item iii) follows from that $\mathfrak{G}_{\rm B}$ factors over the inclusion
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\hookrightarrow{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
whose singular value sequence is $O(k^{-m/(n-1)})$ as $k\to\infty$. Indeed,
using a Weyl law on the $n-1$-dimensional compact manifold $\Sigma$ shows that
the singular values of ${\rm H}^{\rm s+m}(\Sigma;E)\hookrightarrow{\rm H}^{\rm
s}(\Sigma;E)$ is $O(k^{-m/(n-1)})$ as $k\to\infty$ for any $s\in\mathbb{R}$. ∎

######  Remark 7.3.

The statement that $\mu_{k}(Q_{+}+G)=\mu_{k}(D_{\rm
Dir}^{-1})=c_{Q}k^{-\frac{m}{n}}+o(k^{-\frac{m}{n}})$ is a direct consequence
of the spectral asymptotics of the Dirichlet realisation, see [agmonasy].
Compare to [garding53] and [grubb77a, Theorem 1]. Therefore
$\mu_{k}(Q_{+})=c_{Q}k^{-\frac{m}{n}}+o(k^{-\frac{m}{n}})$ follows from that
$\mu_{k}(G)=o(k^{-\frac{m}{n}})$ by [grubb84] and the Weyl-Fan inequality (see
for instance [simontrace, Theorem 1.7]). The proof above is included since it
shows how the different components in the resolvent of $D_{B}$ contributes to
the spectral asymptotics.

######  Theorem 7.4.

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
defining a lower semi-bounded realisation $D_{B}$. Define the constant

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

The operator $D_{\rm B}$ has discrete real spectrum, and its spectrum
$\lambda_{0}(D_{\rm B})\leq\lambda_{1}(D_{\rm B})\leq\lambda_{2}(D_{\rm
B})\leq\cdots$ satisfies that

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

###### Proof.

Using Theorem 7.1, Lemma 7.2 and the Weyl-Fan inequality (see for instance
[simontrace, Theorem 1.7]) we conclude that

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

Here $(\mu_{k}(D_{\rm B}))_{k\in\mathbb{N}}$ denotes the singular value
sequence of the unbounded operator $D_{B}$, so by definition $\mu_{k}(D_{\rm
B})=\sqrt{\mu_{k}((1+D_{\rm B}^{*}D_{\rm B})^{-1})^{-1}-1}$. Since $D_{B}$ is
self-adjoint and lower semi-bounded, $\mu_{k}(D_{\rm B})=\lambda_{k}(D_{\rm
B})$ for $k$ large enough and the theorem follows. ∎

######  Remark 7.5.

The operator $D_{B}$ is lower semi-bounded if and only if the operator
$\mathfrak{T}_{\rm B}$ from Theorem 7.1, item iii), is lower semi-bounded. For
a certain class of elliptic boundary conditions, the operator
$\mathfrak{T}_{\rm B}$ is realised as a pseudodifferential operator in
[grubb74, Section 8] and in this case the asymptotics of the negative
eigenvalues was studied.

######  Remark 7.6.

Theorem 7.4 was proven in [agmonasy] under the additional hypothesis that
there is a $k>n/m$ such that $D_{B}^{k}$ is regular, i.e.
$\mathrm{dom}(D_{B}^{k})\subseteq{\rm H}^{\rm km}(M;E)$. By complex
interpolation, using that $D_{B}$ is self-adjoint, this condition implies
$\mathrm{dom}(D_{B}^{l})\subseteq{\rm H}^{\rm lm}(M;E)$ for $l=1,\ldots,k$.
Note that $\mathrm{dom}(D_{B}^{k})\subseteq{\rm H}^{\rm km}(M;E)$ if and only
if $B$ is $m(k-1)$-semiregular. Let us provide a simple example where the
assumption appearing in [agmonasy] fails, but Theorem 7.4 holds. Our example
is quite similar to Example 6.13 and provided solely for the purpose of giving
intuition for the difference in assumptions between Theorem 7.4 and
[agmonasy].

Assume that $M\subseteq\mathbb{R}^{n}$ is a compact domain with smooth
boundary and consider a shift of the standard Laplacian

$D=\Delta+1=1-\sum_{j=1}^{n}\partial_{x_{j}}^{2}.$

In particular, both the Dirichlet problem and the Neumann problem for $D$ are
uniquely solvable. For a real-valued function $a\in
L^{\infty}(\partial\Omega,\mathbb{R})$, consider the Robin boundary condition

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

A short computation with Proposition 2.26 (or in other words using Green’s
formula) shows that $B_{a}$ is self-adjoint. Since the operator $D_{B_{a}}$ is
defined from the closed lower semibounded quadratic form

$q_{a}(u):=\int_{M}(|u|^{2}+|\nabla
u|^{2})\mathrm{d}V(x)+\int_{\Sigma}a|u|^{2}\mathrm{d}S(x),\quad u\in{\rm
H}^{\rm 1}(M),$

the operator $D_{B_{a}}$ is lower semi-bounded and so satisfies the
assumptions of Theorem 7.4 as soon as $\mathrm{dom}(D_{B_{a}})\subseteq{\rm
H}^{\rm m}(M;E)$, i.e. $B_{a}$ is semi-regular (or equivalently regular, by
self-adjointness).

By the same argument as in Example 6.13, $B_{a}$ is $s$-semiregular if and
only if for any $(\xi_{0},\xi_{1})\in B_{a}$, the property
$(\Lambda_{DN}+a)\xi_{0}\in{\rm H}^{\rm s+1/2}(\Sigma)$ implies that
$\xi_{0}\in{\rm H}^{\rm s+3/2}(\Sigma)$. If $a$ is such that the
multiplication operator $a:{\rm H}^{\rm s+3/2}(\Sigma)\to{\rm H}^{\rm
s+1/2}(\Sigma)$ is compact, then for a generic444In the sense that for
$\varepsilon\in(-1,1)$ outside of a discrete set of points
$\Lambda_{DN}+a+\varepsilon:{\rm H}^{\rm s+3/2}(\Sigma)\to{\rm H}^{\rm
s+1/2}(\Sigma)$ is invertible. such $a$, the boundary condition $B_{a}$ is
$s$-semiregular. Conversely, if $B_{a}$ is $s$-semi-regular, then elliptic
regularity of $\Lambda_{DN}$ and the fact that $(\Lambda_{DN}+a)\xi_{0}\in{\rm
H}^{\rm s+1/2}(\Sigma)\;\Rightarrow\xi_{0}\in{\rm H}^{\rm s+3/2}(\Sigma)$
implies that $a:{\rm H}^{\rm s+3/2}(\Sigma)\to{\rm H}^{\rm s+1/2}(\Sigma)$ is
continuous.

Let us now give a condition on $a$ such that
$\mathrm{dom}(D_{B_{a}}^{k})\subseteq{\rm H}^{\rm km}(M;E)$ holds for $k=1$
but not for $k=2$, and therefore for no $k>1$. We note that
$u\in\mathrm{dom}(D_{B_{a}}^{2})$ if and only if
$D_{B_{a}}u\in\mathrm{dom}(D_{B_{a}})$. Therefore, the inclusion
$\mathrm{dom}(D_{B_{a}}^{2})\subseteq{\rm H}^{\rm 4}(M)$ holds if and only if
for $u\in\mathrm{dom}(D_{B_{a}})$ such that $D_{B_{a}}u\in{\rm H}^{\rm 2}(M)$
it holds that $u\in{\rm H}^{\rm 4}(M)$. From Theorem 6.6 we see that
$\mathrm{dom}(D_{B_{a}}^{2})\subseteq{\rm H}^{\rm 4}(M)$ if and only if
$B_{a}$ is $2$-semiregular. By the argument above, this occurs if the function
$a$ makes the multiplication operator $a:{\rm H}^{\rm 3/2}(\Sigma)\to{\rm
H}^{\rm 1/2}(\Sigma)$ compact but the multiplication operator $a:{\rm H}^{\rm
7/2}(\Sigma)\to{\rm H}^{\rm 5/2}(\Sigma)$ is not even continuous. This occurs
when $a\in{\rm C}^{1}(\Sigma)$ but $a\notin{\rm H}^{\rm 5/2}(\Sigma)$. Indeed,
continuity of $a:{\rm H}^{\rm 7/2}(\Sigma)\to{\rm H}^{\rm 5/2}(\Sigma)$ fails
if $a=a1\notin{\rm H}^{\rm 5/2}(\Sigma)$ and ${\rm C}^{1}(\Sigma)$ acts
continuously ${\rm H}^{\rm 1/2}(\Sigma)$ (by interpolation of the continuous
actions ${\rm H}^{\rm 1}(\Sigma)$ and ${\rm L}^{2}(\Sigma)$) and ${\rm H}^{\rm
3/2}(\Sigma)\to{\rm H}^{\rm 1/2}(\Sigma)$ is compact by Rellich’s theorem. For
an example of an $a\in{\rm C}^{1}(\Sigma)$ with $a\notin{\rm H}^{\rm
5/2}(\Sigma)$, take $M$ to be the unit disc, so $\Sigma=S^{1}$, and define $a$
from the lacunary Fourier series

$a(z):=\sum_{k=0}^{\infty}2^{-2k}(z^{2^{k}}+z^{-2^{k}})\in{\rm
C}^{1}(S^{1})\setminus{\rm H}^{\rm 5/2}(S^{1}).$ (25)

For $a$ as in Equation (25), the regular realisation $(1+\Delta)_{B_{a}}$ on
the unit disc provides a concrete example where the Weyl law holds by Theorem
7.4 but the assumption $\mathrm{dom}(D_{B}^{k})\subseteq{\rm H}^{\rm km}(M;E)$
for some $k>n/m$ from [agmonasy] fails.

######  Corollary 7.7.

Let $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ be an elliptic
differential operator of order $m>0$ and $B\subseteq\check{\mathrm{H}}(D)$ a
regular and $m$-semiregular boundary condition. Then the sequence of singular
values of the operator $D_{\rm B}:{\rm L}^{2}(M;E)\dashrightarrow{\rm
L}^{2}(M;F)$ satisfies that

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

where

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

and $\sigma_{D}$ denotes the interior principal symbol of $D$.

###### Proof.

We have that $\mu_{k}(D_{\rm B})^{2}=\lambda_{k}(D_{\rm B}^{*}D_{\rm B})$ and
by Corollary 6.8, $D_{\rm B}^{*}D_{\rm B}$ is regular if $B$ is regular and
$m$-semiregular. The corollary follows from Theorem 7.4 since $D_{\rm
B}^{*}D_{\rm B}$ is non-negative. ∎

######  Remark 7.8.

A consequence of Corollary 7.7 is that if $D$ is a formally self-adjoint,
elliptic differential operator of order $m>0$ and
$B\subseteq\check{\mathrm{H}}(D)$ is a Lagrangian, regular and $m$-semiregular
boundary condition, then $D_{\rm B}$ has discrete spectrum consisting of real
eigenvalues that asymptotically behaves as

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

where

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

For more details on the spectral asymptotics of the sequences of negative and
positive eigenvalues, see [gerdsgreenbook, Chapter 4.5].

######  Remark 7.9.

The assumption that $B$ is regular in Theorem 7.4 can in general not be
dropped. To give an example thereof, we assume for simplicity that $D$ is as
in Theorem 7.4 and $D_{\rm Dir}$ is invertible. Following the notations of
[grubb68, grubb74], for a small $m^{\prime}<m$ we take a regular
$T_{0}\in\Psi^{\boldsymbol{m^{\prime}}}(\Sigma;E\otimes\mathbb{C}^{m})$ with
$[T_{0},P_{\mathcal{C}}]\in\Psi^{\boldsymbol{m^{\prime}-m}}(\Sigma;E\otimes\mathbb{C}^{m})$
and define $T:=P_{\mathcal{C}}T_{0}P_{\mathcal{C}}$. The operator $T$ is a
densely defined Fredholm operator $\mathcal{C}_{D}\to\mathcal{C}_{D}$ with
$\mathrm{dom}(T)=P_{\mathcal{C}}{\mathbb{H}}^{{m^{\prime}-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
Assume for simplicity that $T$ is invertible. Let $D_{T}$ be the realisation
of $D$ constructed as in [grubb68, Chapter II, Proposition 1.2]. The
realisation $D_{T}$ is not regular, but
$\gamma(\mathrm{dom}(D_{T}))\subseteq{\mathbb{H}}^{{m^{\prime}-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
cf. [grubb74, Corollary 6.10]. By [grubb68, Chapter II, Theorem 1.4], we have
that

$D_{T}^{-1}-D_{\rm Dir}=T^{-1},$

where $T^{-1}$ is “extended by zero” to an operator on ${\rm
L}^{2}(M;E)=\mathcal{C}_{D}\oplus\mathrm{ran}(D_{\rm min})$. If
$m^{\prime}<m(n-1)/n$, the proof of Theorem 7.4 breaks down as it only
produces the lower estimate

$\mu_{k}(D_{T})\gtrsim k^{\frac{m^{\prime}}{n-1}},\quad\mbox{as
$k\to\infty$}.$

##  8\. Rigidity results for the index of elliptic differential operators

In this section we prove a rigidity result for the Fredholm index of Fredholm
realisations $D_{\rm B}$. As noted in Remark 2.13, the index formula of
Theorem 2.12 implies homotopy invariance for the choice of Fredholm boundary
condition when $D$ is fixed. We now extend this observation to more general
homotopies.

We topologise the space of pseudodifferential operators by declaring the
symbol mappings and its local splittings defined from Kohn-Nirenberg
quantisation, as well as the inclusion of the smoothing operators, to be
continuous. The space of order $m$ differential operators ${\rm
C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ will be topologised as linear
operators between the Fréchet topologies of ${\rm C}^{\infty}(M;E)$ and ${\rm
C}^{\infty}(M;F)$. Equivalently, we can topologise the space of order $m$
differential operators ${\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ in
terms of the ${\rm C}^{\infty}$-topology on the coefficients of the operator.

######  Theorem 8.1.

Assume that $(D_{t})_{t\in[0,1]}$ is a family of elliptic differential
operators of order $m>0$ such that

  1. (i)

The family is continuous in the topology on the space of order $m$
differential operators ${\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$.

  2. (ii)

$\mathrm{dom}(D_{t,{\rm max}})$ is independent of $t$.

Let $[0,1]\ni t\mapsto P_{t}\in\mathbb{B}(\check{\mathrm{H}}(D))$ be a norm
continuous family of projectors such that
$B_{t}:=(1-P_{t})\check{\mathrm{H}}(D)$ satisfies that
$(B_{t},\mathcal{C}_{D_{t}})$ is a Fredholm pair in $\check{\mathrm{H}}(D)$
for all $t\in[0,1]$. Then it holds that

$\operatorname{ind}(D_{0,B_{0}})=\operatorname{ind}(D_{1,B_{1}}).$

In order to prove this theorem, we first prove the following two lemmas.

######  Lemma 8.2.

Assume that $(D_{t})_{t\in[0,1]}$ is a family of elliptic differential
operators of order $m>0$ as in Theorem 8.1. Then viewing $D_{t,\max}$ as a
continuous linear mapping $\mathrm{dom}(D_{0,\max})\to{\rm L}^{2}(M;F)$ it
holds that $t\mapsto D_{t,\max}$ defines a norm-continuous mapping
$[0,1]\to\mathbb{B}(\mathrm{dom}(D_{0,\max}),{\rm L}^{2}(M;F))$.

###### Proof.

The proof of this lemma is slightly technical and relies on the construction
of approximate Calderón projectors following Theorem 3.8 and 3.18. The family
$(D_{t})_{t\in[0,1]}$ extends to a continuous family of elliptic differential
operators on a closed manifold containing $M$ as a domain, this is as in the
proof of Theorem 3.18 above. We follow the notation in the proof of Theorem
3.18. Since the family $(D_{t})_{t\in[0,1]}$ is continuous in the topology on
the space of order $m$ differential operators, we can construct a continuous
family $(\hat{T}_{t})_{t\in[0,1]}\subseteq\Psi^{-m}_{\rm cl}(M;F,E)$ such that
$\hat{T}_{t}$ is a parametrix of $\hat{D}_{t}$ for all $t\in[0,1]$. Following
Equation (6), we define an approximate Poisson operator

$\tilde{\mathcal{K}}_{t}\left(\xi_{j}\right)_{j=0}^{m-1}=\sum_{j=0}^{m-1}\sum_{l=0}^{m-1-j}(-i)^{l+1}\hat{T}_{t}[A_{m-l-j-1}\xi_{j}\otimes\delta_{t=0}^{(l)}].$

Using the same arguments as in [seeley65], it holds that
$\tilde{\mathcal{K}}_{t}:\check{\mathrm{H}}(D)\to\mathrm{dom}(D_{t,\rm max})$
continuously for all $t\in[0,1]$. Indeed, letting $\mathcal{K}_{D_{t}}$ denote
the Poisson operator of $D_{t}$ (see [seeley65] or Theorem 3.18) above, we
have that
$\tilde{\mathcal{K}}_{t}-\mathcal{K}_{D_{t}}:\check{\mathrm{H}}(D)\to{\rm
C}^{\infty}(M;E)$. We therefore also have that
$\gamma\circ\tilde{\mathcal{K}}_{t}-P_{\mathcal{C}_{t}}\in\Psi^{-\infty}(\Sigma;E\otimes\mathbb{C}^{m})$
is a smoothing operator, and in particular
$\gamma\circ\tilde{\mathcal{K}}_{t}$ is an approximate Calderón projection as
in Theorem 3.8.

Consider the family of bounded operators
$R_{t}:=\tilde{\mathcal{K}}_{t}\circ\gamma:\mathrm{dom}(D_{\max})\to\mathrm{dom}(D_{\max})$.
We note that for $f\in C^{\infty}(M;E)$,

$D_{t}R_{t}f=\sum_{j=0}^{m-1}\sum_{l=0}^{m-1-j}(-i)^{l+1}(D_{t}\hat{T}_{t}-1)[A_{m-l-j-1}\gamma_{j}(f)\otimes\delta_{t=0}^{(l)}],$

because distributions supported on $\Sigma$ vanish when restricted to $M$.
However, since $\hat{T}_{t}$ is a continuous family of parametrices, the
mapping $[0,1]\ni
t\mapsto(\hat{D}_{t}\hat{T}_{t}-1)\in\Psi^{-\infty}(\hat{M};\hat{F})$ is
continuous. We conclude that $t\mapsto D_{t}R_{t}$ factors over
$\scalebox{1.5}{a}\gamma:\mathrm{dom}(D_{\max})\to\hat{\mathrm{H}}(D^{\dagger})$,
the continuous inclusion
$\hat{\mathrm{H}}(D^{\dagger})\hookrightarrow\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})$
and a continuous family of operators
$\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to{\rm C}^{\infty}(M;F)$.
As such, $[0,1]\ni t\mapsto
D_{t}R_{t}\in\mathbb{B}(\mathrm{dom}(D_{\max}),{\rm H}^{\rm s}(M;F))$ is norm-
continuous for all $s\in\mathbb{R}$, and in particular $[0,1]\ni t\mapsto
D_{t}R_{t}\in\mathbb{B}(\mathrm{dom}(D_{\max}),{\rm L}^{2}(M;F))$ is norm-
continuous.

Since
$\tilde{\mathcal{K}}_{t}-\mathcal{K}_{D_{t}}:\check{\mathrm{H}}(D)\to{\rm
C}^{\infty}(M;E)$, we have that $(1-R_{t}):\mathrm{dom}(D_{\rm max})\to{\rm
H}^{\rm m}(M;E)$. Indeed, we can write $\mathrm{dom}(D_{\rm max})={\rm H}^{\rm
m}(M;E)+\ker(D_{t,\max})$ by Example 3.21 and $1-R_{t}$ is smoothing on
$\ker(D_{t,\max})$ because of
$\tilde{\mathcal{K}}_{t}-\mathcal{K}_{D_{t}}:\check{\mathrm{H}}(D)\to{\rm
C}^{\infty}(M;E)$. Using the semi-classical nature of the arguments in
[seeley65], it follows that $[0,1]\ni t\mapsto
1-R_{t}\in\mathbb{B}(\mathrm{dom}(D_{\rm max}),{\rm H}^{\rm m}(M;E))$ is norm-
continuous. It is clear that $[0,1]\ni t\mapsto D_{t}\in\mathbb{B}({\rm
H}^{\rm m}(M;E),{\rm L}^{2}(M,E))$ is continuous. Therefore $[0,1]\ni t\mapsto
D_{t}(1-R_{t})\in\mathbb{B}(\mathrm{dom}(D_{\rm max}),{\rm L}^{2}(M,E))$.

Since the map $t\mapsto D_{t}=D_{t}R_{t}+D_{t}(1-R_{t})$ is a sum of
continuous maps $[0,1]\to\mathbb{B}(\mathrm{dom}(D_{\rm max}),{\rm
L}^{2}(M,E))$, the map $t\mapsto D_{t,\max}$ defines a norm-continuous mapping
$[0,1]\to\mathbb{B}(\mathrm{dom}(D_{0,\max}),{\rm L}^{2}(M;F))$. ∎

######  Lemma 8.3.

Assume that $(D_{t})_{t\in[0,1]}$ is a family of elliptic differential
operators of order $m>0$ as in Theorem 8.1 and define the family of self-
adjoint operators

$\tilde{D}_{t}:=\begin{pmatrix}0&D_{t,{\rm min}}\\\ D^{\dagger}_{t,{\rm
max}}&0\end{pmatrix},$

for $t\in[0,1]$. Then for each $\lambda\in\mathbb{R}\setminus\\{0\\}$, the
mapping $t\mapsto(\lambda i+\tilde{D}_{t})^{-1}$ is a norm-continuous mapping
$[0,1]\to\mathbb{B}({\rm L}^{2}(M;F\oplus E))$.

###### Proof.

By the assumptions of Theorem 8.1, the domain of $\tilde{D}_{t}$ is constant
in $t$. We have for $t_{1},t_{2}\in[0,1]$ that

$(\lambda i+\tilde{D}_{t_{1}})^{-1}-(\lambda
i+\tilde{D}_{t_{2}})^{-1}=(\lambda
i+\tilde{D}_{t_{1}})^{-1}(\tilde{D}_{t_{2}}-\tilde{D}_{t_{1}})(\lambda
i+\tilde{D}_{t_{2}})^{-1}$

Since $(\lambda i+\tilde{D}_{t})^{-1}:{\rm L}^{2}(M;F\oplus
E)\to\mathrm{dom}(\tilde{D}_{t})=\mathrm{dom}(D_{t,\min})\oplus\mathrm{dom}(D_{t,\max}^{\dagger})$
with the uniform norm bound $|\lambda|^{-1}$ as $t$ varies, the lemma follows
from Lemma 8.2 (applied to $(D_{t}^{\dagger})_{t\in[0,1]}$) and the fact that
$t\mapsto D_{t,\min}$ clearly defines a norm-continuous mapping
$[0,1]\to\mathbb{B}({\rm H}^{\rm m}_{\rm 0}(M;E),{\rm L}^{2}(M;F))$. ∎

###### Proof of Theorem 8.1.

The ellipticity of each $D_{t}$ ensures that $\mathrm{dom}(D_{t,\min})={\rm
H}^{\rm m}_{\rm 0}(M;E)$ is constant in $t$. We remark that item ii ensures
that
$\check{\mathrm{H}}(D_{t})\cong\mathrm{dom}(D_{t,\max})/\mathrm{dom}(D_{t,\min})=\mathrm{dom}(D_{t,\max})/{\rm
H}^{\rm m}_{\rm 0}(M;E)$ is independent of $t$, justifying the notation
$\check{\mathrm{H}}(D)$ for the Cauchy data space.

The only remaining property to verify in order to invoke Theorem B.32 below is
that the mappings

$[0,1]\ni t\mapsto\begin{pmatrix}0&D_{t,{\rm min}}\\\ D^{\dagger}_{t,{\rm
max}}&0\end{pmatrix}\quad\mbox{and}\quad[0,1]\ni
t\mapsto\begin{pmatrix}0&D^{\dagger}_{t,{\rm min}}\\\ D_{t,{\rm
max}}&0\end{pmatrix},$

are continuous in the gap topology. Lemma 8.3 applied to the families
$(D_{t})_{t\in[0,1]}$ and $(D_{t}^{\dagger})_{t\in[0,1]}$ furnishes us with
this fact. ∎

######  Corollary 8.4.

Let $D$ be an elliptic differential operator and $B$ a regular boundary
condition. Assume that $B$ is graphically decomposed with respect to the
Calderon projector by $(V_{+},W_{+},V_{-},W_{-},g)$ as in Definition 5.3. Then
it holds that

$\operatorname{ind}(D_{B})=\dim(W_{+})-\dim(W_{-})+\dim(\ker(D_{\min})-\dim(\ker(D_{\min}^{\dagger}).$

###### Proof.

We define the continuous path of regular boundary conditions
$(B_{t})_{t\in[0,1]}$ from the continuous path of data for a graphical
decomposition $(V_{+},W_{+},V_{-},W_{-},tg)_{t\in[0,1]}$. Note that

$B_{0}=W_{+}\oplus V_{-}\quad\mbox{and}\quad
B_{0}^{*}=\scalebox{1.5}{a}_{\dagger}^{-1}({V}^{\ast}_{+}\cap{\mathbb{H}}^{\frac{1}{2}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\oplus{W}^{\ast}_{-}).$ (26)

We also note that $V_{+}\oplus
W_{+}=\mathcal{C}_{D}\cap{\mathbb{H}}^{m-1/2}(\Sigma;E\otimes\mathbb{C}^{m})$
by the definition of a graphical decomposition with respect to the Calderon
projection. By Theorem 8.1, we have that

$\operatorname{ind}(D_{B})=\operatorname{ind}(D_{B_{0}})=\operatorname{ind}(D_{B_{1}}).$

The proof now follows from Theorem 2.12 upon proving that
$\operatorname{ind}(B_{0},\mathcal{C}_{D})=\dim(W_{+})-\dim(W_{-})$. This
follows from the fact that $B_{0}\cap\mathcal{C}_{D}=W_{+}$ and
$B_{0}^{*}\cap\mathcal{C}_{D^{\dagger}}=\scalebox{1.5}{a}_{\dagger}^{-1}({W}^{\ast}_{-})$
by Equation (26). ∎

######  Theorem 8.5.

Assume that $(D_{t})_{t\in[0,1]}$ is a family of elliptic differential
operators of order $m>0$ which is continuous in the topology on the space of
order $m$ differential operators ${\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(M;F)$. Let
$(P_{t})_{t\in[0,1]}\subseteq\mathbb{B}({\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}))$
be a norm continuous family of projectors such that

  1. (1)

For all $t\in[0,1]$, $P_{t}$ extends to a bounded operator on
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.

  2. (2)

For all $t\in[0,1]$, $\alpha\in\\{-1/2,m-1/2\\}$ the operator

$P_{\mathcal{C}_{t}}-(1-P_{t}):{\mathbb{H}}^{\alpha}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{\alpha}(\Sigma;E\otimes\mathbb{C}^{m}),$
(27)

is a Fredholm operator, where $P_{\mathcal{C}_{t}}$ denotes the Calderon
projector for $D_{t}$.

Then for all $t\in[0,1]$,
$B_{t}:=(1-P_{t}){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
is a regular boundary condition for $D_{t}$ and the index
$\operatorname{ind}(D_{t,B_{t}})$ is independent of $t$. In particular,

$\operatorname{ind}(D_{0,B_{0}})=\operatorname{ind}(D_{1,B_{1}}).$

The proof goes along similar lines as that of Theorem B.32 but adapted to
regular realisations with domains contained in Sobolev spaces. We note that
Theorem 8.5 in particular holds for a continuous family
$(P_{t})_{t\in[0,1]}\subseteq\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ of pseudodifferential projectors such that
for all $t\in[0,1]$

$p_{+}(D_{t})-(1-\sigma_{\boldsymbol{0}}(P_{t}))\in
C^{\infty}(S^{*}\Sigma;\mathrm{Aut}(\pi^{*}E)).$

###### Proof.

We note that each $B_{t}$ is a regular boundary condition by Theorem 4.14.
Since $(D_{t})_{t\in[0,1]}$ is a continuous family of elliptic differential
operators, it defines a bounded adjointable map of $C[0,1]$-Hilbert
$C^{*}$-modules

$\mathbb{D}:{\rm H}^{\rm m}(M;E)\otimes C[0,1]\to{\rm L}^{2}(M;F)\otimes
C[0,1].$

We write
$\check{C}_{t}:=P_{t}{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and let $\check{C}_{C([0,1])}$ denote the $C[0,1]$-Hilbert $C^{*}$-submodule
of ${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\otimes
C([0,1])$ with fibre $\check{C}_{t}$, it is well-defined by norm continuity of
the family $(P_{t})_{t\in[0,1]}$. We write $P\gamma:{\rm H}^{\rm
m}(M;E)\otimes C([0,1])\to\check{C}_{C([0,1])}$ for the bounded adjointable
mapping obtained from composing the trace mapping with the family
$(P_{t})_{t\in[0,1]}$.

Following Proposition B.30 and Theorem B.32, we define the operator

$\check{L}:{\rm H}^{\rm m}(M;E)\otimes C[0,1]\to\begin{matrix}{\rm
L}^{2}(M;F)\otimes C[0,1]\\\ \oplus\\\
\check{C}\end{matrix},\quad\check{L}u=\begin{pmatrix}\mathbb{D}u\\\ P\gamma
u\end{pmatrix}$

This operator is an adjointable $C[0,1]$-linear map because $\mathbb{D}$ and
$P\gamma$ are adjointable. We note that $\check{L}$ if Fredholm in each fibre
by Proposition B.30 (cf. [BB, Proposition A.1]). By an argument along the same
lines as in Theorem B.32, the adjointable $C[0,1]$-linear map $\check{L}:{\rm
H}^{\rm m}(M;E)\otimes C[0,1]\to{\rm L}^{2}(M;F)\otimes C[0,1]\oplus\check{C}$
is Fredholm with a well defined index $\operatorname{ind}(\check{L})\in
K^{0}([0,1])$. Again arguing as in Theorem B.32, homotopy invariance of
$K$-theory implies that

$\operatorname{ind}(D_{t,B_{t}})=\operatorname{ind}(\check{L}_{t})=r_{t}^{*}\operatorname{ind}(\check{L}),$

where $r_{t}$ denotes the inclusion $\\{t\\}\subseteq[0,1]$, is independent of
$t$. ∎

We end this section with an outlook towards index theory. It poses an
interesting problem to compute $\operatorname{ind}(D_{{\rm B}_{P}})$ where $D$
is an elliptic differential operator $D$ of order $m>0$ and
$B_{P}:=(1-P){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
is a regular pseudo-local boundary condition on $D$ defined from
$P\in\Psi^{\boldsymbol{0}}_{\rm cl}(\Sigma;E\otimes\mathbb{C}^{m})$. While
Theorem 2.12 and Corollary 5.7 give formulas, a more worthwhile goal would be
an explicit formula for the index of $D_{\rm B}$ in terms of the interior
principal symbol of $D$, the germ of $D$ on $\Sigma$, the differential
operator  a defining the boundary pairing and the projection $P$. The Atiyah-
Patodi-Singer index theorem and the index theorem in the extended Boutet de
Monvel calculus of [schulzeseiler] indicates that such an explicit computation
is possible. Under some mild assumptions on $P$, the special case of first
order elliptic operators is susceptible to heat kernel techniques by the
results of [G99].

There is an interesting special case that the authors of this paper have not
found an explicit solution to in the literature. A classical situation is that
$\not{D}$ is a Dirac operator and $P$ is a spectral projection of the boundary
operator of $\not{D}$ constructed from a normal vector field to the boundary.
Atiyah-Patodi-Singer [APS] computed the index in the case of product structure
at the boundary and [grubb92] considered the general case. For more details,
see also [bosswojc, melroseAPS]. Assume that we leave this classical realm and
instead use a general vector field transversal to the boundary to define the
adapted boundary operator $A$. Then $A$ is not necessarily self-adjoint, but
only bisectorial (cf. [BBan]). Is there an explicit formula for the index
$\operatorname{ind}(D_{{\rm B}_{P}})$ for $P=\chi^{+}(A)$? Preferably, such a
formula only uses local data and spectral data of the bisectorial operator
$A$. The results of [G99] indicates that there might be a positive answer. The
importance of this question is seen in that similar questions can be asked on
manifolds with Lipschitz boundary, where a smooth vector field transversal to
the boundary (almost everywhere) makes sense.

##  9\. Symbol computations with Calderón projectors in the first order case

In this section, we study the way in which the Calderón projection of a first
order elliptic operator relates to the the positive spectral projection of an
adapted boundary operator. We show that the difference of these projectors are
an operator of order $-1$ and continuous, but in general not compact on the
Cauchy data space.

Near the boundary,

$D=\sigma\partial_{x_{n}}+A_{1},$ (28)

where $A_{1}=(A_{1}(x_{n}))_{x_{n}\in[0,1]}$ is a family of first order
differential operators acting from $E|_{\Sigma}\to\Sigma$ to
$F|_{\Sigma}\to\Sigma$. By ellipticity of $D$, $\sigma$ is a bundle
isomorphism and $A_{1}$ is a family of elliptic differential operators. We
note that an adapted boundary operator for $D$ is precisely a differential
operator $A$ on $E|_{\Sigma}$ such that $A-\sigma^{-1}A_{1}|_{x_{n}=0}$ is of
order zero.

Recall the construction of $E_{+}(D)$ and $p_{+}(D)$ from the subsection 3.1.

######  Lemma 9.1.

Let $D$ be a first order elliptic differential operator and $A$ an adapted
boundary operator on $E|_{\Sigma}$. Then $\chi^{+}(A)$ is a classical order
zero pseudo-differential operator whose principal symbol is $p_{+}(D)$.

###### Proof.

Since $\chi^{+}(A)$ is constructed as a sectorial projection, it is a
classical order zero pseudo-differential operator, see [grubbsec]. Its
principal symbol is the same as $\chi^{+}(a)$ where $a$ is the principal
symbol of $A$. By construction, $a=\sigma^{-1}a_{1}$ where $a_{1}$ is the
principal symbol of $A_{1}|_{x_{n}=0}$ appearing in the collar decomposition
$D=\sigma\partial_{x_{n}}+A_{1}$. We conclude that
$\sigma_{\partial}(D)=\sigma\partial_{x_{n}}+a_{1}=\sigma(\partial_{x_{n}}+a)$.
It is clear that for $(x^{\prime},\xi^{\prime})\in S^{*}\Sigma$ and $v_{0}\in
E_{x}$, Equation (5), that in this case takes the form

$\begin{cases}\sigma(\partial_{t}+a(x^{\prime},\xi^{\prime}))v(t)=0,\;t>0,\\\
v(0)=v_{0},\end{cases}$

has an exponentially decaying solution if and only if
$v_{0}\in\chi^{+}(a(x^{\prime},\xi^{\prime}))E_{x}$, namely
$v(t)=\mathrm{e}^{-a(x^{\prime},\xi^{\prime})t}v_{0}$. Similarly, there is an
exponentially increasing solution if and only if
$v_{0}\in\chi^{-}(a(x^{\prime},\xi^{\prime}))E_{x}$. So
$E_{+}(D)=\chi^{+}(a)E$ and $E_{-}(D)=\chi^{-}(a)E$ and we can conclude that
the principal symbol of $\chi^{+}(A)$ equals $p_{+}(D)$. ∎

######  Remark 9.2.

By Proposition 3.4, we know that $E_{+}(D)\neq 0$ and $E_{-}(D)\neq 0$. In
fact, each fibre is non-zero if the dimension exceeds $1$. For first order
operators, this can be proven more directly using the symmetry
$a(x^{\prime},\xi^{\prime})=-a(x^{\prime},-\xi^{\prime})$ which implies that
$p_{+}(D)(x^{\prime},\xi^{\prime})=1-p_{+}(D)(x^{\prime},-\xi^{\prime})$ so
$p_{+}(D)\neq 0$ and a similar argument implies $p_{-}(D)\neq 0$.

######  Lemma 9.3.

Let $D$ be a Dirac type operator and $A$ an adapted boundary operator on
$E|_{\Sigma}$ such that

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

near the boundary. Then
$\chi^{+}(A)-P_{\mathcal{C}_{D}}\in\Psi^{-\infty}(\Sigma;E)$ is a smoothing
operator.

###### Proof.

It suffices to prove that $\chi^{+}(A)-Q\in\Psi^{-\infty}(\Sigma;E)$ is a
smoothing operator for $Q$ being the approximate Calderón projection from
Theorem 3.8. Since the construction in Theorem 3.8 is microlocal at the
boundary we can assume that $M=\Sigma\times[0,\infty)$. In this case, any
$f\in\ker D_{\rm max}$ can be written in the form

$f(t)=\mathrm{e}^{-tA}f(0).$

As such, $\mathcal{K}g(t)=\mathrm{e}^{-tA}\chi^{+}(A)g$ and the Calderón
projection $P_{\mathcal{C}_{D}}=\chi^{+}(A)$ is an approximate Calderón
projection. ∎

######  Lemma 9.4.

Let $D$ be a first order elliptic differential operator with adapted boundary
operator $A$. The first term in the Calderón projection is given by

$\displaystyle\sigma_{0}(P_{\mathcal{C}_{D}})=$ $\displaystyle
p_{+}(D)=\chi^{+}(\sigma_{1}(A)).$

In coordinates near a point $x_{0}\in\Sigma$, we have that

$\displaystyle\sigma_{-1}(P_{\mathcal{C}_{D}})(x^{\prime},\xi^{\prime})=$
$\displaystyle\sum_{z:\mathrm{Re}(z)>0}\mathrm{Res}_{z=\lambda}t_{1}(x^{\prime},\xi^{\prime},i\lambda),$

where

$\displaystyle t_{1}(x^{\prime},\xi^{\prime},i\lambda)=$
$\displaystyle-(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}(c_{A,0}(x^{\prime})+c_{R_{0},0}(x^{\prime},0))(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}-$ $\displaystyle-(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}c_{A}\left(x^{\prime},(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}\mathrm{d}_{x^{\prime}}c_{A}(x^{\prime},\xi^{\prime}))(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}\right)-$ $\displaystyle-(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-2}\frac{\partial c_{R_{0}}}{\partial
x_{n}}(x^{\prime},0,\xi^{\prime})(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}.$

Here, we have written $D=\sigma(\partial_{x_{n}}+A+R_{0})$ for an
$x_{n}$-dependent first order differential operator $R_{0}$ on $\Sigma$,
$c_{A}$ denotes the principal symbol of $A$, $c_{A,0}$ the sub-leading term
(and similarly for $A$), and where we use the notation
$c_{A}(x^{\prime},\nu\otimes M):=c_{A}(x^{\prime},\nu)M$ for an endomorphism
valued covector $\nu\otimes M$.

###### Proof.

By the formula for the full symbol of the approximate Calderón projection from
Theorem 3.8, we have that

$Qf=\gamma_{0}\circ T_{0}[\sigma f\otimes\delta_{t=0}],$

where $T_{0}$ is a parametrix for $D$. In particular, we note that

$Qf=\gamma_{0}\circ T[f\otimes\delta_{t=0}],$

where $T$ is a parametrix for $\sigma^{-1}D$. The operator $T$ is a classical
pseudodifferential operator of order $-1$ in a neighbourhood of $M$ in an
extended manifold. In coordinates near a point $x_{0}\in\Sigma$, a full symbol
$t$ of $T$ has an asymptotic expansion $t\sim\sum_{j=0}^{\infty}t_{j}$ and
each $t_{j}=t_{j}(x,\xi)$ is smooth in $x$ and homogeneous of degree $-1-j$ in
$\xi$. Using the methodology of [horIII, Chapter XX], cf. Remark 3.9, the full
symbol of $Q$ can be computed by

$q_{j}(x^{\prime},\xi^{\prime})=\sum_{z:\mathrm{Im}(z)>0}\mathrm{Res}_{\xi_{n}=z}t_{j}(x^{\prime},0,\xi^{\prime},\xi_{n}),$

where $(x^{\prime},\xi^{\prime})$ denote the associated coordinates on
$\Sigma$. As such, we shall need to compute the two terms in the asymptotic
expansion of $T$. By an abuse of notation, we will write
$t_{j}(x^{\prime},\xi^{\prime},\xi_{n})$ for
$t_{j}(x^{\prime},0,\xi^{\prime},\xi_{n})$.

To compute the asymptotic expansion of $T$, we write the full symbol of $D$ as
$c_{D}+c_{D,0}$ where $c_{D}$ is the principal symbol (i.e. a homogeneous
degree 1 polynomial) and $c_{D,0}$ is $\xi$-independent. We can write
$D=\sigma(D_{t}+A+R_{0})$ for an $x_{n}$-dependent first order differential
operator $R_{0}$ on $\Sigma$ with $R_{0}|_{x_{n}=0}$ being a multiplication
operator. We use similar notation for $A$ and $R_{0}$ and note that

$\begin{cases}c_{D}(x^{\prime},x_{n},\xi^{\prime},\xi_{n})&=\sigma(x^{\prime},x_{n})(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime})+c_{R_{0}}(x^{\prime},x_{n},\xi^{\prime})),\;\mbox{and}\\\
c_{D,0}(x^{\prime},x_{n})&=\sigma(x^{\prime},x_{n})(c_{A,0}(x^{\prime})+c_{R_{0},0}(x^{\prime},x_{n})).\end{cases}$

The assumption that $A$ is adapted ensures that
$c_{R_{0}}(x^{\prime},0,\xi^{\prime})=0$. To compute the parametrix of
$\sigma^{-1}D$, we use the symbols

$\begin{cases}\tilde{c}_{D}(x^{\prime},x_{n},\xi^{\prime},\xi_{n})&:=\sigma(x^{\prime},x_{n})^{-1}c_{D}(x^{\prime},x_{n},\xi^{\prime},\xi_{n})=i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime})+c_{R_{0}}(x^{\prime},x_{n},\xi^{\prime},\xi_{n}),\\\
\tilde{c}_{D,0}(x^{\prime},x_{n})&:=\sigma(x^{\prime},x_{n})^{-1}c_{D,0}(x^{\prime},x_{n})=c_{A,0}(x^{\prime})+c_{R_{0},0}(x^{\prime},x_{n}).\end{cases}$

From the well known parametrix construction, the full symbol
$t\sim\sum_{j=0}^{\infty}t_{j}$ of $T$ is computed inductively from that
$t_{0}(x,\xi)=c_{D}(x,\xi)^{-1}$ and

$\displaystyle t_{j+1}=$ $\displaystyle-
t_{0}\tilde{c}_{D,0}t_{j}-t_{0}\sum_{k+|\alpha|=j+1,\,k\leq
j}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}\tilde{c}_{D}D^{\alpha}_{x}t_{k}.$

In particular,

$\displaystyle t_{1}=$ $\displaystyle-
t_{0}\tilde{c}_{D,0}t_{0}-t_{0}\sum_{|\alpha|=1}\partial_{\xi}^{\alpha}\tilde{c}_{D}D^{\alpha}_{x}t_{0}.$

At $x_{n}=0$, we have that

$\displaystyle t_{0}(x^{\prime},\xi^{\prime},\xi_{n})=$
$\displaystyle(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1},$

and

$\displaystyle t_{1}$ $\displaystyle(x^{\prime},\xi^{\prime},\xi_{n})=$
$\displaystyle=$
$\displaystyle-(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}(c_{A,0}(x^{\prime})+c_{R_{0},0}(x^{\prime},0))(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}+$
$\displaystyle+(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}\sum_{j=1}^{n-1}c_{A}(x^{\prime},\mathrm{d}x_{j})(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}\frac{\partial
c_{A}}{\partial
x_{j}}(x^{\prime},\xi^{\prime}))(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}+$
$\displaystyle+(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}\tilde{c}_{D}((x^{\prime},0),\mathrm{d}x_{n})(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}\frac{\partial
c_{R_{0}}}{\partial
x_{n}}(x^{\prime},0,\xi^{\prime})(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}=$
$\displaystyle=$
$\displaystyle-(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}(c_{A,0}(x^{\prime})+c_{R_{0},0}(x^{\prime},0))(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}+$
$\displaystyle+(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}c_{A}\left(x^{\prime},(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}\mathrm{d}_{x^{\prime}}c_{A}(x^{\prime},\xi^{\prime}))(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}\right)+$
$\displaystyle+(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-2}\frac{\partial
c_{R_{0}}}{\partial
x_{n}}(x^{\prime},0,\xi^{\prime})(i\xi_{n}+c_{A}(x^{\prime},\xi^{\prime}))^{-1}.\qed$

######  Lemma 9.5.

Let $A$ be an elliptic first order differential operator on $\Sigma$. The
first term in the positive spectral projection is given by

$\displaystyle\sigma_{0}(\chi^{+}(A))=$
$\displaystyle\chi^{+}(\sigma_{1}(A)).$

In coordinates near a point $x_{0}\in\Sigma$, we have that

$\displaystyle\sigma_{-1}(\chi^{+}(A))=$
$\displaystyle\sum_{z:\mathrm{Re}(z)>0}\mathrm{Res}_{\lambda=z}b_{1}(x^{\prime},\xi^{\prime},\lambda),$

where in the notations of Lemma 9.4

$\displaystyle b_{1}(x^{\prime},\xi^{\prime},\lambda)=$
$\displaystyle-(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}c_{A,0}(x^{\prime})(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}+$ (29) $\displaystyle-(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}c_{A}\left(x^{\prime},(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}\mathrm{d}_{x^{\prime}}c_{A}(x^{\prime},\xi^{\prime}))(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}\right).$

###### Proof.

Consider $B:=(\lambda-A)^{-1}$ as a pseudodifferential with a parameter.
Picking coordinates near a point $x_{0}\in\Sigma$, we write the full symbol
with parameter of $B$ as $b\sim\sum_{j=0}^{\infty}b_{j}$, where
$b_{j}=b_{j}(x^{\prime},\xi^{\prime},\lambda)$ is smooth in $x^{\prime}$ and
homogeneous of degree $-1-j$ in $(\xi^{\prime},\lambda)$. By [grubbsec,
Theorem 3.3], the full symbol of $\chi^{+}(A)$ has the asymptotic expansion
$\sum_{j=0}^{\infty}\chi_{j}$ where

$\chi_{j}(x^{\prime},\xi^{\prime})=\sum_{z:\mathrm{Re}(z)>0}\mathrm{Res}_{\lambda=z}b_{j}(x^{\prime},\xi^{\prime},\lambda).$

As such, we shall need to compute the first two terms in the asymptotic
expansion of $B$.

Using the notations of Lemma 9.4 and its proof, we can compute $b_{j}$
inductively from that $b_{0}(x^{\prime},\xi^{\prime},\lambda)=(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}$ and

$\displaystyle b_{j+1}=$ $\displaystyle-
b_{0}c_{A,0}b_{j}-b_{0}\sum_{k+|\alpha|=j+1,\,k\leq
j}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}c_{A}D^{\alpha}_{x}b_{k}.$

In particular,

$\displaystyle b_{1}=$ $\displaystyle-
b_{0}c_{A,0}b_{0}-b_{0}\sum_{|\alpha|=1}\partial_{\xi}^{\alpha}c_{A}D^{\alpha}_{x}b_{0},$

which is readily computed to reproduce the formula in Equation (29). ∎

######  Definition 9.6.

Let $D$ be an elliptic first order differential operator with adapted boundary
operator $A$. We say that $A$ is a well-adapted boundary operator if
$A=\sigma^{-1}A_{1}|_{x_{n}=0}$, where $A_{1}$ is as in Equation (28).

We note that upon fixing a choice of normal to the boundary, there is a
uniquely determined well adapted boundary operator. In the notations of Lemma
9.4, $A$ is well adapted if and only if $c_{R_{0},0}(x^{\prime},0)=0$ which
holds if and only if $R_{0}|_{x_{n}=0}=0$. The following Proposition is
immediate from Lemma 9.4 and 9.5.

######  Proposition 9.7.

Let $D$ be a first order elliptic differential operator with adapted boundary
operator $A$. Then $\chi^{+}(A)-P_{\mathcal{C}_{D}}$ is of order $-1$ and its
principal symbol is in the notation of Lemma 9.4 given by

$\displaystyle\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})=$
$\displaystyle\sum_{z:\mathrm{Re}(z)>0}\mathrm{Res}_{\lambda=z}p(x^{\prime},\xi^{\prime},\lambda),$

where

$\displaystyle p(x^{\prime},\xi^{\prime},\lambda)=$ $\displaystyle(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}c_{R_{0},0}(x^{\prime},0)(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}+$ $\displaystyle+(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-2}\frac{\partial c_{R_{0}}}{\partial
x_{n}}(x^{\prime},0,\xi^{\prime})(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1},$

If $A$ is well adapted, then $\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})$ is
constructed from residues of the simpler symbol

$\displaystyle p(x^{\prime},\xi^{\prime},\lambda)=$ $\displaystyle(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-2}\frac{\partial c_{R_{0}}}{\partial
x_{n}}(x^{\prime},0,\xi^{\prime})(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-1}.$

######  Corollary 9.8.

Let $D$ be a first order elliptic differential operator with well adapted
boundary operator $A$.

  * •

If $\frac{\partial c_{R_{0}}}{\partial x_{n}}|_{\Sigma}$ commutes with
$c_{A}$, then $\chi^{+}(A)-P_{\mathcal{C}_{D}}\in\Psi^{-2}_{\rm cl}(M;E)$ and
acts compactly on $\check{\mathrm{H}}_{A}(D)$.

  * •

If $\frac{\partial c_{R_{0}}}{\partial x_{n}}|_{\Sigma}$ anti-commutes with
$c_{A}$, then

$\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})=\frac{1}{4}c_{A}^{-2}\chi^{-}(c_{A})\frac{\partial
c_{R_{0}}}{\partial x_{n}}|_{\Sigma}.$

###### Proof.

The first item follows from Proposition 9.7 if we can prove that
$\mathrm{Res}_{\lambda=z}p(x^{\prime},\xi^{\prime},\lambda)=0$ for all $z$.
Since $A$ is well adapted, and $c_{A}$ commutes with $\frac{\partial
c_{R_{0}}}{\partial x_{n}}|_{\Sigma}$, Proposition 9.7 ensures that

$\displaystyle p(x^{\prime},\xi^{\prime},\lambda)=$ $\displaystyle(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-3}\frac{\partial c_{R_{0}}}{\partial
x_{n}}(x^{\prime},0,\xi^{\prime}).$

In particular, the residues of $p$ vanish if $\frac{\partial
c_{R_{0}}}{\partial x_{n}}|_{\Sigma}$ commutes with $c_{A}$.

On the other hand, if $\frac{\partial c_{R_{0}}}{\partial x_{n}}|_{\Sigma}$
anti-commutes with $c_{A}$

$\displaystyle p(x^{\prime},\xi^{\prime},\lambda)=$ $\displaystyle(\lambda-
c_{A}(x^{\prime},\xi^{\prime}))^{-2}(\lambda+c_{A}(x^{\prime},\xi^{\prime}))^{-1}\frac{\partial
c_{R_{0}}}{\partial x_{n}}(x^{\prime},0,\xi^{\prime}).$

Using Proposition 9.7 we compute that in this case

$\displaystyle\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})(x^{\prime},\xi^{\prime})=$
$\displaystyle\frac{1}{4}c_{A}(x^{\prime},\xi^{\prime})^{-2}\chi^{-}(c_{A}(x^{\prime},\xi^{\prime}))\frac{\partial
c_{R_{0}}}{\partial x_{n}}(x^{\prime},0,\xi^{\prime}).\qed$

######  Proposition 9.9.

Let $D$ be an elliptic first order differential operator with adapted boundary
operator $A$. Then $\chi^{+}(A)-P_{\mathcal{C}_{D}}$ is compact on
$\check{\mathrm{H}}_{A}(D)$ if and only if

$\sigma_{0}(\chi^{-}(A))\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})\sigma_{0}(\chi^{+}(A))=0,$

as elements of ${\rm C}^{\infty}(S^{*}\Sigma;\mathrm{End}(\pi^{*}E))$.

###### Proof.

We write

$\displaystyle\chi^{+}(A)-P_{\mathcal{C}_{D}}=$
$\displaystyle\chi^{+}(A)(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{+}(A)+\chi^{-}(A)(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{-}(A)+$
$\displaystyle\qquad+\chi^{+}(A)(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{-}(A)+\chi^{-}(A)(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{+}(A)$
$\displaystyle=$
$\displaystyle\chi^{+}(A)(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{+}(A)+\chi^{-}(A)(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{-}(A)+$
$\displaystyle\quad+\chi^{+}(A)(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{-}(A)+\chi^{-}(A)(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{+}(A)$

Since $\chi^{+}(A)-P_{\mathcal{C}_{D}}\in\Psi^{-1}_{\rm cl}(\Sigma;E)$ the
first two terms are compact for any adapted boundary operator $A$. The third
term is a pseudodifferential operator of order $-1$ and it acts compactly on
$\check{\mathrm{H}}_{A}(D)$ if and only if it acts compactly ${\rm
H}^{\rm{\frac{1}{2}}}(\Sigma;E)\to{\rm H}^{\rm-{\frac{1}{2}}}(\Sigma;E)$ and
as such the third term is compact on $\check{\mathrm{H}}_{A}(D)$ for any
adapted boundary operator $A$. The fourth term is a pseudodifferential
operator of order $-1$ and it acts compactly on $\check{\mathrm{H}}_{A}(D)$ if
and only if it acts compactly ${\rm H}^{\rm-{\frac{1}{2}}}(\Sigma;E)\to{\rm
H}^{\rm{\frac{1}{2}}}(\Sigma;E)$ and as such the fourth term is compact on
$\check{\mathrm{H}}_{A}(D)$ if and only if its symbol vanish, i.e.
$\sigma_{0}(\chi^{-}(A))\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})\sigma_{0}(\chi^{+}(A))=0$.
∎

######  Proposition 9.10.

Let $D$ be a first order elliptic differential operator with well adapted
boundary operator $A$ and assume that $\frac{\partial c_{R_{0}}}{\partial
x_{n}}|_{\Sigma}$ anti-commutes with $c_{A}$. Then
$\chi^{+}(A)-P_{\mathcal{C}_{D}}$ is compact on $\check{\mathrm{H}}_{A}(D)$ if
and only if

$\chi^{-}(c_{A})\frac{\partial c_{R_{0}}}{\partial x_{n}}|_{\Sigma}=0.$

###### Proof.

By Proposition 9.9, it suffices to prove that
$\chi^{-}(c_{A})\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{+}(c_{A})=0$
if and only if $\chi^{-}(c_{A})\frac{\partial c_{R_{0}}}{\partial
x_{n}}|_{\Sigma}=0$ when $\frac{\partial c_{R_{0}}}{\partial x_{n}}|_{\Sigma}$
anti-commutes with $c_{A}$. But by Corollary 9.8,

$\displaystyle\chi^{-}(c_{A})\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})\chi^{+}(c_{A})=$
$\displaystyle\frac{1}{4}c_{A}^{-2}\chi^{-}(c_{A})\frac{\partial
c_{R_{0}}}{\partial x_{n}}|_{\Sigma}\chi^{+}(c_{A})$ $\displaystyle=$
$\displaystyle\frac{1}{4}c_{A}^{-2}\chi^{-}(c_{A})\frac{\partial
c_{R_{0}}}{\partial x_{n}}|_{\Sigma},$

where we in the last line used that $\frac{\partial c_{R_{0}}}{\partial
x_{n}}|_{\Sigma}$ anti-commutes with $c_{A}$. The proposition follows from
that $c_{A}$ is invertible. ∎

######  Example 9.11.

Let us give an easy example showcasing when $\chi^{+}(A)-P_{\mathcal{C}_{D}}$
fails to be compact on $\check{\mathrm{H}}_{A}(D)$. Take $M$ to be the unit
disc and $E=F=M\times\mathbb{C}^{2}$.

We first consider the rather straight forward example of the Dirac type
operator $D_{0}$ given in polar coordinates by

$D_{0}=\begin{pmatrix}0&\partial_{r}+\frac{i}{r}\partial_{\theta}\\\
-\partial_{r}+\frac{i}{r}\partial_{\theta}&0\end{pmatrix}=\sigma(\partial_{r}+A+R_{00}),$

where

$\sigma=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\quad
A=\begin{pmatrix}-i\partial_{\theta}&0\\\
0&i\partial_{\theta}\end{pmatrix}\quad\mbox{and}\quad R_{00}=(r^{-1}-1)A.$

It is readily verified that $A$ is a well adapted boundary operator, and in
this case it is self-adjoint. A short computation with the Cauchy-Riemann
equation in polar coordinates show that $P_{\mathcal{C}_{D_{0}}}=\chi^{+}(A)$
for the operator $D_{0}$.

To jazz up this example a bit, we define

$R_{0}:=R_{00}-i\alpha\sigma\partial_{\theta},$

where $\alpha=\alpha(r)$ is a smooth function $\alpha\in{\rm C}^{\infty}_{\rm
c}(0,1]$ with $\alpha(1)=0$. We define

$D_{\alpha}:=\sigma(\partial_{r}+A+R_{0})=\begin{pmatrix}i\alpha\partial_{\theta}&\partial_{r}+\frac{i}{r}\partial_{\theta}\\\
-\partial_{r}+\frac{i}{r}\partial_{\theta}&i\alpha\partial_{\theta}\end{pmatrix}.$

The reader should note that $D_{\alpha}$ is formally self-adjoint if and only
if $\alpha$ is real valued and $A$ is well adapted for any choice of $\alpha$.
Let us compute $\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})$ in this case.
For simplicity, we write $\xi$ for the cotangent variable on $\Sigma=S^{1}$.
We have that

$c_{A}(\xi)=\begin{pmatrix}-\xi&0\\\
0&\xi\end{pmatrix},\quad\mbox{and}\quad(\lambda-
c_{A})^{-1}(\xi)=\begin{pmatrix}(\lambda+\xi)^{-1}&0\\\
0&(\lambda-\xi)^{-1}\end{pmatrix}.$

We have $c_{R_{0}}(r,\xi)=(r^{-1}-1)c_{A}-\alpha\sigma\xi$ and compute that

$\frac{\partial c_{R_{0}}}{\partial
r}|_{\Sigma}(\xi)=-c_{A}(\xi)-\alpha^{\prime}(1)\sigma\xi.$

Although Corollary 9.8 does not give an immediate answer because
$\frac{\partial c_{R_{0}}}{\partial r}|_{\Sigma}$ does not commute nor
anticommute with $c_{A}$ in this case, we have a sum of a commuting and
anticommuting term and a short computation gives us that

$\displaystyle p(\xi,\lambda)=$ $\displaystyle-(\lambda-
c_{A}(\xi))^{-2}\frac{\partial c_{R_{0}}}{\partial r}(r,\xi)|_{r=1}(\lambda-
c_{A}(\xi))^{-1}$ $\displaystyle=$ $\displaystyle(\lambda-
c_{A}(\xi))^{-2}c_{A}(\xi)(\lambda-c_{A}(\xi))^{-1}+(\lambda-
c_{A}(\xi))^{-2}\alpha^{\prime}(1)\sigma\xi(\lambda-c_{A}(\xi))^{-1}$
$\displaystyle=$ $\displaystyle(\lambda-
c_{A}(\xi))^{-3}c_{A}(\xi)+\alpha^{\prime}(1)\xi(\lambda-
c_{A}(\xi))^{-2}(\lambda+c_{A}(\xi))^{-1}\sigma.$

The first term has no residues, and the residues of the second term is
computed as in Corollary 9.8 to produce

$\displaystyle\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})(\xi)=$
$\displaystyle\frac{\alpha^{\prime}(1)\xi}{4}c_{A}^{-2}\chi^{-}(c_{A})\sigma=\frac{\alpha^{\prime}(1)}{4\xi}\chi^{-}(c_{A})\sigma$
$\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}(1)}{4}\begin{pmatrix}0&\frac{\chi_{(0,\infty)}(\xi)}{\xi}\\\
-\frac{\chi_{(-\infty,0)}(\xi)}{\xi}&0\end{pmatrix},$

using that

$\chi^{-}(c_{A})=\sigma_{0}(\chi^{-}(A))=\begin{pmatrix}\chi_{(0,\infty)}(\xi)&0\\\
0&\chi_{(-\infty,0)}(\xi)\end{pmatrix}.$

We see that $\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})\neq 0$ if and only
if $\alpha^{\prime}(1)\neq 0$. Moreover, since

$\displaystyle\chi^{-}(c_{A})\sigma_{-1}(\chi^{+}(A)-$ $\displaystyle
P_{\mathcal{C}_{D}})\chi^{+}(c_{A})$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}(1)}{4}\begin{pmatrix}0&\frac{\chi_{(0,\infty)}(\xi)}{\xi}\\\
-\frac{\chi_{(-\infty,0)}(\xi)}{\xi}&0\end{pmatrix}\begin{pmatrix}\chi_{(-\infty,0)}(\xi)&0\\\
0&\chi_{(0,\infty)}(\xi)\end{pmatrix}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}(1)}{4}\begin{pmatrix}0&\frac{\chi_{(0,\infty)}(\xi)}{\xi}\\\
-\frac{\chi_{(-\infty,0)}(\xi)}{\xi}&0\end{pmatrix},$

we see that $\chi^{+}(A)-P_{\mathcal{C}_{D}}$ fails to be compact on
$\check{\mathrm{H}}_{A}(D)$ if and only if $\alpha^{\prime}(1)\neq 0$ by
Proposition 9.9. Let us summarise the salient features of this example in a
proposition.

######  Proposition 9.12.

Assume that $M$ is the unit disc, $E=F=M\times\mathbb{C}^{2}$ and take
$D_{\alpha}$ as in Example 9.11 for an $\alpha$ with $\alpha^{\prime}(1)\neq
0$. We let $A$ denote its well adapted boundary operator. Then it holds that

  1. (i)

$\sigma_{-1}(\chi^{+}(A)-P_{\mathcal{C}_{D}})\neq 0$.

  2. (ii)

$\chi^{+}(A)-P_{\mathcal{C}_{D}}$ fails to be compact on
$\check{\mathrm{H}}_{A}(D)$.

  3. (iii)

The space $\chi^{+}(A){\rm
H}^{\rm-{\frac{1}{2}}}(S^{1},\mathbb{C}^{2})\cap{\mathcal{C}_{D}}$ is a
finite-dimensional subspace of ${\rm C}^{\infty}(S^{1},\mathbb{C}^{2})$.

###### Proof.

The first two statements were already proven in Example 9.11. It remains to
prove that last.

We define the operator
$\Lambda:=\frac{\alpha^{\prime}(1)}{4}(1+\Delta)^{-{\frac{1}{2}}}\in\Psi^{-1}_{\rm
cl}(S^{1},\mathbb{C}^{2})$. Let us first prove the claim that
$L:=\chi^{+}(A)-P_{\mathcal{C}_{D}}+\sigma\Lambda\chi^{-}(A)\in\Psi^{-1}_{\rm
cl}(S^{1},\mathbb{C}^{2})$ is elliptic. We have that

$\sigma_{-1}(L)=\frac{\alpha^{\prime}(1)}{4}\begin{pmatrix}0&\frac{\chi_{(0,\infty)}(\xi)}{\xi}-\frac{\chi_{(-\infty,0)}(\xi)}{\xi}\\\
-\frac{\chi_{(-\infty,0)}(\xi)}{\xi}-\frac{\chi_{(0,\infty)}(\xi)}{\xi}&0\end{pmatrix}=\frac{\alpha^{\prime}(1)}{4}\begin{pmatrix}0&\frac{1}{|\xi|}\\\
-\frac{1}{\xi}&0\end{pmatrix}.$

This is invertible for all $\xi\neq 0$, so $L$ is elliptic.

Consider the space $\chi^{+}(A){\rm
H}^{\rm-{\frac{1}{2}}}(S^{1},\mathbb{C}^{2})\cap{\mathcal{C}_{D}}$. We have
the following equivalence

$\displaystyle f\in\chi^{+}(A){\rm
H}^{\rm-{\frac{1}{2}}}(S^{1},\mathbb{C}^{2})\cap{\mathcal{C}_{D}}$
$\displaystyle\iff\begin{cases}\chi^{+}(A)f=f\\\
P_{\mathcal{C}_{D}}f=f\end{cases}$
$\displaystyle\iff\begin{cases}\chi^{+}(A)f=f\\\
(\chi^{+}(A)-P_{\mathcal{C}_{D}})f=0\end{cases}$
$\displaystyle\iff\begin{cases}\chi^{+}(A)f=f\\\ Lf=0.\end{cases}$

In the last equivalence we used that if $\chi^{+}(A)f=f$, then
$\chi^{-}(A)f=0$ and $\chi^{-}(A)f=0$ implies that
$(\chi^{+}(A)-P_{\mathcal{C}_{D}})f=Lf$. We can conclude that $\chi^{+}(A){\rm
H}^{\rm-{\frac{1}{2}}}(S^{1},\mathbb{C}^{2})\cap{\mathcal{C}_{D}}=\chi^{+}(A){\rm
H}^{\rm-{\frac{1}{2}}}(S^{1},\mathbb{C}^{2})\cap\ker(L)$. But since $L$ is
elliptic, $\ker(L)\subseteq{\rm C}^{\infty}(S^{1},\mathbb{C}^{2})$ is a
finite-dimensional subspace and item iii follows. ∎

Although Proposition 9.12 shows that $\ker(D_{\chi^{+}(A){\rm
H}^{\rm-\frac{1}{2}}})$ may fail to be infinite dimensional in general, we
have instead the following.

######  Corollary 9.13.

We have that for any spectral cut $r\in\mathbb{R}$ (i.e., $r\in\mathbb{R}$
such that $A_{r}:=A-r$ is invertible bisectorial) and any pseudo-differential
Calderón projection $\mathrm{P}_{{\mathcal{C}_{D}}}$,

$\faktor{{\mathcal{C}_{D}}}{\mathrm{P}_{{\mathcal{C}_{D}}}\chi^{+}(A_{r}){\rm
H}^{\rm-\frac{1}{2}}(\Sigma;E)}\quad\text{and}\quad\faktor{\chi^{+}(A_{r}){\rm
H}^{\rm-\frac{1}{2}}(\Sigma;E)}{\chi^{+}(A){\mathcal{C}_{D}}}$

are finite dimensional.

###### Proof.

Using Corollary 3.28, we have that
$\mathcal{C}_{D}^{c}=\ker\mathrm{P}_{{\mathcal{C}_{D}}}$ defines a regular
boundary condition. Therefore, from Theorem 5.6 with the choice of
$\mathcal{P}_{\pm}=\chi^{\pm}(A_{r})$ we have that
$(\mathcal{C}_{D}^{c},\chi^{+}(A_{r}){\rm H}^{\rm-\frac{1}{2}}(\Sigma;E))$ is
a Fredholm pair in $\check{\mathrm{H}}_{A}(D)$. Therefore,

$\faktor{\check{\mathrm{H}}_{A}(D)}{\mathcal{C}_{D}^{c}+\chi^{+}(A_{r}){\rm
H}^{\rm-\frac{1}{2}}(\Sigma;E)}\cong\faktor{{\mathcal{C}_{D}}}{\mathrm{P}_{{\mathcal{C}_{D}}}\chi^{+}(A_{r}){\rm
H}^{\rm-\frac{1}{2}}(\Sigma;E)}$

via Lemma A4 in [BBan].

For the remaining quotient, we invoke Theorem 5.6 with a choice of
$\mathcal{P}_{+}=\mathrm{P}_{{\mathcal{C}_{D}}}$, and since the generalised
APS condition $\chi^{-}(A_{r}){\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$ is a regular
boundary condition, we have that $(\chi^{-}(A_{r}){\rm
H}^{\rm\frac{1}{2}}(\Sigma;E),{\mathcal{C}_{D}})$ is a Fredholm pair in
$\check{\mathrm{H}}_{A}(D)$. Therefore, using [BBan, Lemma A.4], we obtain
that

$\faktor{\check{\mathrm{H}}_{A}(D)}{\chi^{-}(A_{r}){\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)+{\mathcal{C}_{D}}}\cong\faktor{\chi^{+}(A_{r}){\rm
H}^{\rm-\frac{1}{2}}(\Sigma;E)}{\chi^{+}(A_{r}){\mathcal{C}_{D}}}.$

This finishes the proof. ∎

If a boundary condition $B$ defines a realisation with infinite dimensional
kernel, then this is equivalent to saying that $B\cap{\mathcal{C}_{D}}$ is
infinite dimensional. Although Corollary 9.13 “measures” the amount of
$\chi^{+}(A){\rm H}^{\rm-\frac{1}{2}}(\Sigma;E)$ in ${\mathcal{C}_{D}}$ (as
seen through the projection $\mathrm{P}_{{\mathcal{C}_{D}}}$), it does not
give us information about such a $B$. The following gives a more precise
description in the way of a necessary condition.

######  Proposition 9.14.

Let $B$ be a boundary condition. If $\dim\ker(D_{\rm B})=\infty$, then
$B\not\subset{\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$ and
$\dim\chi^{+}(A_{r})B=\infty$.

###### Proof.

If $B\subset{\rm H}^{\rm\frac{1}{2}}$, then by Proposition 2.6, we have that
$\ker(D_{\rm B})$ is finite. Therefore, the contrapositive yields that for
$\ker(D_{\rm B})$ infinite, we have that $B\not\subset{\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)$.

Next, assume that $\dim\chi^{+}(A_{r})B<\infty$. For $u\in B$, we have that
$u=\chi^{-}(A_{r})u+\chi^{+}(A_{r})u$, and we always have that
$\chi^{-}(A_{r})u\in{\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$. Setting

$B^{\prime}:=\chi^{-}(A_{r}){\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)\oplus\chi^{+}(A_{r})B,$

we therefore have an inclusion $B\subset B^{\prime}$. The space $B^{\prime}$
is a boundary condition since $\chi^{+}(A_{r})B$ was assumed to be finite
dimensional. Now, letting $B_{0}=\chi^{-}(A_{r}){\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)$, the map

$\ker(D_{{\rm B}^{\prime}})/\ker(D_{{\rm
B}_{0}})\hookrightarrow\mathrm{dom}(D_{{\rm B}^{\prime}})/\mathrm{dom}(D_{{\rm
B}_{0}})\cong\chi^{+}(A_{r})B,$

is an injection given that $\mathrm{dom}(D_{{\rm
B}^{\prime}})=\mathrm{dom}(D_{{\rm
B}_{0}})+\mathrm{dom}(D_{\chi^{+}(A_{r})B})$ This allows us to conclude that

$\dim\ker(D_{{\rm B}^{\prime}})/\ker(D_{{\rm
B}_{0}})\leq\dim\chi^{+}(A_{r})B,$

and since

$\ker(D_{{\rm B}^{\prime}})\cong\ker(D_{{\rm B}^{\prime}})/\ker(D_{{\rm
B}_{0}})\oplus\ker(D_{{\rm B}_{0}}),$

we have that

$\dim\ker(D_{\rm B})\leq\dim\ker(D_{{\rm
B}^{\prime}})\leq\dim\chi^{+}(A_{r})B+\dim\ker(D_{{\rm B}_{0}})<\infty.$

Taking the contrapositive and combining with our earlier deduction that
$B\not\subset{\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$ concludes the proof. ∎

##  Appendix A Lemmas on dimensions and subspaces

######  Lemma A.1.

Let $\mathcal{B}$ and $\mathcal{B}^{\ast}$ be Banach spaces and suppose that
$\left\langle\cdot,\cdot\right\rangle:\mathcal{B}\times\mathcal{B}^{\ast}\to\mathbb{C}$
is a perfect pairing. Suppose that $\mathcal{B}=X\oplus Y$ and let $P_{X,Y}$
denote the bounded projection $\mathcal{B}\mapsto X$ and similarly for
$P_{Y,X}=1-P_{X,Y}$. Let $X^{\ast}=P_{X,Y}^{\ast}\mathcal{B}^{\ast}$, the dual
map to $P_{X,Y}$ and similarly $Y^{\ast}=P_{Y,X}^{\ast}\mathcal{B}^{\ast}$.
Then, the following hold:

  1. (i)

$P_{X^{\ast},Y^{\ast}}=P_{X,Y}^{\ast}$ and
$P_{Y^{\ast},X^{\ast}}=P_{Y,X}^{\ast}$. In particular
$\mathcal{B}^{\ast}=X^{\ast}\oplus Y^{\ast}$.

  2. (ii)

The pairing $\left\langle\cdot,\cdot\right\rangle$ restricts to perfect
pairings $X\times X^{\ast}\to\mathbb{C}$, as well as $Y\times
Y^{\ast}\to\mathbb{C}$.

  3. (iii)

The annihilators

$Y^{\perp}:=\left\\{b^{\ast}\in\mathcal{B}^{\ast}:\left\langle
b,y\right\rangle=0\ \forall y\in Y\right\\}=X^{\ast},\ \text{and}\ \quad
X^{\perp}=Y^{\ast}.$

###### Proof.

The item i is immediate from the definition of the dual projectors.

We prove ii. Let $x\in X$ and $x^{\ast}\in X^{\ast}$. Then, there exists
$C<\infty$ such that

$|\left\langle x,x^{\ast}\right\rangle|\leq
C\|x\|_{\mathcal{B}}\|x^{\ast}\|_{\mathcal{B}^{\ast}}=\|P_{X,Y}x\|_{\mathcal{B}}\|P_{X^{\ast},Y^{\ast}}x^{\ast}\|_{\mathcal{B}^{\ast}}=\|x\|_{X}\|x^{\ast}\|_{X^{\ast}}.$

Again, from the fact that $\left\langle\cdot,\cdot\right\rangle$ is a perfect
paring between $\mathcal{B}\times\mathcal{B}^{\ast}$, we have a $C_{1}<\infty$
such that

$\|x\|\leq C_{1}\sup\left\\{\frac{|\left\langle
x,b^{\ast}\right\rangle|}{\|b^{\ast}\|_{\mathcal{B}^{\ast}}}:0\neq
b^{\ast}\in\mathcal{B}^{\ast}\right\\}.$ (30)

Now, note that

$\left\langle x,b^{\ast}\right\rangle=\left\langle
P_{X,Y}x,b^{\ast}\right\rangle=\left\langle
x,P_{X^{\ast},Y^{\ast}}b^{\ast}\right\rangle.$ (31)

Moreover, there exists $C_{2}$ such that

$\|P_{X^{\ast},Y^{\ast}}b^{\ast}\|+\|P_{Y^{\ast},X^{\ast}}b^{\ast}\|\leq
C_{2}\|b^{\ast}\|,$

and therefore,

$\frac{1}{\|b^{\ast}\|}\leq C_{2}\frac{1}{\|P_{X^{\ast},Y^{\ast}}b^{\ast}\|}.$
(32)

Combining (31) and (32) and using (30), we obtain

$|\left\langle x,x^{\ast}\right\rangle|\leq
C_{1}C_{2}\sup\left\\{\frac{|\left\langle
x,b^{\ast}\right\rangle|}{\|P_{X^{\ast},Y^{\ast}}b^{\ast}\|_{\mathcal{B}^{\ast}}}:0\neq
P_{X^{\ast},Y^{\ast}}b^{\ast}\in\mathcal{B}^{\ast}\right\\}=C_{1}C_{2}\sup_{\begin{subarray}{c}x^{\ast}\in
X^{\ast}\\\ 1=\|x^{\ast}\|_{X^{\ast}}\end{subarray}}|\left\langle
x,x^{\ast}\right\rangle|.$

A symmetric argument yields that

$\|x^{\ast}\|\lesssim\sup_{\begin{subarray}{c}x\in X\\\
1=\|x\|_{X}\end{subarray}}|\left\langle x,x^{\ast}\right\rangle|.$

Lastly, we prove iii. First, suppose that that $b^{\ast}\in Y^{\perp}$. That
is, $0=\left\langle b^{\ast},y\right\rangle$ for all $y\in Y$. Since every
$y=P_{Y,X}b$ for some $b\in\mathcal{B}$, $0=\left\langle
P_{Y^{\ast},X^{\ast}}b^{\ast},b\right\rangle$ for all $b\in\mathcal{B}$. This
implies that $b^{\ast}\in\ker P_{Y^{\ast},X^{\ast}}=X^{\ast}$ and therefore,
$Y^{\perp}\subset X^{\ast}$.

Conversely, let $x^{\ast}\in X^{\ast}$. Then, for $y\in Y$, we have that

$\left\langle x^{\ast},y\right\rangle=\left\langle
P^{\ast}_{X,Y}x^{\ast},y\right\rangle=\left\langle
x^{\ast},P_{X,Y}y\right\rangle=0.$

That is, $X^{\ast}\subset Y^{\perp}$. The statement that $X^{\perp}=Y^{\ast}$
follows similarly. ∎

Next we prove two results concerning the dimension of certain subspaces.

######  Lemma A.2.

Let $P\in\Psi^{\boldsymbol{0}}_{\rm cl}(M;\boldsymbol{E})$ be a classical
zeroth order pseudo-differential projection in the Douglis-Nirenberg calculus
acting on sections of an $\mathbb{R}$-graded Hermitian vector bundle
$\boldsymbol{E}$ on a closed manifold $\Sigma$ with $\dim(\Sigma)>0$. If
$\sigma_{\boldsymbol{0}}(P)\neq 0,1$, then for any $s\in\mathbb{R}$, the space
$P{\mathbb{H}}^{s}(\Sigma;\boldsymbol{E})$ has infinite dimension and infinite
codimension.

In particular, in the setting of Lemma 2.28 with $\dim(M)>1$, then for any
admissible cut $r\in\mathbb{R}$, the spaces $\chi^{\pm}(A_{r}){\rm
L}^{2}(\Sigma;E)$ are infinite dimensional. Moreover, the space
$\check{\mathrm{H}}_{A}(D)\neq\left\\{0\right\\}$.

###### Proof.

We have that $P$ acts compactly on ${\mathbb{H}}^{s}(\Sigma;\boldsymbol{E})$
if and only if $\sigma_{\boldsymbol{0}}(P)\neq 0$. We conclude that
$P{\mathbb{H}}^{s}(\Sigma;\boldsymbol{E})$ has infinite dimension if and only
if $\sigma_{\boldsymbol{0}}(P)\neq 0$. The infinite codimensionality of
$P{\mathbb{H}}^{s}(\Sigma;\boldsymbol{E})$ is by a similar argument equivalent
to $\sigma_{\boldsymbol{0}}(P)\neq 1$. ∎

######  Lemma A.3.

Let $A$ be any elliptic first order differential operator acting on sections
of a vector bundle $E$ on a closed manifold $\Sigma$ such that its resolvent
set is non-empty and in some point the principal symbol of $A$ has an
eigenvalue with positive real part. Then there exists two sequences of
eigenvalues $(\lambda_{j}^{+})_{j\in\mathbb{N}}$ and
$(\lambda_{j}^{-})_{j\in\mathbb{N}}$ of $A$ such that

$\mathrm{Re}(\lambda^{\pm}_{j})\to\pm\infty,\quad\mbox{as $j\to\infty$}.$

In particular, in the setting of Lemma 2.28 with $\dim(M)>1$, then for any
admissible cut $r\in\mathbb{R}$ and an $s\in\mathbb{R}$, the spaces
$\chi^{\pm}(A_{r}){\rm H}^{\rm s}(\Sigma;E)$ are infinite dimensional.
Moreover, the space $\check{\mathrm{H}}_{A}(D)\neq\left\\{0\right\\}$.

###### Proof.

Define the zeroth order classical pseudodifferential operator
$F:=A(1+A^{*}A)^{-{\frac{1}{2}}}$. The spectrum of $F$ is contained in the
unit disc. From the fact that the symbol mapping is a $*$-homomorphism, it
follows that the spectrum of $F$ contains the spectrum of its symbol, which we
will denote by $\upsigma_{F}$. In fact, the spectrum of $\upsigma_{F}$
coincides with the essential spectrum of $F$.

From the fact that $A$ is first order, it follows that
$\lambda\in\mathrm{spec}(\upsigma_{A}(x,\xi))$ if and only if
$-\lambda\in\mathrm{spec}(\upsigma_{A}(x,-\xi))$. This implies the spectrum of
$\upsigma_{F}$ is symmetric about the imaginary axis. Since the principal
symbol of $A$ has an eigenvalue with positive real part in some point, it
follows that $F$ has essential spectrum on both sides of the imaginary axis.

Now, $A$ is assumed to have non-empty resolvent set, hence its must be
discrete by [shubinsbook]. The essential spectrum of $F$ consists of all
limits points of the image of the spectrum of $A$ under the homeomorphism
$z\rightarrow z(1+|z|^{2})^{-{\frac{1}{2}}}$ of the complex plane onto the
open unit disc. Since $F$ has essential spectrum on both sides of the
imaginary axis, the spectrum of $A$ has limit points on both sides of the
imaginary axis. It follows that the spectrum of $A$ must go to infinity on
both sides of the imaginary axis. By [shubinsbook], each element of the
spectrum is an eigenvalue and the statement follows.

Now, we have that each eigenspace $\mathrm{Eig}(\lambda_{i})\subset{\rm
C}^{\infty}(\Sigma;E)$ and therefore, $\chi^{\pm}(A_{r}){\rm H}^{\rm
s}(\Sigma;E)$ are infinite-dimensional. Therefore
$\check{\mathrm{H}}_{A}(D)\neq{0}$. ∎

##  Appendix B Bär-Ballmann’s approach from an abstract perspective

Although the main results of this paper concern elliptic differential
operators on manifolds with boundary, a large proportion of the main tools can
be formulated in a general abstract formalism. We believe several of these
results to be known to specialists, but nevertheless have decided to formulate
the Bär-Ballmann approach in complete abstraction. Similar ideas to those
appearing in this appendix can be found in [bossfury]. In particular, several
of our main results have abstract counterparts that hold in potentially non-
elliptic settings.

###  B.1. Preliminaries

Throughout this section, we fix a closed densely defined linear operator
between separable Hilbert spaces $\mathscr{H}_{1}\to\mathscr{H}_{2}$. For
reasons that will later become apparent we shall denote this operator by
$T_{\rm min}$. We also fix an adjoint $T^{\dagger}_{\rm
min}:\mathscr{H}_{2}\to\mathscr{H}_{1}$ which is closed and densely defined.
In the main part of this paper, we were exclusively concerned with $T_{\rm
min}$ being the minimal closure of an elliptic differential operator on a
manifold with boundary and $T^{\dagger}_{\rm min}$ the minimal closure of its
formal adjoint. We introduce the notation

$T_{\rm max}:=(T^{\dagger}_{\rm min})^{*}\quad\mbox{and}\quad T_{\rm
max}^{\dagger}:=(T_{\rm min})^{*}.$

Note that $T_{\rm min}\subseteq T_{\rm max}$ and $T_{\rm
min}^{\dagger}\subseteq T_{\rm max}^{\dagger}$.

######  Definition B.1.

  * •

A pair $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ as above is said to
be a formally adjointed pair (FAP).

  * •

A realisation $T$ of a FAP is an extension $T$ of $T_{\rm min}$ contained in
$T_{\rm max}$. If $T$ is closed as an operator, we say that $T$ is a closed
realisation of $\mathcal{T}$.

  * •

The Cauchy data space of a FAP $\mathcal{T}$ is defined as the Hilbert space

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

The quotient map $\mathrm{dom}(T_{\rm
max})\to\check{\mathscr{H}}_{\mathcal{T}}$ will be denoted by $\gamma$.

The reader can note that also in this abstract setting, whenever
$\check{\mathrm{H}}(\mathcal{T})$ is a Banach space equipped with a surjective
map $\gamma:\mathrm{dom}(T_{\rm max})\to\check{\mathrm{H}}(\mathcal{T})$ with
kernel $\mathrm{dom}(T_{\rm min})$ then
$\check{\mathrm{H}}(\mathcal{T})\cong\check{\mathscr{H}}_{\mathcal{T}}$ via an
isomorphism induced from the boundary mappings. We can therefore speak of
concrete Cauchy data spaces also in this abstract setting.

If $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ is a FAP, then
$\mathcal{T}^{\dagger}:=(T^{\dagger}_{\rm min},T_{\rm min})$ is also a FAP. A
symmetric closed operator $S$ will, unless otherwise indicated, be identified
with the FAP $(S,S)$. In particular, $S_{\rm min}=S=S_{\rm min}^{\dagger}$ and
$S_{\rm max}=S^{*}=S_{\rm max}^{\dagger}$. We use the notation
$\check{\mathscr{H}}_{S}:=\check{\mathscr{H}}_{(S,S)}\equiv\faktor{\mathrm{dom}(S^{*})}{\mathrm{dom}(S)}$.

######  Proposition B.2.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP. Define the
densely defined operator $\tilde{T}$ on $\mathscr{H}\oplus\mathscr{H}$ by

$\tilde{T}:=\begin{pmatrix}0&T_{\rm min}\\\ T_{\rm
min}^{\dagger}&0\end{pmatrix}.$

Then, the following hold.

  1. (i)

The operator $\tilde{T}$ is closed and symmetric, and its adjoint is given by

$\tilde{T}^{*}:=\begin{pmatrix}0&T_{\rm max}\\\ T_{\rm
max}^{\dagger}&0\end{pmatrix}.$

Both $\tilde{T}$ and $\tilde{T}^{*}$ are $\mathbb{Z}/2$-graded in the grading
of $\mathscr{H}\oplus\mathscr{H}$ making the first summand even and the second
odd.

  2. (ii)

The equality $\mathrm{dom}(\tilde{T}^{*})=\mathrm{dom}(T_{\rm
max}^{\dagger})\oplus\mathrm{dom}(T_{\rm max})$ induces an equality

$\check{\mathscr{H}}_{\tilde{T}}=\check{\mathscr{H}}_{\mathcal{T}^{\dagger}}\oplus\check{\mathscr{H}}_{\mathcal{T}}.$

  3. (iii)

The von Neumann decomposition of $\mathrm{dom}(\tilde{T}^{*})$ induces an
isomorphism of Hilbert spaces

$\check{\mathscr{H}}_{\mathcal{T}^{\dagger}}\oplus\check{\mathscr{H}}_{\mathcal{T}}\cong\ker(\tilde{T}^{*}+i)\oplus\ker(\tilde{T}^{*}-i).$

We omit the proof as it follows from standard techniques for unbounded
operators. The reader can find relevant details in [weidmann].

######  Lemma B.3.

Let $S$ be a closed symmetric operator.

  1. (i)

The sesquilinear form

$\omega_{S,0}:\mathrm{dom}(S^{*})\times\mathrm{dom}(S^{*})\to\mathbb{C},\quad\omega_{S,0}(v_{1},v_{2}):=\langle
S^{*}v_{1},v_{2}\rangle_{\mathscr{H}}-\langle
v_{1},S^{*}v_{2}\rangle_{\mathscr{H}},$

descends to a well-defined sesquilinear non-degenerate pairing

$\omega_{S}:\check{\mathscr{H}}_{S}\times\check{\mathscr{H}}_{S}\to\mathbb{C},$

satisfying the symmetry condition

$\omega_{S}(\xi_{1},\xi_{2})=-\overline{\omega_{S}(\xi_{2},\xi_{1})}.$

  2. (ii)

Under the isomorphism
$\check{\mathscr{H}}_{S}\cong\ker(S^{*}+i)\oplus\ker(S^{*}-i)$, the pairing
$\omega_{S}$ corresponds to the sesquilinear form

$\displaystyle\omega_{S,vN}:$
$\displaystyle(\ker(S^{*}+i)\oplus\ker(S^{*}-i))\times(\ker(S^{*}+i)\oplus\ker(S^{*}-i))\to\mathbb{C},$
$\displaystyle\omega_{S,vN}\left(\begin{pmatrix}v_{1,+}\\\
v_{1,-}\end{pmatrix},\begin{pmatrix}v_{2,+}\\\
v_{2,-}\end{pmatrix}\right):=2i\left\langle\begin{pmatrix}v_{1,+}\\\
v_{1,-}\end{pmatrix},\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}v_{2,+}\\\
v_{2,-}\end{pmatrix}\right\rangle_{\mathscr{H}\oplus\mathscr{H}}.$

###### Proof.

The analogous symmetry condition
$\omega_{S,0}(v_{1},v_{2})=-\overline{\omega_{S,0}(v_{2},v_{1})}$ for
$\omega_{S,0}$ is readily verified. To prove item i, we take
$v^{\prime}\in\mathrm{dom}(S^{*})$ and $v\in\mathrm{dom}(S)$ and compute that

$\omega_{S,0}(v,v^{\prime}):=\langle S^{*}v,v^{\prime}\rangle-\langle
v,S^{*}v^{\prime}\rangle=\langle S^{*}v,v^{\prime}\rangle-\langle
Sv,v^{\prime}\rangle=0,$

since $v\in\mathrm{dom}(S)$ and $S^{*}\supseteq S$. This computation, the
symmetry condition on $\omega_{S,0}$ and its sesquilinearity implies that
$\omega_{S}$ is a well-defined sesquilinear form satisfying the stated
symmetry condition. To show that $\omega_{S}$ is non-degenerate, if
$v\in\mathrm{dom}(S^{*})$ satisfies $\omega_{S}([v],\xi)=0$ for all
$\xi\in\check{\mathscr{H}}_{S}$ then $\langle
S^{*}v,v^{\prime}\rangle_{\mathscr{H}}=\langle
v,S^{*}v^{\prime}\rangle_{\mathscr{H}}$ for all
$v^{\prime}\in\mathrm{dom}(S^{*})$ so
$v\in\mathrm{dom}(S^{**})=\mathrm{dom}(S)$ so $[v]=0$ in
$\check{\mathscr{H}}_{S}$.

Let us prove item ii. We use the convention that the inner product is linear
in the second leg. For $v_{1,\pm},v_{2,\pm}\in\ker(S^{*}\pm i)$,

$\displaystyle\omega_{S}(v_{1,+}+v_{1,-},$ $\displaystyle v_{2,+}+v_{2,-})$
$\displaystyle=$ $\displaystyle\langle
S^{*}(v_{1,+}+v_{1,-}),v_{2,+}+v_{2,-}\rangle-\langle
v_{1,+}+v_{1,-},S^{*}(v_{2,+}+v_{2,-})\rangle$ $\displaystyle=$
$\displaystyle\langle-iv_{1,+}+iv_{1,-},v_{2,+}+v_{2,-}\rangle-\langle
v_{1,+}+v_{1,-},-iv_{2,+}+iv_{2,-}\rangle$ $\displaystyle=$ $\displaystyle
2i\langle v_{1,+},v_{2,-}\rangle-2i\langle v_{1,-},v_{2,+}\rangle$
$\displaystyle=$ $\displaystyle\omega_{S,vN}\left(\begin{pmatrix}v_{1,+}\\\
v_{1,-}\end{pmatrix},\begin{pmatrix}v_{2,+}\\\
v_{2,-}\end{pmatrix}\right).\qed$

######  Proposition B.4.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP. The
sesquilinear form

$\omega_{\mathcal{T},0}:\mathrm{dom}(T_{\rm
max}^{\dagger})\times\mathrm{dom}(T_{\rm
max})\to\mathbb{C},\quad\omega_{\mathcal{T},0}(v_{1},v_{2}):=\langle T_{\rm
max}^{\dagger}v_{1},v_{2}\rangle-\langle v_{1},T_{\rm max}v_{2}\rangle,$

descends to a well-defined sesquilinear non-degenerate pairing

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

satisfying the symmetry condition

$\omega_{\mathcal{T}}(\xi_{1},\xi_{2})=-\overline{\omega_{\mathcal{T}^{\dagger}}(\xi_{2},\xi_{1})}.$

###### Proof.

That $\omega_{\mathcal{T}}$ is well-defined and satisfies the symmetry
condition follows from Proposition B.2 and Lemma B.3. The proof that
$\omega_{\mathcal{T}}$ is non-degenerate goes along the same lines as in the
proof of Lemma B.3 and is omitted. ∎

###  B.2. The Cauchy data space and boundary conditions

For a subspace $B\subseteq\check{\mathscr{H}}_{T}$, we define the realisation
$T_{\rm B}$ by

$\mathrm{dom}(T_{\rm B}):=\\{v\in\mathrm{dom}(T_{\rm max}):\gamma(v)\in B\\}.$

######  Proposition B.5.

Let $(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP. The construction $B\mapsto
T_{\rm B}$ sets up a one-to-one correspondence of (closed) realisations of
$(T_{\rm min},T^{\dagger}_{\rm min})$ and (closed) subspaces of
$\check{\mathscr{H}}_{T}$.

###### Proof.

A short argument involving the five lemma shows that given a (closed) subspace
$B\subseteq\check{\mathscr{H}}_{\mathcal{T}}$, $T_{\rm B}$ is the unique
(closed) realisation $T$ with $\gamma\mathrm{dom}(T)=B$. A similar argument
shows that given a (closed) realisation $T$, the subspace
$B:=\gamma(\mathrm{dom}(T))\subseteq\check{\mathscr{H}}_{\mathcal{T}}$ is the
unique (closed) subspace $B_{0}$ for which $T=T_{{\rm B}_{0}}$. ∎

######  Definition B.6.

A subspace $B\subseteq\check{\mathscr{H}}_{\mathcal{T}}$ is called a
_generalised boundary condition_ and a closed subspace
$B\subseteq\check{\mathscr{H}}_{T}$ a _boundary condition_.

If $B$ is a generalised boundary condition, we define its adjoint boundary
condition as

$B^{*}:=\\{w\in\mathscr{H}_{\mathcal{T}^{\dagger}}:\omega_{\mathcal{T}}(w,v)=0\quad\forall
v\in B\\}.$

In this terminology, Proposition B.5 can be phrased as there being a one-to-
one correspondence between closed realisations of a formally adjointed closed
operator and its boundary conditions. The terminology adjoint boundary
condition is motivated by the next proposition.

######  Proposition B.7.

Let $(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP and $B$ a generalised
boundary condition. Then it holds that

$(T_{\rm B})^{*}=T^{\dagger}_{{\rm B}^{*}}.$

###### Proof.

We first show that $(T_{\rm B})^{*}\supseteq T^{\dagger}_{\rm B^{*}}$. If
$v\in\mathrm{dom}T^{\dagger}_{\rm B^{*}}$ then for any
$w\in\mathrm{dom}T_{{\rm B}}$ we have that

$\langle T_{{\rm B}^{*}}^{\dagger}v,w\rangle-\langle v,T_{{\rm
B}}w\rangle=\langle T_{\rm max}^{\dagger}v,w\rangle-\langle v,T_{\rm
max}w\rangle=\omega_{\mathcal{T}}(\gamma(w),\gamma(v))=0.$

Therefore, $v\in\mathrm{dom}(T_{\rm B})^{*}$ and $(T_{\rm B})^{*}v=T_{{\rm
B}^{*}}^{\dagger}v$.

We prove $(T_{\rm B})^{*}\subseteq T^{\dagger}_{{\rm B}^{*}}$. As $T_{\rm
B}\supseteq T_{\min}$, $(T_{\rm B})^{*}\subseteq T^{\dagger}_{\max}$ so
consider a
$v\in\mathrm{dom}(T_{\max}^{\dagger})\setminus\mathrm{dom}(T^{\dagger}_{{\rm
B}^{*}})$, i.e. a $v\in\mathrm{dom}(T_{\max}^{\dagger})$ with $\gamma(v)\notin
B^{*}$. Nondegeneracy of $\omega_{\mathcal{T}}$ implies that there exists a
$w\in\mathrm{dom}(T_{{\rm B}})$ with
$\omega_{\mathcal{T}}(\gamma(v),\gamma(w))\neq 0$, or equivalently that there
exists a $w\in\mathrm{dom}(T_{{\rm B}})$ with

$\langle T_{\rm max}^{\dagger}v,w\rangle-\langle v,T_{\rm
max}w\rangle=\langle(T_{\rm B})^{*}v,w\rangle-\langle v,T_{\rm B}w\rangle\neq
0.$

This shows that $v\in\mathrm{dom}(T_{\rm B})^{*}$ so we conclude that $(T_{\rm
B})^{*}\subseteq T^{\dagger}_{{\rm B}^{*}}$. ∎

######  Proposition B.8.

Let $S$ be a closed symmetric operator.

  1. (i)

For any generalised boundary condition $B$, $S_{\rm B}^{*}=S_{{\rm B}^{*}}$.

  2. (ii)

A boundary condition $B$ for $S$ defines a self-adjoint realisation $S_{\rm
B}$ if and only if $B$ is Lagrangian (i.e. $B^{*}=B$).

  3. (iii)

A boundary condition $B$ for $S$ defines a symmetric realisation $S_{\rm B}$
if and only if $B$ is isotropic (i.e. $B\subseteq B^{*}$).

###### Proof.

By definition, $S^{\dagger}_{\rm min}=S_{\rm min}=S$ for the FAP constructed
from a closed symmetric operator and therefore the proposition follows from
Proposition B.7. ∎

######  Proposition B.9.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP and recall the
notation from Proposition B.2.

  1. (i)

There is a one-to-one correspondence between closed realisations of
$\mathcal{T}$ and self-adjoint $\mathbb{Z}/2$-graded realisations of
$\tilde{T}$ given by

$T_{\rm B}\longleftrightarrow\begin{pmatrix}0&T_{\rm B}\\\ T^{\dagger}_{{\rm
B}^{*}}&0\end{pmatrix}.$

  2. (ii)

There is a one-to-one correspondence between closed subspaces
$B\subseteq\check{\mathscr{H}}_{\mathcal{T}}$ and $\mathbb{Z}/2$-graded
Lagrangian subspaces
$\tilde{B}\subseteq\check{\mathscr{H}}_{\tilde{T}}=\mathrm{dom}(\tilde{T}^{*})/\mathrm{dom}(\tilde{T})$
given by

$B\longleftrightarrow B^{*}\oplus B.$

###### Proof.

By Proposition B.7 and B.8, item i follows from item ii. Let us prove item ii.
We note that

$\omega_{\tilde{T}}\left(\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix},\begin{pmatrix}v_{1}^{\prime}\\\
v_{2}^{\prime}\end{pmatrix}\right)=\omega_{\mathcal{T}}(v_{1},v_{2}^{\prime})+\omega_{\mathcal{T}^{\dagger}}(v_{2},v_{1}^{\prime}),$

for $v_{1},v_{1}^{\prime}\in\check{\mathscr{H}}_{\mathcal{T}^{\dagger}}$ and
$v_{2},v_{2}^{\prime}\in\check{\mathscr{H}}_{\mathcal{T}}$. Therefore, given a
$B\subseteq\check{\mathscr{H}}_{\mathcal{T}}$, $\tilde{B}:=B^{*}\oplus B$ is a
$\mathbb{Z}/2$-graded isotropic subspace (i.e.
$\tilde{B}\subseteq\tilde{B}^{*}$). Given $(v_{1},v_{2})\in\tilde{B}^{*}$, the
condition

$\omega_{\tilde{T}}\left(\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix},\begin{pmatrix}v_{1}^{\prime}\\\ 0\end{pmatrix}\right)=0,$

for all $v_{1}^{\prime}\in B^{*}$ implies that $v_{2}\in B$ and the condition

$\omega_{\tilde{T}}\left(\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix},\begin{pmatrix}0\\\ v_{2}^{\prime}\end{pmatrix}\right)=0,$

for all $v_{2}^{\prime}\in B$ implies that $v_{1}\in B^{*}$ which proves that
$\tilde{B}^{*}\subseteq\tilde{B}$. We conclude that $\tilde{B}$ is a
$\mathbb{Z}/2$-graded Lagrangian subspace. Conversely, given a
$\mathbb{Z}/2$-graded Lagrangian subspaces
$\tilde{B}\subseteq\check{\mathscr{H}}_{\tilde{T}}$ the fact that $\tilde{B}$
is graded implies that $\tilde{B}=B^{\prime}\oplus B$ for closed subspaces
$B^{\prime}\subseteq\check{\mathscr{H}}_{\mathcal{T}\dagger}$ and
$B\subseteq\check{\mathscr{H}}_{\mathcal{T}}$ and the fact that $\tilde{B}$ is
Lagrangian implies that $B^{\prime}=B^{*}$ by an argument as above. We
conclude that $B\leftrightarrow B^{*}\oplus B$ produces a one-to-one
correspondence as stated. ∎

######  Proposition B.10.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP and
$B_{1}\subseteq B_{2}$ be boundary conditions. Then $T_{{\rm B}_{1}}\subseteq
T_{{\rm B}_{2}}$ and there is a short exact sequence of Hilbert spaces

$0\to\mathrm{dom}(T_{{\rm B}_{1}})\to\mathrm{dom}(T_{{\rm
B}_{2}})\to\faktor{B_{2}}{B_{1}}\to 0.$

In particular, the following special cases hold for a boundary condition $B$:

  1. (i)

$\faktor{\mathrm{dom}(T_{\rm B})}{\mathrm{dom}(T_{\rm min})}\cong B$,

  2. (ii)

$\faktor{\mathrm{dom}(T_{\rm max})}{\mathrm{dom}(T_{\rm
B})}\cong\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{B}$.

This follows from standard considerations with the open mapping theorem and
the five lemma.

######  Definition B.11.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP. We define the
Hardy space as

$\mathcal{C}_{\mathcal{T}}:=\gamma\ker T_{\rm
max}\subseteq\check{\mathscr{H}}_{\mathcal{T}}.$

It is readily verified that the Hardy space fits into a short exact sequence
of vector spaces

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

This also follows from the next proposition.

######  Proposition B.12.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP and $B$ be a
boundary condition. Then there is a short exact sequence of vector spaces

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

###### Proof.

This follows from the fact that $\ker(T_{\rm min})=\ker(T_{\rm
max})\cap\mathrm{dom}(T_{\min})$ and $\ker(T_{\rm B})=\ker(T_{\rm
max})\cap\mathrm{dom}(T_{\rm B})$. ∎

Let us briefly comment on the relation to boundary triplets. We compare our
result to the setup of [grubb68]. For more details, history and context for
boundary triplets, see also [malamud, behrndtetal]. We remark that if a FAP
$\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ admits a realisation $T$
which is invertible, which is equivalent to
$0\in\mathrm{res}(T)=\mathrm{res}(T^{*})^{\mathrm{conj}}$, then $T^{*}$ is
also invertible and therefore, $\ker T_{\rm min}=\ker T_{\rm
min}^{\dagger}=0$. Thus, $\gamma|:\ker T_{\rm
max}\to\mathcal{C}_{\mathcal{T}}$ and $\gamma|:\ker T_{\rm
max}^{\dagger}\to\mathcal{C}_{\mathcal{T}^{\dagger}}$ are vector space
isomorphisms.

######  Proposition B.13.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP admitting a
realisation $T_{\beta}$ which is invertible. Define
$\mathrm{pr}_{\beta}:=T_{\beta}^{-1}T_{\rm max}:\mathrm{dom}(T_{\rm
max})\to\mathrm{dom}(T_{\beta})$ and
$\mathrm{pr}_{\beta^{\dagger}}:=(T_{\beta}^{*})^{-1}T_{\rm
max}^{\dagger}:\mathrm{dom}(T_{\rm
max}^{\dagger})\to\mathrm{dom}(T_{\beta}^{*})$. Then, the following hold.

  1. (i)

It holds that

$\displaystyle\mathrm{dom}(T_{\rm max})=\mathrm{dom}(T_{\beta})+\ker T_{\rm
max},\quad\text{and}$ $\displaystyle\mathrm{dom}(T_{\rm
max}^{\dagger})=\mathrm{dom}(T_{\beta}^{*})+\ker T_{\rm max}^{\dagger},$

and the mappings

$\displaystyle\mathrm{dom}(T_{\rm max})$
$\displaystyle\xrightarrow{\mathrm{pr}_{\beta}\oplus\gamma(1-\mathrm{pr}_{\beta})}\mathrm{dom}(T_{\beta})\oplus\mathcal{C}_{\mathcal{T}}\quad\mbox{and}$
$\displaystyle\mathrm{dom}(T_{\rm max}^{\dagger})$
$\displaystyle\xrightarrow{\mathrm{pr}_{\beta^{\dagger}}\oplus\gamma(1-\mathrm{pr}_{\beta^{\dagger}})}\mathrm{dom}(T_{\beta}^{*})\oplus\mathcal{C}_{\mathcal{T}^{\dagger}},$

are isomorphisms.

  2. (ii)

The isomorphisms from i induce isomorphisms

$\displaystyle\check{\mathscr{H}}_{\mathcal{T}}\cong\check{\mathscr{H}}_{\mathcal{T},\beta}\oplus\mathcal{C}_{\mathcal{T}}\quad\mbox{and}\quad\check{\mathscr{H}}_{\mathcal{T}^{\dagger}}\cong\check{\mathscr{H}}_{\mathcal{T}^{\dagger},\beta}\oplus\mathcal{C}_{\mathcal{T}^{\dagger}},$

where $\check{\mathscr{H}}_{\mathcal{T},\beta}:=\gamma\mathrm{dom}(T_{\beta})$
and
$\check{\mathscr{H}}_{\mathcal{T}^{\dagger},\beta}:=\gamma\mathrm{dom}T_{\beta}^{*}$
are closed subspaces.

  3. (iii)

There is a one-to-one correspondence between boundary conditions $B$ of
$\mathcal{T}$ and triples $(W,V,\mathfrak{T})$, where
$V\subseteq\mathcal{C}_{\mathcal{T}}$ and
$W\subseteq\mathcal{C}_{\mathcal{T}^{\dagger}}$ are closed subspaces and
$\mathfrak{T}:V\dashrightarrow W$ is a closed densely defined operator.

###  B.3. Characterising Fredholm boundary conditions

In this section, we show how to characterise Fredholm boundary conditions for
a FAP satisfying the extra condition that $T_{\rm min}$ and $T_{\rm
min}^{\dagger}$ have closed range and $\ker(T_{\rm min})$ and $\ker(T_{\rm
min}^{\dagger})$ being finite-dimensional. For the sake of terminology, we
define this notion as follows.

######  Definition B.14.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP. We say that
$\mathcal{T}$ has closed range if $T_{\rm min}$ and $T_{\rm min}^{\dagger}$
have closed range. We say that $\mathcal{T}$ is a Fredholm FAP if it has
closed range and $\ker(T_{\rm min})$ and $\ker(T_{\rm min}^{\dagger})$ are
finite-dimensional.

The reader should note that by the closed range theorem, $\mathcal{T}=(T_{\rm
min},T^{\dagger}_{\rm min})$ is a FAP with closed range if and only if $T_{\rm
min}$ and $T_{\rm max}$ have closed range. For the purpose of studying
elliptic differential operators, the following sufficient condition for closed
range and Fredholmness will be useful.

######  Proposition B.15.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP. Assume that
the inclusion $\iota:\mathrm{dom}(T_{\rm min})\hookrightarrow\mathscr{H}_{1}$
is a compact inclusion. Then, the following holds.

  1. (i)

$\ker(T_{\rm min})$ is finite-dimensional.

  2. (ii)

$T_{\rm min}$ and $T_{\rm max}^{\dagger}$ have closed range.

  3. (iii)

$\mathscr{H}_{1}=\ker(T_{\min})\oplus^{\perp}\mathrm{ran}(T_{\max}^{\dagger})$
and
$\mathscr{H}_{2}=\ker(T_{\max}^{\dagger})\oplus^{\perp}\mathrm{ran}(T_{\min})$.

In particular, if both $\mathrm{dom}(T_{\rm
min})\hookrightarrow\mathscr{H}_{1}$ and $\mathrm{dom}(T_{\rm
min}^{\dagger})\hookrightarrow\mathscr{H}_{2}$ are compact inclusions, then
$\mathcal{T}$ is a Fredholm FAP and we additionally have that
$\mathscr{H}_{2}=\ker(T_{\min}^{\dagger})\oplus^{\perp}\mathrm{ran}(T_{\max})$.

###### Proof.

For $u\in\mathrm{dom}(T_{\rm min})$, we have the estimate

$\|u\|_{T_{\rm min}}\simeq\|\iota u\|_{\mathscr{H}_{1}}+\|T_{\rm
min}u\|_{\mathscr{H}_{2}}.$

Then, by Proposition A3 in [BB] (ii), we conclude that $T_{\rm min}$ has
finite dimensional kernel and closed image. The fact that $\mathrm{ran}(T_{\rm
min}^{*})=\mathrm{ran}(T_{\rm max}^{\dagger})$ is closed follows from the
closed range theorem. ∎

######  Remark B.16.

Note that, even without the assumption
$\mathrm{dom}(T_{\min}^{\dagger})\hookrightarrow\mathscr{H}_{2}$ is a compact
embedding it still holds that
$\mathscr{H}_{2}=\ker(T^{\dagger}_{\min})\oplus\overline{\mathrm{ran}(T_{\max})}$.
The compactness of the embedding ensures that
$\mathrm{ran}(T_{\max})=\overline{\mathrm{ran}(T_{\max})}$ as well as the
finite dimensionality of $\ker(T_{min}^{\dagger})$.

######  Lemma B.17.

Let $K\subset Z$ be a closed subspace of a Banach space $Z$ and let $X\subset
Z$ a subspace. If $K\cap X$ is closed, the map $\Phi:\faktor{X}{(K\cap
X)}\to\faktor{Z}{K}$ defined by $\Phi[u]_{\faktor{X}{(K\cap
X)}}=[u]_{\faktor{Z}{K}}$ is well-defined and injective.

###### Proof.

To show the map is well-defined, suppose $[x]_{\faktor{X}{(K\cap
X)}}=[y]_{\faktor{X}{(K\cap X)}}$. Then, $x-y\in(K\cap X)\subset K$ and
therefore, $[x]_{\faktor{Z}{K}}=[y]_{\faktor{Z}{K}}$.

For injectivity, suppose that $\varphi[u]_{\faktor{X}{(K\cap X)}}=0$, which is
the same as $[u]_{\faktor{Z}{K}}=0$. Then, $u=u-0\in K$ and since $u\in X$, we
obtain that $u\in K\cap X$. That is, $u\in(K\cap X)$ and therefore,
$[u]_{\faktor{X}{(K\cap X)}}=0$. ∎

######  Proposition B.18.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP and $B$ a
generalised boundary condition. The map

$\Phi_{\rm B}:\faktor{\mathrm{dom}(T_{\rm B})}{\ker(T_{\rm
B})}\to\mathrm{ran}(T_{\rm B}),\qquad\Phi_{\rm B}[u]:=T_{\max}u,$

is a bounded bijection. Moreover, the map

$\faktor{\mathrm{dom}(T_{\rm B})}{\ker(T_{\rm
B})}\to\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}$

given by

$[u]_{\faktor{\mathrm{dom}(T_{\rm B})}{\ker(T_{\rm
B})}}\mapsto[u]_{\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}}$

is well-defined and is a bounded injection.

###### Proof.

To show that this map is bounded, we note that

$\displaystyle\|\Phi_{\rm B}[u]\|$
$\displaystyle=\|T_{\max}u\|\leq\inf_{x\in\ker(T_{\rm
B})}\|u-x\|+\|T_{\max}u\|$ $\displaystyle=\inf_{x\in\ker(T_{\rm
B})}\|u-x\|+\|T_{{\rm B}}u\|=\|[u]\|_{\faktor{\mathrm{dom}(T_{\rm
B})}{\ker(T_{\rm B})}}.$

Surjectivity is clear. For injectivity, note that if $\Phi_{\rm B}[u]=0$, then
$T_{\max}u=0$, so $u\in\ker(T_{\max})$. But since $u\in\mathrm{dom}(T_{\rm
B})$, we have that $u\in\ker(T_{\rm B})=\mathrm{dom}(T_{\rm
B})\cap\ker(T_{max})$.

The map

$\faktor{\mathrm{dom}(T_{\rm B})}{\ker(T_{\rm
B})}\to\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}$

is a well-defined injection since $\ker(T_{\rm B})=\mathrm{dom}(T_{\rm
B})\cap\ker(T_{\max})$ by invoking Lemma B.17. ∎

######  Lemma B.19.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP and
$Y\subset\mathrm{dom}(T_{\max})$ a subspace. Consider the generalised boundary
condition $B:=\gamma(Y)\subseteq\check{\mathscr{H}}_{\mathcal{T}}$. Then,
$\mathrm{dom}(T_{{\rm B}})=\mathrm{dom}(T_{\min})+Y$.

###### Proof.

Since $Y,\mathrm{dom}(T_{\min})\subset\mathrm{dom}(T_{{\rm B}})$, we have that
$\mathrm{dom}(T_{{\rm B}})\supset\mathrm{dom}(T_{\min})+Y$.

To prove the opposite containment, let $u\in\mathrm{dom}(T_{\rm B})$, which is
if and only if $\gamma(u)\in B=\gamma(Y)$. Therefore, there exists $y\in Y$
such that $\gamma(u)=\gamma(y)$. So, $u-y\in\mathrm{dom}(T_{\min})$ and $y\in
Y$ so $u=(u-y)+y\in\mathrm{dom}(T_{\min})+Y$. ∎

######  Lemma B.20.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP. For a
generalised boundary condition $B$, we have that $\mathrm{ran}(T_{\rm
B})=\mathrm{ran}(T_{{\rm B}+\mathcal{C}_{\mathcal{T}}})$ and that
$\mathrm{dom}(T_{{\rm B}+\mathcal{C}_{\mathcal{T}}})=\mathrm{dom}(T_{\rm
B})+\ker(T_{\max})$.

###### Proof.

It is clear that $\mathrm{ran}(T_{\rm B})\subset\mathrm{ran}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})$. Set $Y=\mathrm{dom}(T_{\rm
B})+\ker(T_{\max})$ and apply Lemma B.19 to obtain that $\mathrm{dom}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})=\mathrm{dom}(T_{\rm B})+\ker(T_{\max})$, since
$\mathrm{dom}(T_{\min})\subset\mathrm{dom}(T_{\rm B})$. Then, for
$u\in\mathrm{ran}(T_{{\rm B}+\mathcal{C}_{\mathcal{T}}})$, we can find
$v\in\mathrm{dom}(T_{{\rm B}+\mathcal{C}_{\mathcal{T}}})$ such that
$u=T_{\max}v$. By what we have just proved, we have $v=x_{\rm
B}+y\in\mathrm{dom}(T_{\rm B})+\ker(T_{\max})$ and therefore, $u=T_{\rm
max}v=T_{\rm max}x_{{\rm B}}$. I.e., $u\in\mathrm{ran}(T_{\rm B})$. ∎

######  Lemma B.21.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP. For a
generalised boundary condition $B^{\prime}$ such that
$\mathcal{C}_{\mathcal{T}}\subset B^{\prime}$, we have that the inclusion map

$\faktor{\mathrm{dom}(T_{{\rm
B}^{\prime}})}{\ker(T_{\max})}\hookrightarrow\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}$

is an isometry. If $\mathcal{T}$ has closed range, the normed space
$\faktor{\mathrm{dom}(T_{{\rm B}^{\prime}})}{\ker(T_{\max})}$ is closed if and
only if $B^{\prime}$ is a boundary condition, i.e., closed in
$\check{\mathscr{H}}_{\mathcal{T}}$.

###### Proof.

Note that the induced norm on $\faktor{\mathrm{dom}(T_{{\rm
B}^{\prime}})}{\ker(T_{\max})}$ is

$\|[u]\|^{2}=\inf_{x\in\ker(T_{\max})}\|u-x\|^{2}+\|T_{{\rm
B}^{\prime}}u\|^{2}=\inf_{x\in\ker(T_{\max})}\|u-x\|^{2}+\|T_{\max}u\|^{2},$

where the norm after the ultimate equality is the norm on
$\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}$. If $B^{\prime}$ is closed,
then $\mathrm{dom}(T_{{\rm B}^{\prime}})$ is closed and therefore,
$\faktor{\mathrm{dom}(T_{{\rm B}^{\prime}})}{\ker(T_{\max})}$ is a Banach
space, i.e. it is closed in $\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}$.
On the other hand, if $\faktor{\mathrm{dom}(T_{{\rm
B}^{\prime}})}{\ker(T_{\max})}$ is closed, then $\mathrm{dom}(T_{{\rm
B}^{\prime}})$ is closed since $\mathrm{dom}(T_{{\rm
B}^{\prime}})\cong\faktor{\mathrm{dom}(T_{{\rm
B}^{\prime}})}{\ker(T_{\max})}\oplus\ker(T_{\max})$. ∎

######  Lemma B.22.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP with closed
range. For a generalised boundary condition $B$, $\mathrm{ran}(T_{\rm B})$ is
closed if and only if the image of the map $\Xi:\mathrm{dom}(T_{\rm
B})\to\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}$ given by

$\Xi:\mathrm{dom}(T_{\rm B})\to\faktor{\mathrm{dom}(T_{\rm B})}{\ker(T_{\rm
B})}\hookrightarrow\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}$

has closed range.

###### Proof.

We first start by noting that, by Proposition B.18, the open mapping theorem
yields that the we have that the map $\Phi_{\max}$ as defined in Proposition
B.18 is homeomorphism between
$\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}$ and
$\mathrm{ran}(T_{\max})$.

Moreover, by the definition of $\Phi_{\rm B}$, we have the following commuting
diagram.

${\faktor{\mathrm{dom}(T_{\rm B})}{\ker(T_{\rm
B})}}$${\faktor{\mathrm{dom}(T_{\max})}{\ker(T_{\max})}}$${\mathrm{ran}(T_{\rm
B})}$${\mathrm{ran}(T_{\max})}$$\scriptstyle{\Phi_{\rm
B}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\Phi_{\max}}$$\scriptstyle{\iota_{2}}$

Note that $\mathrm{ran}(T_{\rm B})\subset\mathrm{ran}(T_{\max})$, so
$\iota_{2}$ is simply the inclusion map. Since $\Phi_{\max}$ is a Banach space
isomorphism, we have that

$\Phi_{\max}^{-1}{{\lvert}}_{\mathrm{ran}(T_{\rm B})}:\mathrm{ran}(T_{\rm
B})\to\iota_{1}\left(\faktor{\mathrm{dom}(T_{\rm B})}{\ker(T_{\rm B})}\right)$

is bounded. Therefore, $\mathrm{ran}(T_{\rm B})$ is closed if and only if
$\iota_{1}\left(\faktor{\mathrm{dom}(T_{\rm B})}{\ker(T_{\rm B})}\right)$ is
closed. The latter is, indeed, the image of $\Xi$ so this proves the lemma. ∎

######  Theorem B.23.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP with closed
range. For a generalised boundary condition $B$, $\mathrm{ran}(T_{\rm B})$ is
closed if and only if $B+\mathcal{C}_{\mathcal{T}}$ is a boundary condition.

###### Proof.

Lemma B.20 yields that $\mathrm{ran}(T_{\rm B})=\mathrm{ran}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})$. Then, Lemma B.22 says that this is closed if
and only if the range of the map $\Xi$ is closed. Since
$\mathcal{C}_{\mathcal{T}}\subset B+\mathcal{C}_{\mathcal{T}}$, it is easy to
see that the range of $\Xi$ is precisely $\faktor{\mathrm{dom}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})}{\ker(T_{\max})}$. Therefore, by Lemma B.21, we
obtain that $\mathrm{ran}(T_{\rm B})$ is closed if and only if
$B+\mathcal{C}_{\mathcal{T}}$ is closed, i.e., that
$B+\mathcal{C}_{\mathcal{T}}$ is a boundary condition. ∎

######  Corollary B.24.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP with closed
range and $B$ a generalised boundary condition. Then the Hardy space
$\mathcal{C}_{\mathcal{T}}$ is closed in $\check{\mathscr{H}}_{\mathcal{T}}$.

###### Proof.

The corollary follows directly from Theorem B.23 with $B=0$. ∎

######  Lemma B.25.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP with closed
range. When $\mathrm{ran}(T_{\rm B})$ is closed,

$\faktor{\mathscr{H}_{2}}{\mathrm{ran}(T_{\rm
B})}\cong\faktor{\mathscr{H}_{2}}{\mathrm{ran}(T_{\max})}\oplus\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B+\mathcal{C}_{\mathcal{T}})}\cong\ker(T_{\min}^{\dagger})\oplus\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B+\mathcal{C}_{\mathcal{T}})}.$

###### Proof.

By assumption $\mathrm{ran}(T_{\max})$ is closed and therefore,

$\displaystyle\mathscr{H}_{2}$
$\displaystyle\cong\faktor{\mathscr{H}_{2}}{\mathrm{ran}(T_{\max})}\oplus\mathrm{ran}(T_{\max})$
$\displaystyle\cong\faktor{\mathscr{H}_{2}}{\mathrm{ran}(T_{\max})}\oplus\faktor{\mathrm{ran}(T_{\max})}{\mathrm{ran}(T_{\rm
B})}\oplus\mathrm{ran}(T_{\rm B}).$

From Lemma B.20, we have that $\mathrm{ran}(T_{\rm B})=\mathrm{ran}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})$ and $T_{{\rm B}+\mathcal{C}_{\mathcal{T}}}$ is
a closed operator from Lemma B.21. Thus, note that $\ker(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})=\ker(T_{\max})$ and therefore,

$\faktor{\mathrm{dom}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})}{\ker(T_{\max})}=\faktor{\mathrm{dom}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})}{\ker(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})}\cong\mathrm{ran}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})=\mathrm{ran}(T_{\rm B}).$

Thus,

$\faktor{\mathrm{ran}(T_{\max})}{\mathrm{ran}(T_{\rm
B})}\cong\faktor{\mathrm{dom}(T_{\max})}{\mathrm{dom}(T_{{\rm
B}+\mathcal{C}_{\mathcal{T}}})}\cong\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B+\mathcal{C}_{\mathcal{T}})}$

from Proposition B.10. ∎

######  Proposition B.26.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a Fredholm FAP. For a
general boundary condition $B$, the following are equivalent:

  1. (i)

$\mathrm{ran}(T_{\rm B})$ has finite algebraic codimension in
$\mathscr{H}_{2}$;

  2. (ii)

$T_{\rm B}$ has closed range and $\mathrm{ran}(T_{\rm B})^{\perp}$ is finite
dimensional;

  3. (iii)

$T_{\rm B}$ has closed range and $\ker(T_{\rm B}^{\ast})$ is finite
dimensional;

  4. (iv)

$B+\mathcal{C}_{\mathcal{T}}$ has finite algebraic codimension in
$\check{\mathscr{H}}_{\mathcal{T}}$;

  5. (v)

$B+\mathcal{C}_{\mathcal{T}}$ is closed and has finite algebraic codimension
in in $\check{\mathscr{H}}_{\mathcal{T}}$.

If any of these hold, then

$\ker(T_{\rm B}^{\ast})\cong{\faktor{\mathscr{H}_{2}}{\mathrm{ran}(T_{\rm
B})}}\cong\ker(T_{\min}^{\dagger})\oplus{\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B+\mathcal{C}_{\mathcal{T}})}}.$

###### Proof.

The equivalence of i-iii, as well as iv-v are standard. The statements v and
ii are equivalent because of Lemma B.20 coupled with Lemma B.25. The formula
follows from the fact that $\mathrm{ran}(T_{\rm B})^{\perp}\cong\ker(T_{\rm
B}^{\ast})$ along with Lemma B.25. ∎

Before proceeding with studying Fredholm properties of realisations, we make a
remark about the maximal domain in terms of the Hardy space and a suitable
realisation.

######  Corollary B.27.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a FAP with closed
range and $B$ be a boundary condition. Then $B$ is a complement to
$\mathcal{C}_{\mathcal{T}}$ in $\check{\mathscr{H}}_{\mathcal{T}}$ if and only
if $T_{\rm B}$ has closed range and satisfies that

$\ker(T_{\rm min})=\ker(T_{\rm B})\quad\mbox{and}\quad\ker(T_{\rm
min}^{*})=\ker(T_{{\rm B}^{*}}^{\dagger}).$

In this case,

$\mathrm{dom}(T_{\rm max})=\mathrm{dom}(T_{\rm B})\oplus_{\ker T_{\rm
min}}\ker(T_{\rm max}).$

The reader should note that closed complements of the Hardy space in
$\check{\mathscr{H}}_{\mathcal{T}}$ always exist for a FAP with closed range
since $\check{\mathscr{H}}_{\mathcal{T}}$ is a Hilbert space and the Hardy
space is closed by Corollary B.24.

###### Proof.

Assume that $B$ is a complement to $\mathcal{C}_{\mathcal{T}}$ in
$\check{\mathscr{H}}_{\mathcal{T}}$. The operator $T_{\rm B}$ has closed range
by Theorem B.23. Let
$P:\check{\mathscr{H}}_{\mathcal{T}}\to\mathcal{C}_{\mathcal{T}}$ denote the
projection along $B$. We then arrive at a short exact sequence

$0\to\mathrm{dom}(T_{\rm B})\to\mathrm{dom}(T_{\rm
max})\xrightarrow{P\circ\gamma}\mathcal{C}_{\mathcal{T}}\to 0.$ (33)

In particular, we have the short exact sequence

$0\to\ker(T_{\rm B})\to\ker(T_{\rm
max})\xrightarrow{\gamma}\mathcal{C}_{\mathcal{T}}\to 0,$

showing that $\ker(T_{\rm min})=\ker(T_{\rm B})$. Moreover, the short exact
sequence (33) implies that the map $\Phi_{\rm B}$ from Proposition B.18 is a
bijection, and so $\mathrm{ran}(T_{\rm B})=\mathrm{ran}(T_{\max})$. Since $B$
is a complement to $\mathcal{C}_{\mathcal{T}}$, $B+\mathcal{C}_{\mathcal{T}}$
is closed, so $\mathrm{ran}(T_{\rm B})=\mathrm{ran}(T_{\max})$ implies that
$\ker(T_{\rm min}^{*})=\ker(T_{{\rm B}^{*}}^{\dagger})$.

Assume now that $T_{\rm B}$ has closed range and satisfies that

$\ker(T_{\rm min})=\ker(T_{\rm B})\quad\mbox{and}\quad\ker(T_{\rm
min}^{*})=\ker(T_{{\rm B}^{*}}^{\dagger}).$

The first equality ensures that $B\cap\mathcal{C}_{\mathcal{T}}=0$. The second
equality and Proposition B.26 implies that
$B+\mathcal{C}_{\mathcal{T}}=\check{\mathscr{H}}_{\mathcal{T}}$. Therefore $B$
is a complement to $\mathcal{C}_{\mathcal{T}}$.

The equality $\mathrm{dom}(T_{\rm max})=\mathrm{dom}(T_{\rm B})\oplus_{\ker
T_{\rm min}}\ker(T_{\rm max})$ follows from Equation (33). ∎

Now, we characterise Fredholm boundary conditions via the language of Fredholm
pairs.

######  Theorem B.28.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a Fredholm FAP and
$B$ be a boundary condition. The following are equivalent:

  1. (i)

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

  2. (ii)

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

If either of these equivalent conditions hold, we have that

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

Moreover,

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

###### Proof.

Since $B$ is closed, $T_{\rm B}$ is a closed operator and $\ker(T_{\rm B})$ is
a closed subspace of $\mathscr{H}_{1}$. If i holds, then
$B+\mathcal{C}_{\mathcal{T}}$ is closed and by Theorem B.23, we have that
$\mathrm{ran}(T_{\rm B})$ is closed. Similarly, if i holds then
$B+\mathcal{C}_{\mathcal{T}}$ has finite codimension and $\mathrm{ran}(T_{\rm
B})^{\perp}$ has finite codimension in $\mathscr{H}_{2}$ by Proposition B.26.
Moreover, $B\cap\mathcal{C}_{\mathcal{T}}$ is finite dimensional, so
Proposition B.12 yields that $\ker(T_{\rm B})$ is finite dimensional and
therefore, $T_{\rm B}$ is Fredholm.

On the other hand if ii holds, i.e., $T_{\rm B}$ is Fredholm, then
$\mathrm{ran}(T_{\rm B})=\mathrm{ran}(T_{{\rm B}+\mathcal{C}})$ is closed, so
Theorem B.23 implies that $B+\mathcal{C}_{\mathcal{T}}$ is closed. Moreover,
by Proposition B.12 we have that
$B\cap\mathcal{C}_{\mathcal{T}}\cong\faktor{\ker(T_{\rm B})}{\ker(T_{\min})}$
is finite dimensional. Moreover $\mathrm{ran}(T_{\rm B})^{\perp}$ has finite
codimension and so by Proposition B.26 v, $B+\mathcal{C}_{\mathcal{T}}$ has
finite codimension in $\check{\mathscr{H}}_{\mathcal{T}}$. That is,
$(B,\mathcal{C}_{\mathcal{T}})$ is a Fredholm pair in
$\check{\mathscr{H}}_{\mathcal{T}}$.

From a similar calculation, we obtain that

$\ker(T_{\rm B}^{\ast})=\ker(T^{\dagger}_{{\rm
B}^{\ast}})\cong\ker(T_{\min}^{\dagger})\oplus(B^{\ast}\cap\mathcal{C}_{\mathcal{T}^{\dagger}}).$

By the formula in Proposition B.26, we obtain the desired formula. Plugging
this into the index, we obtain

$\displaystyle\operatorname{ind}(T_{\rm B})$ $\displaystyle=\dim\ker(T_{\rm
B})-\dim\ker(T_{\rm B}^{\ast})$
$\displaystyle=\dim\ker(T_{\min})+\dim(B\cap\mathcal{C}_{\mathcal{T}})-\dim\ker(T_{\min}^{\dagger})-\dim(B^{\ast}\cap\mathcal{C}_{\mathcal{T}^{\dagger}})$
$\displaystyle=\operatorname{ind}(B,\mathcal{C}_{\mathcal{T}})+\dim\ker(T_{\min})-\dim\ker(T_{\min}^{\dagger}).\qed$

We say that a boundary condition of a FAP is a Fredholm boundary condition if
the associated realisation is a Fredholm operator.

######  Corollary B.29.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a Fredholm FAP. Let
$B_{1}\subseteq B_{2}$ be generalised boundary conditions with $B_{1}$ being a
Fredholm boundary condition. Then, $B_{2}$ is a Fredholm boundary condition if
and only if $B_{2}/B_{1}$ is finite dimensional. In this case, the following
index formula holds:

$\operatorname{ind}(T_{{\rm B}_{2}})=\operatorname{ind}(T_{{\rm
B}_{1}})+\dim\faktor{B_{2}}{B_{1}}.$

###### Proof.

Assume that $B_{1}$ has finite codimension in $B_{2}$, so clearly $B_{2}$ will
be closed. It is easy to see that $B_{2}\cap\mathcal{C}_{\mathcal{T}}$ is
finite dimensional and since $B_{1}\subset B_{2}$, we have that

$\dim\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{2}+\mathcal{C}_{\mathcal{T}})}\leq\dim\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{1}+\mathcal{C}_{\mathcal{T}})}.$

That is, $(B_{2},\mathcal{C}_{\mathcal{T}})$ is a Fredholm pair and therefore,
we have that $B_{2}$ is a Fredholm boundary condition.

Assume now that $B_{2}$ is a Fredholm boundary condition, i.e. that $B_{2}$ is
closed and $(B_{2},\mathcal{C}_{\mathcal{T}})$ is a Fredholm pair. Note that

$\displaystyle\check{\mathscr{H}}_{\mathcal{T}}$
$\displaystyle\cong(B_{2}+\mathcal{C}_{\mathcal{T}})\oplus\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{2}+\mathcal{C}_{\mathcal{T}})}$
$\displaystyle\cong\faktor{(B_{2}+\mathcal{C}_{\mathcal{T}})}{(B_{1}+\mathcal{C}_{\mathcal{T}})}\oplus(B_{1}+\mathcal{C}_{\mathcal{T}})\oplus\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{2}+\mathcal{C}_{\mathcal{T}})}$

and therefore,

$\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{1}+\mathcal{C}_{\mathcal{T}})}\cong\faktor{(B_{2}+\mathcal{C}_{\mathcal{T}})}{(B_{1}+\mathcal{C}_{\mathcal{T}})}\oplus\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{2}+\mathcal{C}_{\mathcal{T}})}\cong\faktor{B_{2}}{B_{1}}\oplus\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{2}+\mathcal{C}_{\mathcal{T}})}.$

Since $(B_{2},\mathcal{C}_{\mathcal{T}})$ and
$(B_{1},\mathcal{C}_{\mathcal{T}})$ are Fredholm pairs,
$\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{2}+\mathcal{C}_{\mathcal{T}})}$
and
$\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{1}+\mathcal{C}_{\mathcal{T}})}$
are finite-dimensional and we conclude that $\faktor{B_{2}}{B_{1}}$ is finite
dimensional.

For the index formula, we note that it suffices to prove that

$\operatorname{ind}(B_{2},\mathcal{C}_{\mathcal{T}})=\operatorname{ind}(B_{1},\mathcal{C}_{\mathcal{T}})+\dim\faktor{B_{2}}{B_{1}}.$

This identity follows from the computation

$\displaystyle\operatorname{ind}(B_{2},\mathcal{C}_{\mathcal{T}})$
$\displaystyle=\dim(B_{2}\cap\mathcal{C}_{\mathcal{T}})-\dim\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{2}+\mathcal{C}_{\mathcal{T}})}$
$\displaystyle=\dim(B_{1}\cap\mathcal{C}_{\mathcal{T}})+\dim\faktor{(B_{2}\cap\mathcal{C}_{\mathcal{T}})}{(B_{1}\cap\mathcal{C}_{\mathcal{T}})}$
$\displaystyle\qquad-\dim\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{(B_{1}+\mathcal{C}_{\mathcal{T}})}+\dim\faktor{(B_{2}\cap\mathcal{C}_{\mathcal{T}})^{\perp,B_{2}}}{(B_{1}\cap\mathcal{C}_{\mathcal{T}})^{\perp,B_{1}}}$
$\displaystyle=\operatorname{ind}(B_{1},\mathcal{C}_{\mathcal{T}})+\dim\faktor{(B_{2}\cap\mathcal{C}_{\mathcal{T}})}{(B_{1}\cap\mathcal{C}_{\mathcal{T}})}+\dim\faktor{(B_{2}\cap\mathcal{C}_{\mathcal{T}})^{\perp,B_{2}}}{(B_{1}\cap\mathcal{C}_{\mathcal{T}})^{\perp,B_{1}}}.$

This concludes the proof. ∎

In addition to Theorem B.28, a useful way to obtain Fredholm boundary
conditions is given by the following Proposition which follows from [BB,
Proposition A.1]. It is adapted from [BB, Corollary 8.8], although [BB] only
considered the context of regular boundary conditions. The reader can, in the
context of elliptic boundary value problems, compare the construction of the
next proposition to standard constructions in the Boutet de Monvel calculus
[boutetdemonvel, rempelschulze, gerdsgreenbook]. Again, the provenance of the
next result dates back further than [BB] and can be found in for instance
[gerdsgreenbook, Lemma 4.3.1].

######  Proposition B.30.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a Fredholm FAP with a
boundary condition $B$. Choose a closed complementary subspace $\check{C}$ of
$B$ in $\check{\mathscr{H}}_{\mathcal{T}}$ with projection
$\check{P}:\check{\mathscr{H}}_{\mathcal{T}}\to\check{C}$ defined along $B$
(for instance the orthogonal complement). The map

$\check{L}_{\check{P}}:\mathrm{dom}(T_{\max})\to\begin{matrix}\mathscr{H}_{2}\\\
\oplus\\\
\check{C}\end{matrix},\quad\check{L}_{\check{P}}u=\begin{pmatrix}T_{\max}u\\\
\check{P}\gamma u\end{pmatrix},$

satisfies:

  1. (i)

$T_{\rm B}$ has closed range if and only if $\check{L}_{\check{P}}$ has closed
range.

  2. (ii)

The range of $T_{\rm B}$ has finite codimension if and only if the range of
$\check{L}_{\check{P}}$ has finite codimension, in which case

$\dim\mathrm{coker}(T_{\rm B})=\dim\mathrm{coker}(\check{L}_{\check{P}}).$

  3. (iii)

$\ker T_{\rm B}$ has finite dimension if and only if
$\ker\check{L}_{\check{P}}$ has finite dimension, in which case

$\dim\ker(T_{\rm B})=\dim\ker(\check{L}_{\check{P}}).$

  4. (iv)

$\check{L}_{\check{P}}$ is Fredholm if and only if $T_{\rm B}$ is Fredholm.

We also note the following: Fredholm boundary conditions must necessarily be
infinite dimensional.

######  Proposition B.31.

Let $\mathcal{T}=(T_{\rm min},T^{\dagger}_{\rm min})$ be a Fredholm FAP such
that $\mathcal{C}_{\mathcal{T}}$ has infinite codimension in
$\check{\mathscr{H}}_{\mathcal{T}}$. If $B$ is a Fredholm boundary condition
then $\dim B=\dim B^{*}=\infty$.

###### Proof.

We prove via the contrapositive. Let
$B\subset\check{\mathscr{H}}_{\mathcal{T}}$ be a finite dimensional subspace.
Indeed, this is a boundary condition and $B+\mathcal{C}_{\mathcal{T}}$ is
closed since $\mathcal{C}_{\mathcal{T}}$ is closed. Now,

$\displaystyle\check{\mathscr{H}}_{\mathcal{T}}$
$\displaystyle=(B\cap\mathcal{C}_{\mathcal{T}})\oplus(B\cap\mathcal{C}_{\mathcal{T}})^{\perp}$
$\displaystyle\mathcal{C}_{\mathcal{T}}$
$\displaystyle=(B\cap\mathcal{C}_{\mathcal{T}})\oplus(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,\mathcal{C}_{\mathcal{T}}}$
$\displaystyle B+\mathcal{C}_{\mathcal{T}}$
$\displaystyle=B\oplus(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,\mathcal{C}_{\mathcal{T}}}=(B\cap\mathcal{C}_{\mathcal{T}})\oplus(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,B}\oplus(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,\mathcal{C}_{\mathcal{T}}}.$

From this, we obtain

$\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{\mathcal{C}_{\mathcal{T}}}\cong\faktor{(B\cap\mathcal{C}_{\mathcal{T}})^{\perp}}{(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,\mathcal{C}_{\mathcal{T}}}}$

and

$\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{B+\mathcal{C}_{\mathcal{T}}}\cong\faktor{(B\cap\mathcal{C}_{\mathcal{T}})^{\perp}}{(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,B}\oplus(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,\mathcal{C}_{\mathcal{T}}}}.$

Therefore,

$\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{B+\mathcal{C}_{\mathcal{T}}}\oplus(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,B}\cong\faktor{(B\cap\mathcal{C}_{\mathcal{T}})^{\perp}}{(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,\mathcal{C}_{\mathcal{T}}}}\cong\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{\mathcal{C}_{\mathcal{T}}}.$

Since $(B\cap\mathcal{C}_{\mathcal{T}})^{\perp,B}$ is finite dimensional and
$\mathcal{C}_{\mathcal{T}}$ has infinite codimension in
$\check{\mathscr{H}}_{\mathcal{T}}$, we have that
$\faktor{\check{\mathscr{H}}_{\mathcal{T}}}{B+\mathcal{C}_{\mathcal{T}}}$ is
infinite dimensional. That is, $B$ is not Fredholm. By the contrapositive, if
$B$ is Fredholm, then $\dim B=\infty$. Since $B^{\ast}$ is Fredholm if and
only if $B$ is Fredholm, a symmetric argument yields that $\dim
B^{\ast}=\infty$. ∎

###  B.4. Rigidity results for the index

Let us prove a homotopy invariance result for the index of Fredholm
realisations of Fredholm FAPs.

######  Theorem B.32.

Assume that $\mathcal{T}_{t}=(T_{t,{\rm min}},T^{\dagger}_{t,{\rm
min}})_{t\in[0,1]}$ is a family of Fredholm FAPs
$\mathscr{H}_{1}\to\mathscr{H}_{2}$ such that

  * (i)

The mappings

$[0,1]\ni t\mapsto\begin{pmatrix}0&T_{t,{\rm min}}\\\ T^{\dagger}_{t,{\rm
max}}&0\end{pmatrix}\quad\mbox{and}\quad[0,1]\ni
t\mapsto\begin{pmatrix}0&T^{\dagger}_{t,{\rm min}}\\\ T_{t,{\rm
max}}&0\end{pmatrix}$

are continuous in the gap topology on the space of densely defined self-
adjoint operators (i.e. the resolvents are norm continuous with respect to
$t$).

  * ii)

$\mathrm{dom}(T_{t,{\rm min}})$ and $\mathrm{dom}(T_{t,{\rm max}})$ are
constant in $t$, so $\check{\mathscr{H}}_{\mathcal{T}}:=\mathrm{dom}(T_{t,{\rm
max}})/\mathrm{dom}(T_{t,{\rm min}})$ is independent of $t$ up to a canonical
Banach space isomorphism.

Let $[0,1]\ni t\mapsto P_{t}\in\mathbb{B}(\check{\mathscr{H}}_{\mathcal{T}})$
be a norm continuous family of projectors such that
$B_{t}:=(1-P_{t})\check{\mathscr{H}}_{\mathcal{T}}$ satisfies that
$(B_{t},\mathcal{C}_{\mathcal{T}_{t}})$ is a Fredholm pair in
$\check{\mathscr{H}}_{\mathcal{T}}$ for all $t\in[0,1]$. Then it holds that

$\operatorname{ind}(T_{0,B_{0}})=\operatorname{ind}(T_{1,B_{1}}).$

The proof uses standard techniques from $KK$-theory and the index theory of
Fredholm operators on Hilbert $C^{*}$-modules. For an overview, see
[knudsenjensenthomsen] and [varillygraciabondia].

###### Proof.

We write $\mathscr{H}_{\mathcal{T}}$ for the Hilbert space
$\mathrm{dom}(T_{t_{0},{\rm max}})$ for some fixed $t_{0}$ and note that
$T_{t,{\rm max}}:\mathscr{H}_{\mathcal{T}}\to\mathscr{H}_{2}$ is continuous
for each $t$. By assumption i) above, the family $(T_{t,{\rm
max}})_{t\in[0,1]}$ defines an adjointable $C[0,1]$-linear map that we denote
by $T_{{\rm
max}}:\mathscr{H}_{\mathcal{T},C([0,1])}\to\mathscr{H}_{2,C([0,1])}$ where
$\mathscr{H}_{\mathcal{T},C([0,1])}$ and $\mathscr{H}_{2,C([0,1])}$ denote the
$C[0,1]$-Hilbert $C^{*}$-module defined from the indicated Hilbert spaces. We
also set $\check{C}_{t}:=P_{t}\check{\mathscr{H}}_{\mathcal{T}}$ and let
$\check{C}_{C([0,1])}$ denote the $C[0,1]$-Hilbert $C^{*}$-submodule of
$C([0,1])\otimes\check{\mathscr{H}}_{\mathcal{T}}$ with fibre $\check{C}_{t}$,
it is well-defined by norm continuity of the family $(P_{t})_{t\in[0,1]}$.

Following Proposition B.30, we define the operator

$\check{L}_{t}:\mathscr{H}_{\mathcal{T}}\to\begin{matrix}\mathscr{H}_{2}\\\
\oplus\\\
\check{C}_{t}\end{matrix},\quad\check{L}_{t}u=\begin{pmatrix}T_{t,\max}u\\\
P_{t}\gamma u\end{pmatrix},$

for $t\in[0,1]$. This operator defines an adjointable $C[0,1]$-linear map
$\check{L}:\mathscr{H}_{\mathcal{T},C([0,1])}\to\mathscr{H}_{2,C([0,1])}\oplus\check{C}_{C([0,1])}$
because $T_{{\rm
max}}:\mathscr{H}_{\mathcal{T},C([0,1])}\to\mathscr{H}_{2,C([0,1])}$ and
$\xi\mapsto(t\mapsto P_{t}\xi)$ is an adjointable map
$\check{\mathrm{H}}(D)_{C([0,1])}\to\check{C}_{C([0,1])}$.

We claim that the adjointable $C[0,1]$-linear map
$\check{L}:\mathscr{H}_{\mathcal{T},C([0,1])}\to\mathscr{H}_{2,C([0,1])}\oplus\check{C}$
is Fredholm. For each $t\in[0,1]$, $\check{L}_{t}$ is Fredholm, so we can find
finite-dimensional vector bundles $E_{t,1},E_{t,2}\to[0,1]$ with $E_{t,1}$
being a subbundle of $\mathscr{H}_{\mathcal{T},C([0,1])}$ such that
$E_{t,1}|_{t}=\ker\check{L_{t}}$ and $E_{t,2}$ being a subbundle of
$\mathscr{H}_{2,C([0,1])}$ such that the induced map
$E_{t,2}|_{t}\to\mathrm{coker}\check{L_{t}}$ is surjective. In particular,
there is a finite-rank perturbation of $\check{L}$ as a mapping
$\mathscr{H}_{\mathcal{T},C([0,1])}\oplus
E_{t,2}\to\mathscr{H}_{2,C([0,1])}\oplus\check{C}\oplus E_{t,1}$ which is
invertible when restricted to the fibre over $t$. By an openness argument,
this finite-rank perturbation of $\check{L}$ is invertible on a small open
neighbourhood $U_{t}$ of $t$. In particular, $\check{L}$ is Fredholm when
restricted to $U_{t}$. By compactness of $[0,1]$, $\check{L}$ is Fredholm.

Since, $\check{L}$ is a Fredholm operator between $C[0,1]$-Hilbert
$C^{*}$-modules, it has a well-defined index in $K^{0}([0,1])$. Under the
isomorphisms $K^{0}([0,1])\cong K^{0}(pt)\cong\mathbb{Z}$, homotopy invariance
of $K$-theory implies that

$\operatorname{ind}(\check{L}_{0})=r_{0}^{*}\operatorname{ind}_{C([0,1])}(\check{L})=r_{1}^{*}\operatorname{ind}_{C([0,1])}(\check{L})=\operatorname{ind}(\check{L}_{1}),$

where $r_{t}$ denotes the inclusion $\\{t\\}\subseteq[0,1]$. We can now
conclude the theorem from Proposition B.30 which implies that

$\operatorname{ind}(T_{0,B_{0}})=\operatorname{ind}(\check{L}_{0})=\operatorname{ind}(\check{L}_{1})=\operatorname{ind}(T_{1,B_{1}}).\qed$

## References