E$ to a closed manifold $\hat{M}$
containing $M$ as a smooth domain, and satisfies that

$\sigma_{-m}(\hat{Q})|_{T^{*}M\setminus M}=\sigma_{D}^{-1}.$

  2. (ii)

$G$ is a singular Green operator of order $-m$ and class $0$ (cf.
[gerdsgreenbook]).

  3. (iii)

$\mathfrak{G}_{\rm B}$ is an operator that factors over continuous mappings

$\displaystyle{\rm L}^{2}(M;E)\xrightarrow{P_{\ker(D_{\rm max}+\lambda)}}$
$\displaystyle\ker(D_{\rm
max}+\lambda)\xrightarrow{\gamma}\mathcal{C}_{D+\lambda}\xrightarrow{P_{\overline{P_{\zeta,\partial}(\lambda)B}^{\check{\mathrm{H}}(D)}}}\overline{P_{\zeta,\partial}(\lambda)B}^{\check{\mathrm{H}}(D)}\xrightarrow{\mathfrak{T}_{\rm
B}^{-1}}$ $\displaystyle\xrightarrow{\mathfrak{T}_{\rm
B}^{-1}}P_{\zeta,\partial}(\lambda)B\hookrightarrow{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\hookrightarrow{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\xrightarrow{\mathcal{K}}{\rm
L}^{2}(M;E),$

where $P_{\ker(D_{\rm max}+\lambda)}$ and
$P_{\overline{P_{\zeta,\partial}(\lambda)B}^{\check{\mathrm{H}}(D)}}$ denotes
the orthogonal projectors and $\mathfrak{T}_{\rm
B}:P_{\zeta,\partial}(\lambda)B\to\overline{P_{\zeta,\partial}(\lambda)B}^{\check{\mathrm{H}}(D)}$
is the operator constructed in [grubb68, Chapter II] (and there denoted by the
densely defined $T:V\to W$ for
$V=W=\overline{P_{\zeta,\partial}(\lambda)B}^{\check{\mathrm{H}}(D)}$ with
$\mathrm{dom}(T)=P_{\zeta,\partial}(\lambda)B$).

We remark that all three operators $Q_{+}$, $G$ and $\mathfrak{G}_{\rm B}$
depend on the choice of $\lambda$. However, $Q_{+}$ and $G$ are independent of
the boundary condition $B$ (a priori there is an implicit dependence via the
choice of $\lambda$, but it is irrelevant for the structural statements we are
interested in).

###### Proof.

It follows from [gerdsgreenbook, Section 3.3] that we, for $\lambda>0$ outside
a discrete set of real numbers, can write

$(D_{\rm Dir}+\lambda)^{-1}=Q_{+}+G,$

where $Q_{+}$ and $G$ are as in the statement of the theorem. The operator
$D_{\rm B}$ is self-adjoint because $D$ is formally self-adjoint and $B$ is
Lagrangian. Since $\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm m}(M;E)$, the
inclusion $\mathrm{dom}(D_{\rm B})\hookrightarrow{\rm L}^{2}(M;E)$ is compact.
Therefore $D_{\rm B}$ has only discrete spectrum. For $\lambda$ outside a
discrete subset so that $D_{\rm B}+\lambda$ and $D_{\rm Dir}+\lambda$ are
invertible, it follows from [grubb68, Chapter II, Theorem 1.4] that

$(D_{\rm B}+\lambda)^{-1}-(D_{\rm Dir}+\lambda)^{-1}=\mathfrak{G}_{\rm B},$

where $\mathfrak{G}_{\rm B}$ is as in the statement of the theorem. This
proves the theorem. ∎

######  Lemma 7.2.

The singular values of the operators appearing in Theorem 7.1 have the
following asymptotics:

  1. (i)

$\mu_{k}(Q_{+})=c_{Q}k^{-\frac{m}{n}}+o(k^{-\frac{m}{n}})$ where

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

  2. (ii)

$\mu_{k}(G)=O(k^{-\frac{m}{n-1}})$

  3. (iii)

$\mu_{k}(\mathfrak{G}_{\rm B})=O(k^{-\frac{m}{n-1}})$

###### Proof.

Item i) follows from [gerdsgreenbook, Proposition 4.5.3]. One can also prove
item i) by hand by applying the Weyl law on a closed manifold, see e.g.
[horIV, Chapter XXIX] or [shubinsbook, Chapter III], to suitable perturbations
of $G_{+}$. The statement of item ii) follows from [grubb84]. The statement of