f_{\alpha}\cdot f_{\beta}\bigr{)}\cdot_{g}f_{I}\Bigr{)}$
$\displaystyle=\Bigl{(}\sum_{\alpha<\beta}g(\nabla_{e_{i}}^{L.C.,g}\tau_{g_{0},g}e_{\alpha},\tau_{g_{0},g}e_{\beta})\cdot
e_{\alpha}\cdot e_{\beta}\Bigr{)}\cdot e_{I}.$

We already know that
$g(\nabla_{e_{i}}^{L.C.,g}e_{\alpha},e_{\beta})={}^{g}\Gamma_{i\alpha}^{\beta}$
depends smoothly on $g$. Thus, the assignment

$g\mapsto{p\mathcal{G}_{g,g_{0}}}_{\ast}\nabla^{\mathfrak{S}_{g}}-\nabla^{\mathfrak{S}_{g_{0}}}$

is a smooth map
$\mathrm{Riem}(M)\rightarrow\Gamma\bigl{(}M;\mathrm{Hom}(\mathfrak{S}_{g_{0}},T^{\vee}M\otimes\mathfrak{S}_{g_{0}})\bigr{)}$
and hence it is also a smooth map

$\mathrm{Riem}(M)\rightarrow\Psi\mathrm{DO}^{0}(\mathfrak{S}_{g},T^{\vee}M\otimes\mathfrak{S}_{g}).$

This implies that $g\mapsto{\not{\mathfrak{D}}}_{g}$ is smooth as it is a
composition of smooth maps. ∎

The next result is classical and follows immediately from the theory developed
in [LawsonMichelsonSpin]*Chapter III.5.

######  Proposition 4.1.33.

Let $M$ be a closed manifold and let $P\in\Psi\mathrm{Dir}(M)$ be a pseudo
Dirac operator with underlying Riemannian metric $g\in\mathrm{Riem}(M)$. Then
$P$ is a self-adjoint, unbounded Fredholm operator on
$L^{2}(\mathfrak{S}_{g})$ with domain $H^{1}(\mathfrak{S}_{g})$.

###### Proof.

By definition, $P$ is symmetric, so $P-{\not{\mathfrak{D}}}_{g}$ is a
symmetric pseudo differential operator of order zero on $\mathfrak{S}_{g}$.
Thus the difference extends to a bounded, symmetric, and hence self-adjoint,
operator on $L^{2}(\mathfrak{S}_{g})$. Since the sum of a self-adjoint
operator with a self-adjoint bounded operator is again self-adjoint
[higson2000analytic]*Exercise 1.9.20,
$P={\not{\mathfrak{D}}}_{g}+(P-{\not{\mathfrak{D}}}_{g})$ is self-adjoint.

Gårdings inequality, see [LawsonMichelsonSpin]*Thm III.5.2, implies that the
minimal domain of $P$ agrees with $H^{1}(\mathfrak{S}_{g})$. It is classical
that elliptic pseudo differential operators on vector bundles over closed
manifolds are unbounded Fredholm operators, see [LawsonMichelsonSpin]*Thm
III.5.2. ∎

##### Relations to K-theory

Our interests in pseudo Dirac operators arise from their connection to
$KO$-theory, which we will outline in this section. All presented results have
already appeared in [ebert2017indexdiff] in a different language and we do not
claim originality for the results presented here; rather we translate his
results into our language, fill some gaps, and give more details. Needless to
say, the presentation here closely follows [ebert2017indexdiff]. Throughout
this section, we assume that $M^{d}$ is a _closed_ spin manifold of positive
dimension.

We start with recalling the connection between $KO$-theory and Clifford-linear
Fredholm operators. Let $H$ be a separable real Hilbert space with
$\mathord{\mathbb{Z}}_{2}$-grading $\iota$ and graded $Cl_{d,0}$-right action
$\mathbf{r}$, which we assume to be a $\ast$-homomorphism. For brevity, we
simply call it a $Cl_{d,0}$-Hilbert space. Call
$\omega_{d,0}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\iota\mathbf{r}(e_{1}\cdots e_{d})$ the _Chirality element_. Recall that
$Cl_{d,0}$ has, up to isomorphism, exactly one irreducible representation if
$d\not\equiv 0\mod 4$, and exactly two irreducible representations if
$d\equiv-1\mod 4$. In the latter case, the irreducible representations are
distinguished as different eigenspaces of the Chirality element.

The triple $(H,\iota,\mathbf{r})$ is called _ample_ if $H$ contains all
irreducible representations with infinite multiplicity.

######  Definition 4.1.34.

Let $(H,\iota,\mathbf{r})$ be an ample $Cl_{d,0}$-Hilbert space. We call an
operator $F$ on $H$ an $Cl_{d,0}$_-Fredholm operator_ if it is Clifford-
linear, odd, and self-adjoint. If $d\equiv-1\mod 4$, we further require that
$\omega_{p,0}F\iota$ is neither essentially positive nor essentially negative,
that means, there exist infinite dimensional, closed subspaces that are
orthogonal to each other and on those $F$ restricts to a positive or negative
operator, respectively.

Let $\mathrm{Fred}^{d,0}(H)$ be the space of all $Cl_{d,0}$-Fredholm operators
on $H$ equipped with the norm topology.

By a classical result of Atiyah and Singer, these spaces represent real
$K$-theory.

######  Theorem 4.1.35 (Theorem [atiyah1969indexskew]).

If $H$ is ample, then $\mathrm{Fred}^{d,0}(H)$ is a classifying space for
$KO^{-d}$. That means there is a natural bijection

$\mathrm{ind}\colon[X,\mathrm{Fred}^{d,0}(H)]\rightarrow KO^{-d}(X)$

for every compact $CW$-complex $X$.

A detailed construction of this map was carried out in the master’s thesis of
Jonathan Glöckle [gloeckle2019master]*Theorem 2.17. For more details on this
space like Bott-periodicity or Morita equivalences, we refer the reader to
[ebert2017indexdiff]. To identify the space of invertible pseudo Dirac
operators as a classifying space for $KO$-theory, we first need to introduce
auxiliary spaces.

######  Definition 4.1.36.

Let $\Psi\mathrm{Dir}^{\times}(M)$ be the subspace of $\Psi\mathrm{Dir}(M)$
consisting of all invertible pseudo Dirac operators and denote $[-1,1]$ by
$I$. Provided $\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$ is non-empty for a fixed
metric $g_{0}$, we define, for a fixed choice of base point
$B\in\Psi\mathrm{Dir}^{\times}(M)$, the spaces

${\mathcal{X}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{P\colon I\rightarrow\Psi\mathrm{Dir}(M)\,:\,P\text{ smooth,
}P(-1)=B,\ P(1)\in\Psi\mathrm{Dir}^{\times}(M)\\}$

and

${\mathcal{X}}_{g_{0}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{P\colon I\rightarrow\Psi\mathrm{Dir}(M)_{g_{0}}\,:\,P\text{ smooth,
}P(-1)=B,\ P(1)\in\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}\\}.$

######  Lemma 4.1.37.

If $\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$ is non-empty, then the evaluation
map

$\mathrm{ev}_{1}\colon{\mathcal{X}}\rightarrow\Psi\mathrm{Dir}^{\times}(M)$

is a weak homotopy equivalence that also restricts to a weak homotopy
equivalence between ${\mathcal{X}}_{g_{0}}$ and
$\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$.

###### Proof.

We only prove the first statement; for the second statement, one simply needs
to add the subscript ${}_{g_{0}}$ to $\Psi\mathrm{Dir}^{\times}(M)$ and the
space $\mathcal{Y}$ defined below.

Let $\mathcal{Y}$ be the set of all _continuous_ paths $P\colon
I\rightarrow\Psi\mathrm{Dir}(M)$ having the same boundary conditions as
elements in ${\mathcal{X}}$. Smoothing theory yields that the inclusion of
${\mathcal{X}}$ into $\mathcal{Y}$ is a weak homotopy equivalence.

The map
$\mathrm{ev}_{1}\colon\mathcal{Y}\rightarrow\Psi\mathrm{Dir}^{\times}(M)$ is a
fibration. Indeed, given a commutative square

$\textstyle{A\times
0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{1}}$$\textstyle{A\times
I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\Psi\mathrm{Dir}^{\times}(M)}$

we define the lift $\mathcal{H}\colon A\times I\rightarrow\mathcal{Y}$ by

$\mathcal{H}(a,s)(t)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\frac{(1-t)}{2}\cdot F(a)+\frac{(1+t)}{2}\cdot h(a,s).$

This is a continuous map because the affine structure on $\Psi\mathrm{Dir}(M)$
is (jointly) continuous with respect to the Fréchet structure. It is also a
lift for $\mathrm{ev}_{1}(\mathcal{H})=h$, showing that $\mathrm{ev}_{1}$ is a
fibration.

The homotopy fibre of $\mathrm{ev}_{1}$ is therefore the preimage of $B$,
which is the set of all closed curves in $\Psi\mathrm{Dir}(M)$ that start (and
end) at $B$. This subspace is clearly convex and therefore contractible,
implying that $\mathcal{Y}$ and $\Psi\mathrm{Dir}^{\times}(M)$ are weak
homotopy equivalent. ∎

The auxiliary path spaces ${\mathcal{X}}$ and ${\mathcal{X}}_{g_{0}}$ help to
identify $\Psi\mathrm{Dir}^{\times}(M)$ as classifying space for $KO$-theory.

We remark that, for a closed manifold $M^{d}$ of positive dimension $d$, the
$Cl_{d,0}$-Hilbert space $H=L^{2}(M,\mathfrak{S}_{g_{0}})$ is always ample.
Indeed, the right action of the Chirality element decomposes
$\mathfrak{S}_{g_{0}}$ into non-zero subbundles $\mathfrak{S}_{g_{0}}^{\pm}$
on which the Chirality element acts as $\pm\mathrm{id}$, hence

$L^{2}(M,\mathfrak{S}_{g_{0}})\cong L^{2}(M,\mathfrak{S}_{g_{0}}^{+})\oplus
L^{2}(M,\mathfrak{S}_{g_{0}}^{-})$

is a decomposition into infinite dimensional eigenspaces of the eigenvalues
$\pm 1$ of the Chirality element.

The following theorem is a collection of results proved by Johannes Ebert in
[ebert2017indexdiff]*Section 4.1:

######  Theorem 4.1.38.

Assume that $\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$ is non-empty. Then the
diagram of continuous maps

$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega\mathrm{Fred}^{d,0}\bigl{(}L^{2}(M,\mathfrak{S}_{g_{0}})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}_{g_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Omega\mathrm{Fred}^{d,0}\bigl{(}L^{2}(M,\mathfrak{S}_{g_{0}})\bigr{)}}$

commutes. The horizontal maps are given by

$P_{t}\mapsto\frac{\mathcal{G}_{g_{t},g_{0}}P_{t}\mathcal{G}_{g_{t},g_{0}}^{-1}}{(1+(\mathcal{G}_{g_{t},g_{0}}P_{t}\mathcal{G}_{g_{t},g_{0}}^{-1})^{2})^{1/2}}$

and the lower map is a weak homotopy equivalence. Here
$\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))$ denotes the space of
paths into $\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))$ whose start and
end-points are invertible operators.

###### Proof.

For fixed $t$, these maps certainly take values in the space of self-adjoint,
odd, Clifford-linear Fredholm operators. From the discussion in
[ebert2017indexdiff]*Section 4.1 it is not clear why the horizontal map takes
values in the subspace with the additional condition that $\omega F\iota$ is
neither essentially positive nor essentially negative if $d\equiv-1\mod 4$.

In order to show this, first recall from [LawsonMichelsonSpin]*Theorem III.5.8
that the spectrum of every self-adjoint elliptic pseudo differential operator
$P$ of positive order on a bundle over a closed manifold consists of countably
many eigenvalues only, that the eigenspaces are finite dimensional, and that
the eigenvalues go to infinity. The operator $P/\sqrt{1+P^{2}}$ is the unique
operator that acts as $\lambda/\sqrt{1+\lambda^{2}}\cdot\mathrm{id}$ on the
$\lambda$-eigenspace of $P$.

The Chirality element satisfies

$\omega_{d,0}^{2}=(-1)^{\frac{d(d+1)}{2}}\quad\text{ and
}\quad\omega_{d,0}^{\ast}=(-1)^{\frac{d(d+1)}{2}}\omega_{d,0}.$

Thus, it is a self-adjoint, odd involution if $d\equiv-1\mod 4$. Since $P$,
and therefore $P/\sqrt{1+P^{2}}$ too, commute with $\omega_{d,0}$, we can
simultaneously diagonalise the two operators.

The grading $\mathrm{e/o}$ anti-commutes with $P$, so the grading sends the
$\lambda$-eigenspace of $P$ to the $-\lambda$-eigenspaces of $P$. The
corresponding statement holds true for the $\pm 1$-eigenspaces of
$\omega_{d,0}$ for the same reason.

Abbreviate $P/\sqrt{1+P^{2}}$ to $F$. Since the grading $\mathrm{e/o}$
commutes with $\omega_{d,0}F$, the operator $\omega_{d,0}F\mathrm{e/o}$ is
positive definite on

$H^{+}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\bigoplus_{\lambda>0}\mathrm{Eig}(P;\lambda)\cap\mathrm{Eig}(\omega_{d,0};1)\oplus\mathrm{Eig}(P;-\lambda)\cap\mathrm{Eig}(\omega_{d,0};-1)$

and negative definite on

$H^{-}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\bigoplus_{\lambda>0}\mathrm{Eig}(P;\lambda)\cap\mathrm{Eig}(\omega_{d,0};-1)\oplus\mathrm{Eig}(P;-\lambda)\cap\mathrm{Eig}(\omega_{d,0};1).$

By the discussion above, the two subspaces are infinite dimensional. Thus,
$\omega_{d,0}F\mathrm{e/o}$ cannot be essentially positive or essentially
negative.

The upper horizontal map is continuous because conjugation with invertible
functions is a continuous operation as
$\Psi\mathrm{DO}^{\bullet}(\mathfrak{S}_{g_{0}})$ is a Fréchet algebra, and
bounded functional calculus is continuous by [ebert2017indexdiff]*Proposition
3.7. The weak equivalence statement is [ebert2017indexdiff]*Proposition 4.3. ∎

The space $\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$ is unsuitable for our
purposes. Because we want to compare different metrics, we need the bigger
space $\Psi\mathrm{Dir}^{\times}(M)$. Luckily, the canonical inclusion is a
weak homotopy equivalence. This is the main result of
[ebert2017indexdiff]*Section 4.2 formulated in our language.

######  Theorem 4.1.39.

The inclusion
$\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}\hookrightarrow\Psi\mathrm{Dir}^{\times}(M)$
is a weak homotopy equivalence.

The identification between $\Psi\mathrm{Dir}^{\times}(M)$ and
$\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))$ is the space-level
version of the index difference of Hitchin [hitchin1974harmonic].

######  Definition 4.1.40.

Fix base points $g_{0}\in\mathrm{Riem}(M)$ and
$B\in\Psi\mathrm{Dir}^{\times}(M)$. For any choice of section
$s\colon\Psi\mathrm{Dir}^{\times}(M)\rightarrow\mathcal{X}$ of
$\mathrm{ev}_{1}$, we call the composition

$\textstyle{\mathrm{inddif}(\text{--},B)\colon\Psi\mathrm{Dir}^{\times}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{ev}_{1}}$$\textstyle{\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))}$

the _(operator) index difference_.

If $g_{0}$ is a psc-metric then we call the pre-composition with the Dirac
operator

$\textstyle{\alpha(\text{--},g_{0})\colon\mathrm{Riem}^{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\not{\mathfrak{D}}}_{(\text{--})}}$$\textstyle{\Psi\mathrm{Dir}^{\times}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{inddif}(\text{--},{\not{\mathfrak{D}}}_{g_{0}})}$$\textstyle{\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))}$

the _index difference_.

We summarise our achievement of this subsection in the following theorem.

######  Theorem 4.1.41.

Let $M^{d}$ be a closed spin manifold of positive dimension. If the space of
all invertible pseudo Dirac operators $\Psi\mathrm{Dir}^{\times}(M)$ is non-
empty, then it is a classifying space for $KO^{-(d+1)}(\mathrm{pt})$. The
natural transformation is given by the _Hitchin index difference_

$\textstyle{[X;\Psi\mathrm{Dir}^{\times}(M)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{inddif}(\text{--},B)_{\ast}}$$\textstyle{[X;\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{adjunction}}$$\scriptstyle{\cong}$$\textstyle{KO^{-(d+1)}(X)}$$\textstyle{KO^{-d}(\Sigma
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{[\Sigma
X;\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ind}}$$\scriptstyle{\cong}$

Our primary goal is to show that the index difference factors through the
concordance set. To this end, we will construct another model for real
$K$-theory that is built out of $\Psi\mathrm{Dir}^{\times}(M)$ in the same way
as $\widetilde{\mathcal{R}^{+}_{\bullet}}(M)$ is built out of
$\mathrm{Riem}^{+}(M)$. We will pursue this construction in Section 4.4 by
“blockifying” the objects of this section. In the next section, we lay the
necessary analytical groundwork.

###  4.2 Block Operators

In this section, we discuss the foundations of _block pseudo differential
operators._ These are operators on $M\times\mathord{\mathbb{R}}^{n}$ that can
be formally thought of as pseudo differentials operators with the property
that if we think of $\mathord{\mathbb{R}}^{n}$ as a cube of infinite length,
then they decompose into a product operator near each face. We will discuss
the required combinatorial language to deal with those operators, present a
construction of a partition of unity that respects the decomposition of a
block pseudo differential operator near infinity, and discuss the application
of these partitions to the support of these block operators. Finally, we
define the operator suspension of a family of pseudo Dirac operator and show
that it is a block operator.

We assume that the reader is familiar with the concept of pseudo differential
operators and their extension of Atiyah and Singer, otherwise he or she may
consult Appendix B and the references therein.

#### Block Bundles and Basics of Block Operators

The seemingly harmless space $\mathord{\mathbb{R}}^{n}$ can be decomposed into
products $\mathord{\mathbb{R}}^{p}\times\mathord{\mathbb{R}}^{n-p}$ in various
ways. It will be important to keep track of these decompositions, which
requires notation that seems quite heavy (and maybe unnecessary) at the
beginning. For
${\mathrm{dom}\,}{\bm{\varepsilon}}\subseteq\mathbf{n}=\\{1,\dots,n\\}$ and
${\bm{\varepsilon}}\colon{\mathrm{dom}\,}{\bm{\varepsilon}}\rightarrow\mathord{\mathbb{Z}}_{2}$
we define, for a given $M$ the sets

$\displaystyle
U_{R}({\bm{\varepsilon}})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=M\times\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}>R\text{
for }i\in{\mathrm{dom}\,}{\bm{\varepsilon}}\\},$
$\displaystyle\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{Map}(\mathbf{n}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}},\mathord{\mathbb{R}})\cong\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}=R\\}$
$\displaystyle\mathord{\mathbb{R}}^{{\bm{\varepsilon}}}_{R}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{x\in\mathord{\mathbb{R}}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}\,:\,{\bm{\varepsilon}}(i_{j})x_{j}>R,\,i_{j}\in{\mathrm{dom}\,}{\bm{\varepsilon}},\,i_{j}<i_{j+1}\\}.$

We would like to emphasise that the identification between
$\mathrm{Map}(\mathbf{n}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}},\mathord{\mathbb{R}})$
and $\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}=R\\}$ is
only canonical once $R$ is fixed.

For such an ${\bm{\varepsilon}}$ let

$c({\bm{\varepsilon}})^{-1}\colon\mathbf{n}\rightarrow\mathbf{n}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}}\sqcup{\mathrm{dom}\,}{\bm{\varepsilon}}$

be the unique map that partitions $\mathbf{n}$ into
$\mathbf{n}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}}$ and
${\mathrm{dom}\,}{\bm{\varepsilon}}$ in an order preserving manner. We write
${\bm{\eta}}\geq{\bm{\varepsilon}}$ if
${\mathrm{dom}\,}{\bm{\eta}}\supseteq{\mathrm{dom}\,}{\bm{\varepsilon}}$ and
${\bm{\eta}}|_{{\mathrm{dom}\,}{\bm{\varepsilon}}}={\bm{\varepsilon}}$. For
all ${\bm{\eta}}\geq{\bm{\varepsilon}}$ let
$c({\bm{\eta}},{\bm{\varepsilon}})^{-1}$ be the unique map that partitions
${\mathrm{dom}\,}{\bm{\eta}}^{c}$ into
${\mathrm{dom}\,}{\bm{\eta}}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}}$ and
${\mathrm{dom}\,}{\bm{\varepsilon}}^{c}$ in an order preserving manner and let
$\mathrm{per}({\bm{\eta}},{\bm{\varepsilon}})^{-1}$ be the unique map that
partitions ${\mathrm{dom}\,}{\bm{\eta}}$ into
${\mathrm{dom}\,}{\bm{\eta}}-{\bm{\varepsilon}}$ and
${\mathrm{dom}\,}{\bm{\varepsilon}}$ in an order preserving manner. Here,
${\bm{\eta}}-{\bm{\varepsilon}}$ denotes ${\bm{\eta}}$ restricted to
${\mathrm{dom}\,}{\bm{\eta}}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}}$.

For each ${\bm{\eta}}\geq{\bm{\varepsilon}}$ we have a commutative diagram

$\textstyle{\mathbf{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\eta}})^{-1}}$$\textstyle{{\mathrm{dom}\,}{\bm{\eta}}^{c}\sqcup{\mathrm{dom}\,}{\bm{\eta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\sqcup\mathrm{per}({\bm{\eta}},{\bm{\varepsilon}})^{-1}}$$\textstyle{{\mathrm{dom}\,}{\bm{\eta}}^{c}\sqcup{\mathrm{dom}\,}{\bm{\eta}}-{\bm{\varepsilon}}\sqcup{\mathrm{dom}\,}{\bm{\varepsilon}}}$$\textstyle{\mathbf{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\varepsilon}})^{-1}}$$\textstyle{{\mathrm{dom}\,}{\bm{\varepsilon}}^{c}\sqcup{\mathrm{dom}\,}{\bm{\varepsilon}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{c({\bm{\eta}},{\bm{\varepsilon}})^{-1}\sqcup\mathrm{id}}$

These maps induce linear maps on Euclidean spaces by pulling back their
inverse. For example,

$\displaystyle c({\bm{\varepsilon}})\colon\mathord{\mathbb{R}}^{n}$
$\displaystyle\rightarrow\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}^{{\bm{\varepsilon}}}$
$\displaystyle(x_{1},\dots,x_{n})$
$\displaystyle\mapsto(x_{c({\bm{\varepsilon}})^{-1}(1)},\dots,x_{c({\bm{\varepsilon}})^{-1}(|{\mathrm{dom}\,}{\bm{\varepsilon}}^{c}|)};x_{c({\bm{\varepsilon}})^{-1}(|{\mathrm{dom}\,}{\bm{\varepsilon}}^{c}|+1)},\dots,x_{c({\bm{\varepsilon}})^{-1}(n)}).$

We make the following notational convention: For a block metric $g$ on
$M\times\mathord{\mathbb{R}}^{n}$, we denote by $g({\bm{\varepsilon}})$ the
restriction to $M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ if we
interpret $\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ as
$\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}=R\\}$ and by
$\partial^{{\bm{\varepsilon}}}g$ the restriction to
$M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ if we interpret
$\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ as
$\mathord{\mathbb{R}}^{|{\mathrm{dom}\,}({\bm{\varepsilon}})^{c}|}$. The
identification between these two is the pull-back with
${\delta^{R{\bm{\varepsilon}}}}$.

The pull-back of $c({\bm{\varepsilon}})$ coordinates yields an isometry

$c({\bm{\varepsilon}})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{id}_{M}\times
c({\bm{\varepsilon}})\colon\left(U_{R}({\bm{\varepsilon}}),g|_{U({\bm{\varepsilon}})}\right)\rightarrow\bigl{(}M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}_{R}^{\bm{\varepsilon}},\underbrace{g({\bm{\varepsilon}})\oplus\langle\cdot,\cdot\rangle}_{=c({\bm{\varepsilon}})_{\ast}g|_{U_{R}({\bm{\varepsilon}})}}\bigr{)}.$

This fits into the following diagram of isometries:

$\textstyle{U_{R}({\bm{\eta}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\eta}})}$$\textstyle{(M\times\mathord{\mathbb{R}}^{n})({\bm{\eta}})\times\mathord{\mathbb{R}}_{R}^{{\bm{\eta}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\times\mathrm{per}({\bm{\eta}},{\bm{\varepsilon}})}$$\textstyle{(M\times\mathord{\mathbb{R}}^{n})({\bm{\eta}})\times\mathord{\mathbb{R}}_{R}^{{\bm{\eta}}-{\bm{\varepsilon}}}\times\mathord{\mathbb{R}}_{R}^{\bm{\varepsilon}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\eta}},{\bm{\varepsilon}})}$$\textstyle{U_{R}({\bm{\varepsilon}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\varepsilon}})}$$\textstyle{(M\times\mathord{\mathbb{R}}^{n})({\bm{\varepsilon}})\times\mathord{\mathbb{R}}_{R}^{\bm{\varepsilon}}.}$

Since all of these maps are induced by permutations on Euclidean spaces, they
are Pin-structure preserving isometries because each Euclidean space has, up
to equivalence, only one $\mathrm{Pin}^{-}$-structure.

Functoriality of the spinor bundle construction, see Lemma 4.1.17, yields a
splitting on the associated spinor bundles:

$\Phi_{\bm{\varepsilon}}=\mathfrak{S}(c({\bm{\varepsilon}}))\colon\mathfrak{S}_{U_{R}({\bm{\varepsilon}})}\rightarrow\mathfrak{S}_{g({\bm{\varepsilon}})}\boxtimes
Cl_{{\bm{\varepsilon}},0}=\mathfrak{S}_{g({\bm{\varepsilon}})\oplus\langle\cdot,\cdot\rangle},$

where $Cl_{{\bm{\varepsilon}},0}$ denotes $Cl_{|{\bm{\varepsilon}}|,0}$. This
gives rise to morphisms between spinor bundles:

$\textstyle{\mathfrak{S}|_{M\times
U_{R}({\bm{\eta}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{{\bm{\eta}}}}$$\textstyle{\mathfrak{S}_{g({\bm{\eta}})}\boxtimes
Cl_{{\bm{\eta}},0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\times\mathfrak{S}(\mathrm{per}({\bm{\varepsilon}},{\bm{\eta}}))}$$\textstyle{\mathfrak{S}_{g({\bm{\eta}})}\boxtimes
Cl_{{\bm{\eta}}-{\bm{\varepsilon}},0}\boxtimes
Cl_{{\bm{\varepsilon}},0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{C}_{{\bm{\eta}},{\bm{\varepsilon}}}}$$\textstyle{\mathfrak{S}|_{M\times
U_{R}({\bm{\varepsilon}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{{\bm{\varepsilon}}}}$$\textstyle{\mathfrak{S}_{g({\bm{\varepsilon}})}\boxtimes
Cl_{{\bm{\varepsilon}},0}.}$

######  Definition 4.2.1.

Let $g$ be a block metric on $M\times\mathord{\mathbb{R}}^{n}$ and let
$A\in\\{\mathrm{id},\mathrm{e/o}\\}$. A pseudo differential operator
$P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ is of _block form_ (_of
type_ $A$) if there are $R>r>0$ such that:

  * (1)

The block metric $g$ decomposes outside of $M\times rI^{n}$.

  * (2)

If $\sigma\in\Gamma_{c}(\mathfrak{S}_{g})$ is supported within
$M\times(RI^{n})^{\circ}$, then $P\sigma$ is supported within
$M\times(RI^{n})^{\circ}$.

  * (3)

The operator $P$ restricts555This means that if a section is supported within
$U_{r}({\bm{\varepsilon}})$, then $P\sigma$ is also supported within this set.
to $M\times U_{r}({\bm{\varepsilon}})$ and there is a
$P({\bm{\varepsilon}})\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g({\bm{\varepsilon}})})$
such that

$(\Phi_{\bm{\varepsilon}})_{\ast}P|_{M\times
U_{r}({\bm{\varepsilon}})}=P({\bm{\varepsilon}})\boxtimes\mathrm{id}+A\boxtimes
D({\bm{\varepsilon}}),$

where
$P({\bm{\varepsilon}})\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g({\bm{\varepsilon}})})$
and $D({\bm{\varepsilon}})$ is a differential operator of order $m$ on the
trivial bundle
$Cl_{{\bm{\varepsilon}},0}\rightarrow\mathord{\mathbb{R}}^{\bm{\varepsilon}}_{r}$
with constant coefficients.

We call $M\times RI^{n}$ _the core_ of $P$.

The constants $R$ and $r$ are not unique. We will see in Lemma 4.2.6 below
that property (2) and (3) hold for all larger constants $R^{\prime}>R$ and
$r^{\prime}>r$ as well.

######  Lemma 4.2.2.

Let $P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ be a block operator
of type $A$. Then the operators $P({\bm{\varepsilon}})$ and
$D({\bm{\varepsilon}})$ are uniquely determined by $P$.

###### Proof.

We prove the statement by induction over
$|{\mathrm{dom}\,}{\bm{\varepsilon}}|$. On
${\mathrm{dom}\,}{\bm{\varepsilon}}=\emptyset$, there is nothing to prove. For
the induction step, it suffices to prove the following statement: Every
operator of the form $P=Q\boxtimes\mathrm{id}+A\boxtimes
D\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g_{M}}\boxtimes Cl_{n,0})$ on
a _not necessarily closed_ spin manifolds $M$ with product metrics
$g=g_{M}\oplus\langle\cdot,\cdot\rangle$ uniquely deterimes $Q$ and $D$. We
will prove this statement by expressing $Q$ and $D$ through $P$.

Let $x^{1},\dots,x^{n}$ be the standard coordinates of
$\mathord{\mathbb{R}}^{n}$ and $\partial_{j}$ the derivation into the $j$-th
direction. The operator $D$ can be written as

$D=\sum_{|{\bm{\alpha}}|\leq m}a_{\bm{\alpha}}\partial^{\bm{\alpha}},$

for some constants $a_{\bm{\alpha}}\in\mathrm{End}(Cl_{n,0})$. For a point
$p\in\mathord{\mathbb{R}}^{n}$, choose smooth functions $f_{1},\dots,f_{n}$
with compact support satisfying $f_{j}(x)=(x_{j}-p_{j})$ in some neighbourhood
$V$ of $p$. Choose further $u\in\Gamma_{c}(M,\mathfrak{S}_{g_{M}})$ and
$v\in\Gamma_{c}(V,Cl_{n,0})$. Finally, for each multiindex ${\bm{\alpha}}$
with $|{\bm{\alpha}}|\leq m$, define the multiplication operator
$\mu_{\bm{\alpha}}$ by

$\displaystyle\mu_{\bm{\alpha}}\colon\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g_{M}}\boxtimes
Cl_{n,0})$
$\displaystyle\rightarrow\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g_{M}}\boxtimes
Cl_{n,0})$ $\displaystyle\sigma$
$\displaystyle\mapsto\prod_{j=1}^{n}f_{j}^{\alpha_{j}}\cdot\sigma.$

The Leibniz rule implies

$[P,\mu_{\bm{\alpha}}](u\otimes v)(m,p)=A(u)(m)\otimes a_{\bm{\alpha}}v(p),$

so $D$ is uniquely determined by $P$.

It remains to determine $Q$ from $P$ and $D$. Since $P-A\boxtimes D$ is
$\mathcal{C}^{\infty}(\mathord{\mathbb{R}}^{n})$-linear on
$\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g_{M}}\boxtimes
Cl_{n,0})$ we can define a linear map

$\tilde{Q}\colon\Gamma_{c}(M,\mathfrak{S}_{g_{M}})\rightarrow\Gamma(M,\mathfrak{S}_{g_{M}})$

as follows: Fix a point $p\in\mathord{\mathbb{R}}^{n}$ and a function
$f\in\mathcal{C}^{\infty}(\mathord{\mathbb{R}}^{n},\mathord{\mathbb{R}})$ that
is identically $1$ near $x$ and compactly supported. Set then

$\tilde{Q}(\sigma)(m)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(P-A\boxtimes D)(\sigma\otimes f\cdot 1)(m,p).$

Plugging in the decomposition gives

$\displaystyle\tilde{Q}(\sigma)(m)$ $\displaystyle=(P-A\boxtimes
D)(\sigma)(m,p)$ $\displaystyle=Q\boxtimes\mathrm{id}(\sigma\otimes f\cdot
1)(m,p)$ $\displaystyle=Q(\sigma)(m)\otimes f(p)=Q(\sigma)(m)\otimes 1,$

which can be identified with $Q(\sigma)$ under the inclusion
$\mathfrak{S}_{g_{M}}\subseteq\mathfrak{S}_{g_{M}}\boxtimes Cl_{n,0}$. ∎

######  Corollary 4.2.3.

If $P$ is a block operator on $\mathfrak{S}_{g}$, then $P({\bm{\varepsilon}})$
is a block operator on $\mathfrak{S}_{g({\bm{\varepsilon}})}$. Equivalently,
if
${\delta^{2\rho{\bm{\varepsilon}}}}\colon\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\xrightarrow{\cong}\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}=2\rho\\}$
denotes the isometric inclusion that plugs $2{\bm{\varepsilon}}(j)\rho$ into
the $j$-th coordinate for all $j\in{\mathrm{dom}\,}{\bm{\varepsilon}}$, then
${\delta^{2\rho{\bm{\varepsilon}}}}^{\ast}P({\bm{\varepsilon}})=({\delta^{2\rho{\bm{\varepsilon}}}})^{-1}_{\ast}P({\bm{\varepsilon}})$
is a block operator on $\mathfrak{S}_{\partial^{{\bm{\varepsilon}}}g}$ for all
$\rho>R$, where $R$ denotes the size of the core of $P$. These block operators
satisfy conditions (1)-(3) for the same constants as $P$.

###### Proof.

Since $g$ decomposes outside of $rI^{n}$, it also decomposes outside of $\rho
I^{n}$. If we identify $\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ with the
affine subspace
$\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(j)x_{j}=\rho\\}$, then
the restricted metric
$g({\bm{\varepsilon}})=g\mathord{\upharpoonleft}_{\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})}$
is again a block metric that decomposes outside of
$rI^{n}\cap\mathord{\mathbb{R}}^{n}{\bm{\varepsilon}})\cong
rI^{n-|{\mathrm{dom}\,}{\bm{\varepsilon}}|}$.

Set
$m\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=n-|{\mathrm{dom}\,}{\bm{\varepsilon}}|$. We verify axiom $(2)$ for
${{\delta^{2r{\bm{\varepsilon}}}}}^{\ast}P({\bm{\varepsilon}})$ by contra-
position. Assume there is a section
$\sigma\in\Gamma_{c}(\mathfrak{S}_{\partial^{\varepsilon}_{{\bm{\varepsilon}}}g({\bm{\varepsilon}})})$
with support in $M\times(RI^{m})^{\circ}$ such that
${{\delta^{2r{\bm{\varepsilon}}}}}^{\ast}P({\bm{\varepsilon}})(\sigma)$ is not
supported within $M\times(RI^{m})^{\circ}$. Pick a smooth function
$f\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ that is supported within $(r,R)$
and define on $R_{r}^{\bm{\varepsilon}}$ the real valued function

$f^{\bm{\varepsilon}}(x)=\prod_{j=1}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}f(x_{j}).$

Then the section $\sigma\otimes f^{\bm{\varepsilon}}$ is supported within the
interior of $M\times RI^{m}\times RI^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}$
but, by assumption, the support of
$\bigl{(}P({\bm{\varepsilon}})\boxtimes\mathrm{id}+A\boxtimes
D\bigr{)}(\sigma\otimes f^{\bm{\varepsilon}})$ is not supported within
$M\times RI^{m}\times RI^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}$. Thus
$\tau\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={\Phi_{\bm{\varepsilon}}}^{-1}_{\ast}(\sigma\otimes
f^{\bm{\varepsilon}})$ is a section of $\mathfrak{S}_{g}$ that is supported
within the interior of $M\times RI^{n}$. But the support of $P(\tau)$ does not
lie in $M\times RI^{n}$, contradicting axiom $(2)$ of $P$.

We first give an informal proof for (3). We know that

$\Phi_{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})\,\ast}P|_{U_{r}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))}=P\bigl{(}{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})))\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes
D({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\bigr{)}.$

If we permute the (fixed) coordinates parametrised by
${\mathrm{dom}\,}\,{\bm{\varepsilon}}$ “to the right”, using
$\mathrm{per}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})),{\bm{\varepsilon}})$
we can identify
$\mathord{\mathbb{R}}^{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))}_{r}$ with
$\mathord{\mathbb{R}}^{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))-{\bm{\varepsilon}}}_{r}\times\mathord{\mathbb{R}}^{{\bm{\varepsilon}}}_{r}$.
The uniqueness statement of Lemma 4.2.2 implies (with abuse of notation)

$P({\bm{\varepsilon}})=P\bigl{(}{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})\bigr{)}\otimes\mathrm{id}+\mathrm{per}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}),{\bm{\varepsilon}})_{\ast}D\bigl{(}{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})\bigr{)}.$

Now the formal proof of (3): The inclusion ${\delta^{2r{\bm{\varepsilon}}}}$
maps $U^{m}_{r}({\bm{\eta}})$ into
$U^{n}_{r}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))$. Furthermore, the
following diagram commutes:

$\textstyle{M\times
U^{m}_{r}({\bm{\eta}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{m}({\bm{\eta}})}$$\scriptstyle{{\delta^{2r{\bm{\varepsilon}}}}}$$\textstyle{M\times\mathord{\mathbb{R}}^{m}({\bm{\eta}})\times\mathord{\mathbb{R}}^{{\bm{\eta}}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\times{\delta^{{\bm{\varepsilon}}}}}$$\textstyle{M\times
U^{n}_{r}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))}$$\textstyle{M\times\mathord{\mathbb{R}}^{n}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\times\mathord{\mathbb{R}}_{r}^{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{per}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}),{\bm{\varepsilon}})}$$\textstyle{M\times\mathord{\mathbb{R}}^{n}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\times\mathord{\mathbb{R}}^{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})-{\bm{\varepsilon}}}_{r}\times\mathord{\mathbb{R}}_{r}^{{\bm{\varepsilon}}}}$

This implies

$(\Phi_{\bm{\eta}})_{\ast}({{\delta^{2r{\bm{\varepsilon}}}}}^{\ast}P({\bm{\varepsilon}})))=(\mathrm{id}\times{\delta^{{\bm{\varepsilon}}}})^{\ast}\bigl{(}P({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\otimes\mathrm{id}+A\otimes\mathrm{per}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}),{\bm{\varepsilon}})_{\ast}D({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\bigr{)},$

so ${{\delta^{2r{\bm{\varepsilon}}}}}^{\ast}P({\bm{\varepsilon}})$ is indeed
of block form. ∎

The converse of this lemma is also true.

######  Lemma 4.2.4.

Let $m>0$ and $P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ be of
block form of type $A$. For each differential operator $D$ of order $m$ with
constant coefficients on
$\mathcal{C}_{c}^{\infty}(\mathord{\mathbb{R}},Cl_{1,0})$, the operator

$Q\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=P\boxtimes\mathrm{id}+A\boxtimes
D\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g}\boxtimes
Cl_{1,0})=\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g\oplus\mathrm{d}x_{n+1}^{2}})$

is also block operator of type $A$.

###### Proof.

If $m>0$, then $Q$ is an element of
$\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g\oplus\mathrm{d}x_{n+1}^{2}})$
by Theorem B.0.28.

Axiom (1) is trivially satisfied. Axiom (2) also holds because $A\boxtimes D$
acts as a differential operator in the additional dimension. Axiom (3)
requires some definitions. Let
${\bm{\varepsilon}}\colon\mathbf{n+1}\rightarrow\mathord{\mathbb{Z}}_{2}$ be
given.

If $n+1\in{\mathrm{dom}\,}{\bm{\varepsilon}}$, then
$U_{r}({\bm{\varepsilon}})=U_{r}(p_{n+1}({\bm{\varepsilon}}))\times\\{{\bm{\varepsilon}}(n+1)x_{n+1}>r\\}$
and
$\Phi_{{\bm{\varepsilon}}}=\Phi_{p_{n+1}({\bm{\varepsilon}})}\times\mathrm{id}$.
In this case, we have

$P({\bm{\varepsilon}})=Q(p_{n+1}({\bm{\varepsilon}}))\boxtimes\mathrm{id}_{\mathord{\mathbb{R}}}\text{
and
}D^{Q}({\bm{\varepsilon}})=D^{P}(p_{n+1}({\bm{\varepsilon}}))\boxtimes\mathrm{id}.$

If $n+1\notin{\mathrm{dom}\,}{\bm{\varepsilon}}$, then
$U_{r}({\bm{\varepsilon}})=U_{r}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}$,
where $U_{r}^{n}({\bm{\varepsilon}})\subseteq\mathord{\mathbb{R}}^{n}$, and
$\Phi_{\bm{\varepsilon}}$ is given by the decomposition

$\textstyle{\mathfrak{S}|_{M\times
U_{r}({\bm{\varepsilon}})}=\mathfrak{S}\mathord{\upharpoonleft}_{M\times
U_{r}^{n}({\bm{\varepsilon}})}\boxtimes
Cl_{1,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{\bm{\varepsilon}}\boxtimes\mathrm{id}}$$\textstyle{\mathfrak{S}_{g({\bm{\varepsilon}})}\boxtimes
Cl_{{\bm{\varepsilon}},0}\boxtimes
Cl_{1,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{swap}}$$\scriptstyle{\mathrm{id}\boxtimes}$$\textstyle{\mathfrak{S}_{g({\bm{\varepsilon}})}\boxtimes
Cl_{1,0}\boxtimes Cl_{{\bm{\varepsilon}},0}.}$

In this case, we have

$Q({\bm{\varepsilon}})=P({\bm{\varepsilon}})\boxtimes\mathrm{id}+A\boxtimes
D\quad\text{ and }\quad D^{Q}({\bm{\varepsilon}})=D^{P}({\bm{\varepsilon}}).$

inline]Figure out if $\mathrm{Cycl}(1:1+|{\mathrm{dom}\,}{\bm{\varepsilon}}|$
or $\mathrm{Cycl}(1:1+|{\mathrm{dom}\,}{\bm{\varepsilon}}|^{-1}$. Define the
correct map in an ad-hoc manner as the conceptual notation will be used later.

With these definitions the equation

${\Phi_{\bm{\varepsilon}}}_{\ast}Q|_{M\times
U_{r}({\bm{\varepsilon}})}=Q({\bm{\varepsilon}})\boxtimes\mathrm{id}+A\boxtimes
D^{Q}({\bm{\varepsilon}})$

is easily verified so that $Q$ is a block operator. ∎

Pseudo differential operators are not local by nature. Block operators on the
other hand are local in certain directions on special classes of open sets. In
order to use local-to-global principles for block operators, we need partition
of unities that are adapted to the specific block form of a block operator.

######  Definition 4.2.5.

Let $\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a compactly supported,
smooth, and even function that is identically $1$ near the origin. We denote
the extensions by zero of

$\chi^{-1}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(1-\chi)|_{\mathord{\mathbb{R}}_{\leq
0}},\quad\chi^{0}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\chi,\quad\text{ and
}\quad\chi^{1}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(1-\chi)|_{\mathord{\mathbb{R}}_{\geq 0}}$

with the same symbols. More generally, we define on $\mathord{\mathbb{R}}^{n}$
or $M\times\mathord{\mathbb{R}}^{n}$ the partition of unity

$\biggl{\\{}\chi^{\bm{\alpha}}(m,x)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\prod_{j=1}^{n}\chi^{{\bm{\alpha}}(j)}(x_{j})\biggr{\\}}.$

parameterised ${\alpha\colon\mathbf{n}\rightarrow\mathord{\mathbb{Z}}_{3}}$.

As an application of this partition, we prove that the constants in the
definition of a block operator are actually lower bounds.

######  Lemma 4.2.6.

Let $P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ be a block operator
and let $R>r$ be constants such that the conditions (1)-(3) of $P$ hold. Then
these conditions also hold for all $R^{\prime}>R$ and $r^{\prime}>r$ with
$R^{\prime}>r^{\prime}$,

###### Proof.

Condition (1) is clearly satisfied for all $r^{\prime}>r$.

We continue with the proof of condition (3). Let
$\sigma\in\Gamma_{c}(\mathfrak{S}_{g})$ be a section with
$\mathrm{supp}\,\,\sigma\subseteq U_{r^{\prime}}({\bm{\varepsilon}})$. Since
the support is compact, we find a slightly bigger
$r^{\prime\prime}>r^{\prime}$ such that $\mathrm{supp}\,\,\sigma\subseteq
U_{r^{\prime\prime}}({\bm{\varepsilon}})$. Let
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth function that is
identically $1$ near $\mathord{\mathbb{R}}_{\geq r^{\prime\prime}}$ and
identically zero near $\mathord{\mathbb{R}}_{\leq r^{\prime}}$.

For
${\bm{\varepsilon}}\colon{\mathrm{dom}\,}{\bm{\varepsilon}}\subseteq\mathbf{n}\rightarrow\mathord{\mathbb{Z}}_{2}$
define

$\chi^{\bm{\varepsilon}}(x)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\prod_{j\in{\mathrm{dom}\,}{\bm{\varepsilon}}}\chi({\bm{\varepsilon}}(j)x_{j}).$

This function is supported within $U_{r^{\prime}}({\bm{\varepsilon}})$. If we
do not notationally distinguish $\chi^{\bm{\varepsilon}}$ and
$(\Phi_{\bm{\varepsilon}})_{\ast}\chi^{\bm{\varepsilon}}=\chi^{\bm{\varepsilon}}\circ\Phi_{\bm{\varepsilon}}^{-1}$,
we have

$\displaystyle(\Phi_{\bm{\varepsilon}})_{\ast}(P\chi^{\bm{\varepsilon}}\sigma)$
$\displaystyle=P({\bm{\varepsilon}})\boxtimes\mathrm{id}(\chi^{\bm{\varepsilon}}\sigma)+A\boxtimes
D(\chi^{\bm{\varepsilon}}\sigma)$
$\displaystyle=\chi^{\bm{\varepsilon}}\cdot(P({\bm{\varepsilon}})\boxtimes\mathrm{id})(\sigma)+D(\chi^{\bm{\varepsilon}})\cdot(A\boxtimes\mathrm{id})(\sigma)+\chi^{\bm{\varepsilon}}\cdot(A\boxtimes
D)(\sigma),$

which is supported within $U_{r^{\prime}}({\bm{\varepsilon}})$, too. Thus, $P$
also restricts to $U_{r^{\prime}}({\bm{\varepsilon}})$ and necessarily
decomposes there as in (3).

Condition (2) uses the construction of Definition 4.2.5. We will prove
condition (2) inductively over $n$, the dimension of the Euclidean factor of
$M\times\mathord{\mathbb{R}}^{n}$. For $n=0$ there is nothing to prove, and we
can continuue with $n\geq 1$. Let
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth and even function
that is identically $1$ near $[-r,r]$ and that vanishes near
$\mathord{\mathbb{R}}\setminus(-R,R)$. Let
$\\{\chi^{\bm{\alpha}}\\}_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}}$ be
the corresponding partition of unity on $M\times\mathord{\mathbb{R}}^{n}$ from
Definition 4.2.5. We clearly have

$P(\sigma)=P(\chi^{\mathbf{0}}\sigma)+\sum_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}\setminus\mathbf{0}}P(\chi^{\bm{\alpha}}\sigma).$

The summand $\chi^{\mathbf{0}}\sigma$ is supported within
$M\times(RI^{n})^{\circ}$, so, by condition (2) of $P$, also
$P(\chi^{\mathbf{0}}\sigma)$ is supported within $M\times(RI^{n})^{\circ}$.

Recall the notation
$\hat{{\bm{\alpha}}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}}\colon\mathrm{supp}\,\,{\bm{\alpha}}\rightarrow\mathord{\mathbb{Z}}_{2}$.
The section $\chi^{\bm{\alpha}}\sigma$ is supported within
$U_{r}(\hat{{\bm{\alpha}}})$. If ${\bm{\alpha}}\neq\mathbf{0}$, then
$\hat{{\bm{\alpha}}}$ is not the empty map so that
$P(\chi^{\bm{\alpha}}\sigma)$ is also supported within
$U_{r}(\hat{{\bm{\alpha}}})$ by the proof of condition (3) above.

Since $\mathrm{supp}\,\sigma\subseteq M\times(R^{\prime}I^{n})^{\circ}$ we can
find a smooth and even function
$\psi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ that vanishes near
$\mathord{\mathbb{R}}\setminus(-R^{\prime},R^{\prime})$ and such that
$\psi^{\mathbf{0}}(x)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\prod_{j=1}^{n}\psi(x_{j})$ is identically $1$ on
$\mathrm{supp}\,\,\sigma$. The block form of $P$ implies

$\displaystyle P(\chi^{\mathbf{{\bm{\alpha}}}}\sigma)$
$\displaystyle=P(\chi^{\mathbf{{\bm{\alpha}}}}\psi^{\mathbf{0}}\sigma)$
$\displaystyle\begin{split}&=\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\cdot
P\left(\Bigl{(}\prod_{j\in\mathrm{Null}({\bm{\alpha}})}\psi\circ\mathord{\mathrm{pr}}_{j}\Bigr{)}\cdot\chi^{\bm{\alpha}}\sigma\right)\\\
&\qquad\quad+D(\hat{{\bm{\alpha}}})\left(\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\right)\cdot
P\left(\Bigl{(}\prod_{j\in\mathrm{Null}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\Bigr{)}\cdot\chi^{\bm{\alpha}}\sigma\right).\end{split}$

By Corollary 4.2.3, $P({\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}})$ is a
block operator on
$\mathfrak{S}_{g({\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}})}$ that satisfy
conditions (1) - (3) for the same constants as $P$ and $A\boxtimes
D({\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}})$ is a differential operator
and hence only support-decreasing. This and the induction assumption, implies
that the section

$P\left(\Bigl{(}\prod_{j\in\mathrm{Null}({\bm{\alpha}})}\psi\circ\mathord{\mathrm{pr}}_{j}\Bigr{)}\cdot\chi^{\bm{\alpha}}\sigma\right)$

must be supported within
$\\{|x_{j}|<R^{\prime}\,:\,j\in\mathrm{Null}\,{\bm{\alpha}}\\}\cap
U_{r}({\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}})$. It follows that

$\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\cdot
P\left(\prod_{j\in\mathrm{Null}({\bm{\alpha}})}\psi\circ\mathord{\mathrm{pr}}_{j}\chi^{\bm{\alpha}}\sigma\right)$

and

$D(\hat{{\bm{\alpha}}})\left(\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\right)\cdot
P\left(\prod_{j\in\mathrm{Null}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\chi^{\bm{\alpha}}\sigma\right)$

are supported within $M\times(R^{\prime}I^{n})^{\circ}\cap
U_{r}({\bm{\alpha}}|_{\mathrm{supp}\,\,{\bm{\alpha}}})$, which finishes the
proof of (2). ∎

######  Corollary 4.2.7.

Let $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ be a block operator
of type $A$ whose core is contained in $M\times RI^{n}$. For
${\bm{\alpha}}\colon\mathbf{n}\rightarrow\mathord{\mathbb{Z}}_{3}$ and
$\rho^{\prime}>\rho>R$ define

$V_{\rho<\rho^{\prime}}({\bm{\alpha}})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=M\times\\{x\in\mathord{\mathbb{R}}^{n}\,:\,x_{j}\in(-\rho^{\prime},\rho^{\prime})\text{
if }j\in\mathrm{Null}({\bm{\alpha}}),\alpha_{j}x_{j}>\rho\text{
otherwise}\\}.$

If $\sigma\in\Gamma_{c}(\mathfrak{S}_{g})$ is supported within
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})$, then $P\sigma$ is also supported
there.

###### Proof.

If ${\bm{\alpha}}=\mathbf{0}$, then
$V_{\rho<\rho^{\prime}}(0)=M\times\rho^{\prime}I^{n}$ and the claim follows
from Lemma 4.2.6.

Assume now that ${\bm{\alpha}}\neq\mathbf{0}$, so that
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})\subseteq
U_{\rho}(\hat{{\bm{\alpha}}})$. For each $u\in\Gamma_{c}(\mathfrak{S}_{g})$
supported with in $V_{\rho<\rho^{\prime}}({\bm{\alpha}})$ we deduce with abuse
of notation

$\displaystyle
Pu=(P(\hat{{\bm{\alpha}}})\boxtimes\mathrm{id})(\chi^{\bm{\alpha}}u)+\mathrm{e/o}\boxtimes
D(\hat{{\bm{\alpha}}})(\chi^{\bm{\alpha}})(\chi^{\bm{\alpha}}u)$

The second summand is a differential operator and hence support-decreasing.

We know from Corollary 4.2.3 that $P(\hat{{\bm{\alpha}}})$ is again a block
operator whose core is contained in
$M\times\rho^{\prime}I^{|\mathrm{Null}({\bm{\alpha}})|}$. Since
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})\cong\rho^{\prime}I^{|\mathrm{Null}({\bm{\alpha}})|}\times\mathord{\mathbb{R}}_{\rho}^{\hat{{\bm{\alpha}}}}$,
it follows from the previous Lemma that

$\mathrm{supp}\,\,P({\bm{\alpha}})(u(\text{--},y))\subseteq\rho^{\prime}I^{|\mathrm{Null}({\bm{\alpha}})|}$

and hence

$\mathrm{supp}\,\,P({\bm{\alpha}})(u)\subseteq\rho^{\prime}I^{|\mathrm{Null}({\bm{\alpha}})|}\times\mathord{\mathbb{R}}_{\rho}^{\hat{{\bm{\alpha}}}}=V_{\rho<\rho^{\prime}}({\bm{\alpha}}).$

∎

The technique used in the previous proofs can be used to prove that block
operators are local in the following limited sense.

######  Corollary 4.2.8.

Let $P$ be a block operator whose core is contained in $M\times\rho I^{n}$. If
$\rho^{\prime}>\rho$ and $V\subseteq U_{\rho^{\prime}}({\bm{\varepsilon}})$ is
an open set that corresponds under $U_{\rho^{\prime}}({\bm{\varepsilon}})\cong
M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}^{{\bm{\varepsilon}}}_{\rho^{\prime}}$
to $M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times
V^{\bm{\varepsilon}}$, then $(Pu)|_{V}$ only depends on (the germ of)
$u|_{\bar{V}}$.

###### Proof.

It suffices to show that if $u$ vanishes identically near $\bar{V}$, then $Pu$
vanishes identically on $V$. Let $\phi\colon
M\times\mathord{\mathbb{R}}^{n}\rightarrow\mathord{\mathbb{R}}_{\geq 0}$ be a
smooth function such that $\phi^{-1}(\mathord{\mathbb{R}}_{>0})=V$ and such
that $\phi$ only depends on variables indexed by
${\mathrm{dom}\,}({\bm{\varepsilon}})$. Let
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth even function
such that $\chi$ is identically $1$ on $[-\rho,\rho]$ and whose support lies
in $(-\rho^{\prime},\rho^{\prime})$. Denote by $\chi^{\bm{\alpha}}$ the
partition of unity from Definition 4.2.5.

Since $\mathrm{supp}\,\phi\subseteq U_{\rho^{\prime}}({\bm{\varepsilon}})$, we
conclude that $\mathrm{supp}\,\,\phi$ and
$V_{\rho<\rho^{\prime}}(\hat{{\bm{\alpha}}})$ have non-empty intersection only
if $\hat{{\bm{\alpha}}}\geq{\bm{\varepsilon}}$. We know from Corollary 4.2.7
that $\mathrm{supp}\,(P\chi^{\bm{\alpha}}u)\subseteq
V_{\rho<\rho^{\prime}}({\bm{\alpha}})$, so we conclude $\phi\cdot
P\chi^{\bm{\alpha}}u=0$ if $\hat{{\bm{\alpha}}}\geq{\bm{\varepsilon}}$ is
false. On the other hand, if $\hat{{\bm{\alpha}}}\geq{\bm{\varepsilon}}$, then
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})\subseteq U_{\rho}({\bm{\varepsilon}})$
and the block decomposition of $P$ implies

$\displaystyle\phi\cdot P\chi^{\bm{\alpha}}u$
$\displaystyle=P\phi\chi^{\bm{\alpha}}u+[\phi,P]\chi^{\bm{\alpha}}u=P\chi^{\bm{\alpha}}\underbrace{\phi
u}_{=0}-i^{-1}\underbrace{\mathrm{symb}_{1}(P)(\text{--},\mathrm{d}\phi)\chi^{\bm{\alpha}}u}_{=0\text{
on }V}$ $\displaystyle=0\qquad\text{ on }V$

because $u|_{V}=0$. Since
$\\{\chi^{\bm{\alpha}}\\}_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}}$ is a
partition of unity, the previous calculations imply $\phi Pu|_{V}=0$ from
which we deduce $Pu|_{V}=0$. ∎

######  Definition 4.2.9.

A block operator $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ is
called _block operator of Dirac-type_ , if $A=\mathrm{e/o}$ and
$D({\bm{\varepsilon}})={\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{\bm{\varepsilon}}}$
is the Dirac operator of the Euclidean metric on
$\mathord{\mathbb{R}}^{\bm{\varepsilon}}$ for all
${\bm{\varepsilon}}\colon{\mathrm{dom}\,}{\bm{\varepsilon}}\rightarrow\mathord{\mathbb{Z}}_{2}$,
and if the principal symbol satisfies
$\mathrm{symb}_{1}(P)=\mathrm{symb}_{1}({\not{\mathfrak{D}}}_{g})$.

The motivating examples for this definition are the Dirac operator of a block
metric and the adjoint operator of a block map of pseudo Dirac operators.

######  Example 4.2.10.

We apply the discussion of Example 4.1.18 to show that
${\not{\mathfrak{D}}}_{g}$ is a block operator (of Dirac type), if $g$ is a
block metric on $M\times\mathord{\mathbb{R}}^{n}$. Assume that the block
metric decomposes outside of $M\times rI^{n}$. Then, since differential
operators are only support decreasing, we can choose any $R>r$ so that
condition (1) and (2) are trivially satisfied. It remains to show that
${\not{\mathfrak{D}}}_{g}$ decomposes appropriately on
$U_{r}({\bm{\varepsilon}})$. Since the defining sequence of the Dirac operator
(4.1.10) is natural with respect to Pin-structure preserving, isometric
diffeomorphisms, we have

$\displaystyle(\Phi_{{\bm{\varepsilon}}})_{\ast}{\not{\mathfrak{D}}}_{g}|_{M\times
U_{r}({\bm{\varepsilon}})}$
$\displaystyle={\not{\mathfrak{D}}}_{c({\bm{\varepsilon}})_{\ast}g}={\not{\mathfrak{D}}}_{g({\bm{\varepsilon}})+\langle\cdot,\cdot\rangle_{\mathord{\mathbb{R}}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}}}={\not{\mathfrak{D}}}_{g({\bm{\varepsilon}})}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}}.$

The decomposition of the spinor bundle in Example 4.1.18 has a weaker version.
If $U\subseteq\mathord{\mathbb{R}}^{n}$ is open and $g\colon
U\rightarrow\mathrm{Riem}(M)$ is a smooth map, then $TU$ is still an
orthogonal complement of $TM$ in $T(M\times U)$ with respect to
$\mathrm{susp}(g)$. This induces a splitting of the spinor bundle
$\mathfrak{S}_{\mathrm{susp}(g)}\cong\mathfrak{S}_{g}\otimes Cl_{n,0}$.

######  Definition 4.2.11.

Let $P\colon U\rightarrow\Psi\mathrm{DO}^{m}(\mathfrak{S})$ be a smooth map,
where the target carries the (subspace topology of the) amplitude topology. It
comes with a smooth map $g\colon U\rightarrow\mathrm{Riem}(M)$. Define on
$\mathfrak{S}_{\mathrm{susp}(g)}$ the operators that are given under the
isomorphism $\mathfrak{S}_{\mathrm{susp}(g)}\cong\mathfrak{S}_{g}\otimes
Cl_{n,0}$ by

$\displaystyle P^{\mathrm{ex}}(\sigma\otimes
e_{I})(m,t)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\left(P_{t}\sigma(\cdot,t)\right)\otimes e_{I},$
$\displaystyle\mathrm{susp}(P)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=P^{\mathrm{ex}}+\sum_{k=1}^{n}\partial_{t_{k}}\cdot\nabla_{\partial_{t_{k}}}^{\mathfrak{S}_{\mathrm{susp}\,g}}.$

We call $\mathrm{susp}(P)$ the _operator suspension of_ $P$.

By the parametrised version of the Atiyah-Singer exterior tensor theorem
B.0.29 we have
$\mathrm{susp}(P)\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{\mathrm{susp}(g)})$
if $m>0$.

We are mostly interested in the case where $U=\mathord{\mathbb{R}}^{n}$ and
$P$ is a smooth block map. Note that this implies that the underlying map of
Riemannian metrics is also a smooth block map. However, in some proofs, we
need the operator suspension on more general open subsets.

######  Lemma 4.2.12.

Let $m>0$ and let
$P\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{DO}^{m}(\mathfrak{S})$
be a smooth block map, where the target carries the (subspace topology of the)
amplitude topology and underlying block map $g$.

Then $P^{\mathrm{ex}}$ and $\mathrm{susp}(P)$ are a block operators of type
$\mathrm{id}$ and $\mathrm{e/o}$, respectively. If $P$ takes values in
$\Psi\mathrm{Dir}(M)$, then $\mathrm{susp}(P)$ is a block operator of Dirac
type.

###### Proof.

We will only prove the statement for $\mathrm{susp}(P)$ in the case that $P$
takes values in $\Psi\mathrm{Dir}(M)$, for the other cases are similar. Since
$P$ and $g$ are block maps, there is an $r>0$ such that $P$ and $g$ are
independent of $x_{i}$ if $|x_{i}|>r$. Then $\mathrm{susp}(g)$ decomposes
outside of $M\times rI^{n}$. Since
$\mathrm{susp}(P)-\mathrm{e/o}\otimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{n}}$
is $\mathcal{C}^{\infty}(\mathord{\mathbb{R}}^{n})$-linear, the operator
restricts to open subsets of the form $M\times U$ with
$U\subseteq\mathord{\mathbb{R}}^{n}$ open, and thus $P$ does it, too.

It remains to show that $P$ decomposes appropriately on
$U_{r}({\bm{\varepsilon}})$. On this subset, a permutation of coordinates
$c({\bm{\varepsilon}})\colon M\times U_{r}({\bm{\varepsilon}})\rightarrow
M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}^{\bm{\varepsilon}}_{r}$
is an isometry if the domain carries the metric
$g|_{U_{r}({\bm{\varepsilon}})}$ and the right hand side has
$\mathrm{susp}(g)({\bm{\varepsilon}})\oplus\langle\cdot,\cdot\rangle$. Thus
the spinor bundle decomposes into

$\mathfrak{S}_{\mathrm{susp}(g)({\bm{\varepsilon}})}\boxtimes
Cl_{{\bm{\varepsilon}},0}\cong\mathfrak{S}_{g}\otimes
Cl_{n-|{\bm{\varepsilon}}|,0}\boxtimes Cl_{{\bm{\varepsilon}},0}$

over
$(M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}}))\times\mathord{\mathbb{R}}^{\bm{\varepsilon}}_{r}$
via $\Phi_{\bm{\varepsilon}}=\mathfrak{S}(c({\bm{\varepsilon}}))$. Since
$P|_{U_{r}({\bm{\varepsilon}})}$ is independent of the coordinates indexed by
${\mathrm{dom}\,}{\bm{\varepsilon}}$, we derive the equation

$\displaystyle(\Phi_{\bm{\varepsilon}})_{\ast}(\mathrm{susp}(P)|_{M\times
U_{r}({\bm{\varepsilon}})})$
$\displaystyle=(\Phi_{\bm{\varepsilon}})_{\ast}\bigl{(}P\otimes\mathrm{id}\bigr{)}+(\Phi_{\bm{\varepsilon}})_{\ast}\bigl{(}\mathrm{e/o}_{\mathfrak{S}_{g}}\otimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{n}}\bigr{)}$
$\displaystyle\begin{split}&=P_{c({\bm{\varepsilon}})^{-1}(\text{--})}\otimes\mathrm{id}+(\mathrm{e/o}_{\mathfrak{S}_{g}}\otimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})})\boxtimes\mathrm{id}_{Cl_{{\bm{\varepsilon}},0}}\\\
&\qquad+\mathrm{e/o}_{\mathfrak{S}_{g}}\otimes\mathrm{e/o}_{Cl_{n-|{\bm{\varepsilon}}|,0}}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{\bm{\varepsilon}}_{r}}\end{split}$
$\displaystyle\begin{split}&=\Bigl{(}(P|_{M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})})_{c({\bm{\varepsilon}})^{-1}(\text{--})}\otimes\mathrm{id}_{Cl_{n-|{\bm{\varepsilon}}|,0}}+\mathrm{e/o}_{\mathfrak{S}_{g}}\otimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})}\Bigr{)}\boxtimes\mathrm{id}\\\
&\qquad+\mathrm{e/o}_{\mathfrak{S}_{\mathrm{susp}(g)({\bm{\varepsilon}})}}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{\bm{\varepsilon}}},\end{split}$

which shows that $\mathrm{susp}(P)$ decomposes as required. ∎

So far, we have only studied block operators on the level of vector bundles
over manifolds. Before we are going to construct the block version of
$\Psi\mathrm{Dir}^{\times}(M)$, we will study its (functional) analytic
properties in the next section.

###  4.3 Analytical Properties of Block Operators

The purpose of this section is to derive the needed analytical properties of
block operators. We will show that each block operator realises to a bounded
operator on positive Sobolev spaces and that the realisations of block
operators of Dirac type satisfy analogous versions of the fundamental elliptic
estimate. Moreover, we will show that if they are self-adjoint and invertible
near infinity, then they are also Fredholm operators. Many of these properties
rely on the structure of a block metric near infinity because we are working
on non-compact manifolds.

In the following, we only consider pseudo differential operators of order
$m\in\\{0,1\\}$.

######  Lemma 4.3.1.

Let $g$ be a block metric on $M\times\mathord{\mathbb{R}}^{n}$ and let
$P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ be a block operator.
Then $P$ extends to a bounded linear operator

$P\colon H^{s}(\mathfrak{S}_{g})\rightarrow H^{s-m}(\mathfrak{S}_{g})$

for all $s\in\mathord{\mathbb{Z}}$.

###### Proof.

We prove this lemma by induction over $n$. For $n=0$, this is classical
because $M$ is closed, see Theorem B.0.18. For $n>0$, we pick appropriate
$R>r>0$ such that $P$ restricts to $M\times(RI^{n})^{\circ}$ and decomposes on
$\\{\varepsilon x_{i}>r\\}$ for all $i\in\\{1,\dots,n\\}$ and all
$\varepsilon\in\mathord{\mathbb{Z}}_{2}$. Let
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth, symmetric
function that is identically $1$ on $(-r,r)$ and vanishes near
$\mathord{\mathbb{R}}\setminus(-R,R)$. Let
$\\{\chi^{\bm{\alpha}}\\}_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}}$ be
the partition of unity obtained from Definition 4.2.5. By the triangle
inequality, it suffices to show that $P\cdot\chi^{\bm{\alpha}}$ extends to a
bounded operator.

If ${\bm{\alpha}}=\mathbf{0}$, then $P\chi^{\mathbf{0}}$ restricts to an
operator on the relatively compact set $M\times(RI^{n})^{\circ}$. Thus, by
Theorem B.0.18, $P\chi^{\mathbf{0}}$ extends to a bounded operator
$H^{s}(\mathfrak{S}_{g})\rightarrow H^{s-1}(\mathfrak{S}_{g})$.

If ${\bm{\alpha}}\neq\mathbf{0}$, then there is a tupel
$(i,\varepsilon)\in\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$ such that
$\mathrm{supp}\,\chi^{\bm{\alpha}}$ is contained in $U_{r}((i,\varepsilon))$.
The induction hypothesis and [palais1965seminar]*Corollary XIV.1 applied to
the discrete Hilbert chains
$H^{s}(M\times\mathord{\mathbb{R}}^{n-1};\mathfrak{S}_{g(i,\varepsilon)})$ and
$H^{s}(\mathord{\mathbb{R}};Cl_{1,0})$ imply

$P(i,\varepsilon)\boxtimes\mathrm{id}+A\boxtimes D(i,\varepsilon)\in
Op^{1}(H^{s}(\mathfrak{S}_{g(i,\varepsilon)}\boxtimes Cl_{1,0})),$

which means that this operator induces a bounded linear map

$H^{s}(\mathfrak{S}_{g(i,\varepsilon)}\boxtimes Cl_{1,0})\rightarrow
H^{s-1}(\mathfrak{S}_{g(i,\varepsilon)}\boxtimes Cl_{1,0})$

for all $s\in\mathord{\mathbb{Z}}$. Since $P$ decomposes into
$P(i,\varepsilon)\boxtimes\mathrm{id}+A\boxtimes D(i,\varepsilon)$ on
$\mathfrak{S}_{g}|_{U_{r}((i,\varepsilon))}\cong\mathfrak{S}_{g(i,\varepsilon)}\boxtimes
Cl_{1,0}|_{\varepsilon\mathord{\mathbb{R}}_{>r}}$, it follows
$P\chi^{\bm{\alpha}}\in Op^{1}(\mathfrak{S}_{g})$. ∎

Recall that a sequence of operators $P_{n}$ converges to $P$ in the Atiyah-
Singer topology if the $P_{i}$ converge uniformly on compact subsets to $P$.
Thus, it is not clear that the fundamental ellitpic estimate carries over to
elements in the Atiyah-Singer closure. However, this is the case, if the
operator $P$ increases the support in a controlled manner.

######  Lemma 4.3.2.

Let $M$ be a not necessarily closed manifold and $E,F\rightarrow M$ be vector
bundles over $M$. Let $P\in\overline{\Psi\mathrm{DO}}^{1}(E,F)$ be an elliptic
pseudo differential operator whose principal symbol is the principal symbol of
an actual pseudo differential operator (i.e. smooth) and that has the
following property: For all compact neighbourhoods $K\subseteq M$ there is a
relatively compact open subset $W$ such that if $\mathrm{supp}\,u\subseteq K$
then $\mathrm{supp}\,Pu\subseteq W$.

Then, for all compact $K\subseteq M$, there is a compact $L\subseteq M$ such
that $P$ induces a bounded map $H_{K}^{s}(E)\rightarrow H_{L}^{s-1}(F)$, where
$H_{K}^{s}(E)$ denotes the subspace of all section with support in $K$. The
operator $P$ furthermore satisfies the elliptic estimate:

$||u||_{s}\leq C\left(||u||_{s-1}+||Pu||_{s-1}\right).$

###### Proof.

The subspace $\Gamma_{c}(E)$ is dense in $H^{s}(E)$ for all
$s\in\mathord{\mathbb{Z}}$ and the multiplication with a smooth compactly
supported function induces a continuous endomorphism on all Sobolev spaces.
Thus, if $\chi$ is a smooth compactly supported function that is identically
$1$ on $K$, then we can approximate each element of $H_{K}^{s}(E)$ by an
element of $\chi\cdot\Gamma_{c}(E)$. If $L$ is the compact subset from the
assumption assigned to $\mathrm{supp}\,\,\chi$, then a simple approximation
argument shows that $P\colon H_{K}^{s}(E)\rightarrow H^{s-1}_{L}(F)$ is well
defined and continuous, because $H^{s-1}_{L}(F)$ is a closed subspace.

To prove the elliptic estimate we argue as follows. Pick smooth functions
$\rho_{0}$ and $\rho_{1}$ such that $\rho_{0}$ is identically $1$ on $K$ and
supported within $L^{\circ}$ and $\rho_{1}$ is identically $1$ on
$\mathrm{supp}\,\rho_{0}$ and supported within $L^{\circ}$. Pick further a
pseudo-inverse $R$ for $P$, i.e., an actual pseudo differential operator of
order $-1$ whose principal symbol is the inverse of $\mathrm{symb}_{1}(P)$.
Then $\rho_{0}-\rho_{0}RP\rho_{1}\in\overline{\Psi\mathrm{DO}}^{0}_{L}(E)$ and
its principal symbol vanishes. Since $\Psi\mathrm{DO}^{-1}(E)$ lies dense in
$\ker\mathrm{symb}_{0}$, we can approximate it by actual pseudo differential
operators of order $-1$, see Lemma B.0.24, so there is a
$Q\in\Psi\mathrm{DO}^{-1}_{L}(E)$ such that
$||\rho_{0}-\rho_{0}RP\rho_{1}-Q||_{s,s}<1/2$.

With that knowledge we can derive the required elliptic estimate from the
elliptic estimate of actual pseudo differential operators. For any $u\in
H_{K}^{s}(E)$ we have $\rho_{j}u=u$ and thus

$\displaystyle||u||_{s}$
$\displaystyle=||(\rho_{0}-\rho_{0}RP\rho_{1})u+\rho_{0}RP\rho_{1}u||_{s}$
$\displaystyle\leq||(\rho_{0}-\rho_{0}RP\rho_{1})u||_{s}+||\rho_{0}RPu||_{s}$
$\displaystyle\leq||(\rho-\rho_{0}RP\rho_{1})u-Qu||_{s}+||Qu||_{s}+||\rho_{0}R||_{s-1,s}||Pu||_{s-1}$
$\displaystyle\leq
1/2||u||_{s}+||Q||_{s-1,s}||u||_{s-1}+||\rho_{0}R||_{s-1,s}||Pu||_{s-1},$

where we used the fact that $\rho_{0}R$ is an actual pseudo differential
operators of order $-1$ that maps sections to sections that are supported
within $\mathrm{supp}\,\rho_{0}$, to deduce that $\rho_{0}R$ induces bounded
maps between Sobolev spaces. The chain of inequalities can be rewritten as

$\displaystyle||u||_{s}$ $\displaystyle\leq
2\left(||Q||_{s-1,s}||u||_{s-1}+||\rho_{0}R||_{s-1,s}||Pu||_{s-1}\right)$
$\displaystyle\leq
2\max\\{||Q||_{s-1,s},||\rho_{0}R||_{s-1,s}\\}\left(||u||_{s-1}+||Pu||_{s-1}\right),$

which gives the second claim. ∎

Since the constants in the fundamental elliptic estimate may depend on the
compact subsets, the result does not immediately carry over to block
operators. To carry the result over, we use the following partition of unity
which is due to Bunke [bunke2009index], who introduced it in special local
coordinates for manifolds with corners.

######  Lemma 4.3.3.

Let $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ be a block operator
of Dirac type whose core is contained in $M\times(-\rho,\rho)^{n}$ and that
decomposes on all $U_{\rho-1/2}({\bm{\varepsilon}})$. For
${\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}$, set

$V_{\rho}({\bm{\alpha}})=\bigl{\\{}(m,x)\in
M\times\mathord{\mathbb{R}}^{n}\,:\,\alpha_{j}x_{j}>\rho-1/2\text{ if
}\alpha_{j}\neq 0,\,x_{j}\in(-\rho,\rho)\text{ otherwise}\bigr{\\}}.$

There is a continuous partition of unity $(\phi_{0},\dots,\phi_{2n})$ that
satisfies the following three properties:

  1. (i)

Each $\phi_{l}$ is almost everywhere differentiable, has bounded derivative
that is zero at infinity, which means
$\lim_{s\to\infty}\mathrm{sup}\\{|\mathrm{d}\phi_{l}|(x)\,|\,x\in
M\times\mathord{\mathbb{R}}^{n}\setminus(-s,s)^{n}\\}=0$.

  2. (ii)

If $u$ is supported within $V_{\rho}({\bm{\alpha}})$ with ${\bm{\alpha}}\neq
0$, then $[P,\phi_{l}]u=\mathbf{c}(\mathrm{grad}(\phi_{l}))(u)$.

  3. (iii)

$\phi_{0}$ is compactly supported and for each $l>0$ there is a unique tupel
$(i,\varepsilon)\in\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$ such that
$\phi_{l}$ is supported within $\\{\varepsilon x_{i}>\rho\\}$.

###### Proof.

Fix $0<a<(2\sqrt{n})^{-1}$. Let $\chi\colon\mathord{\mathbb{R}}_{\geq
0}\rightarrow[0,1]$ be a smooth function whose zero set is
$\mathord{\mathbb{R}}_{\leq a}$ and that is identically $1$ near
$\mathord{\mathbb{R}}_{\geq 2a}$. We denote its mirror symmetric extension to
$\mathord{\mathbb{R}}$ again with $\chi$. For $i\in\\{1,\dots,n\\}$, we define
$\widetilde{\chi}_{i}\colon S^{n-1}\rightarrow[0,1]$ via
$\widetilde{\chi}_{i}(x)=\chi(\mathord{\mathrm{pr}}_{i}(x))$ and set

$\chi_{i}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\frac{\widetilde{\chi_{i}}}{\sum_{j=1}^{n}\widetilde{\chi}_{j}}.$

The denominator is always non-zero because it vanishes at $x$ if and only if
all summands vanish at $x$, which implies $||x||^{2}<n\cdot a<1$. The
$n$-tuple of functions $(\chi_{1},\ldots,\chi_{n})$ has the following three
evident properties:

  1. 1.

$\chi_{i}(x)=0$ if $|x_{i}|<a$,

  2. 2.

$\sigma^{\ast}\chi_{i}=\chi_{\sigma^{-1}(i)}$ for each permutation
$\sigma\in\Sigma_{n}$,

  3. 3.

$\sum_{j}\chi_{j}=1$.

For each
${\bm{\alpha}}=(\alpha_{1},\ldots,\alpha_{n})\in\mathord{\mathbb{Z}}_{3}^{n}$,
we define the set

$\displaystyle A_{\bm{\alpha}}$
$\displaystyle\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{(m,x)\in
M\times\mathord{\mathbb{R}}^{n}\,:\,x_{j}\in[-\rho,\rho]\text{ if
}\alpha_{j}=0,\ \alpha_{j}x_{j}\geq\rho\\}$ $\displaystyle\cong
M\times[-\rho,\rho]^{|\mathrm{Null}({\bm{\alpha}})|}\times\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\alpha_{j}\mathord{\mathbb{R}}_{\geq\rho}.$

The complement of $M\times(-\rho,\rho)^{n}$ is the union of all
$A_{\bm{\alpha}}$ with ${\bm{\alpha}}\neq 0$. For
$(i,\varepsilon)\in\\{1,\ldots,n\\}\times\mathord{\mathbb{Z}}_{2}$, we define

$\hat{\phi}^{\varepsilon}_{i}|_{A_{\bm{\alpha}}}(m,x)=\begin{cases}\chi_{i}\left(\frac{\hat{x}}{||\hat{x}||}\right),&\text{if
}\alpha_{i}\neq 0,\\\ 0,&\text{if }\alpha_{i}=0,\end{cases}$

where
$\hat{x}=(|\alpha_{1}|x_{1}-\alpha_{1}\rho,\dots,|\alpha_{n}|x_{n}-\alpha_{n}\rho)$.
The assignments agree on the overlaps $A_{\bm{\alpha}}\cap A_{\bm{\beta}}$
yielding well defined functions on the complement of
$M\times(-\rho,\rho)^{n}$. Each $\hat{\phi}^{\varepsilon}_{i}$ is continuous
because it is continuous on all $A_{\bm{\alpha}}$. For $0<\delta<1$, we pick a
smooth function $\xi\colon\mathord{\mathbb{R}}_{\geq 0}\rightarrow[0,1]$ whose
zero set is $[0,\rho+\delta]$ and which is identically $1$ near
$\mathord{\mathbb{R}}_{\geq\rho+1}$. Its mirror-symmetric extension to
$\mathord{\mathbb{R}}$ is again denoted by $\xi$. Set
$\xi_{i}(x):=\xi(\mathord{\mathrm{pr}}_{i}(x))$ and extend
$\hat{\phi_{i}^{\varepsilon}}$ to $M\times\mathord{\mathbb{R}}^{n}$ via
$\phi_{i}^{\varepsilon}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\xi_{i}\hat{\phi}_{i}^{\varepsilon}$.

By construction, all $\phi_{i}^{\varepsilon}$ sum up to $1$ on the complement
of $M\times(-\rho+1,\rho+1)^{n}$ and the sum is bounded from above by 1
everywhere. Each $\phi^{\varepsilon}_{i}$ is supported within $\\{\varepsilon
x_{i}\geq\rho+\delta\\}$. Thus, by setting
$\phi_{0}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=1-\sum_{(i,\varepsilon)}\phi_{i}^{\varepsilon}$, we have constructed a
continuous partition of unity.

The function $\phi_{0}$ is supported within $M\times(-\rho-1,\rho+1)^{n}$,
which proves property (iii).

The points at which the partition of unity might not be differentiable lie in
$\bigcup_{{\bm{\alpha}}\neq{\bm{\beta}}}A_{\bm{\alpha}}\cap A_{{\bm{\beta}}}$,
which is a zero-set. Outside this set, the differential is given by

$\mathrm{d}\phi_{i}^{\varepsilon}=\hat{\phi}_{i}^{\varepsilon}(x)\xi^{\prime}(x_{i})\mathrm{d}x_{i}+\xi(x_{i})\mathrm{d}\chi_{i}\cdot\left(\frac{\hat{x}\cdot\hat{x}^{T}-||\hat{x}||^{2}\mathrm{id}}{||\hat{x}||^{3}}\right)\mathrm{diag}(|\alpha_{1}|,\dots,|\alpha_{n}|).$

Since $\mathrm{d}\chi_{i}$ is bounded, we conclude

$|\mathrm{d}\phi_{i}^{\varepsilon}|\in
L^{\infty}(M\times\mathord{\mathbb{R}}^{n})\text{ and
}\lim_{s\to\infty}\mathrm{sup}\\{|\mathrm{d}\phi_{i}^{\varepsilon}|(x)\,|\,x\in
M\times\mathord{\mathbb{R}}^{n}\setminus(-s,s)^{n}\\}=0,$

showing property (i).

By construction, the restriction of each $\phi^{\varepsilon}_{i}$ to
$V_{\rho}({\bm{\alpha}})$ with ${\bm{\alpha}}\neq 0$ depends only on the
coordinates $x_{j}$ indexed by $j\in\mathrm{supp}\,{\bm{\alpha}}$. If
$u\in\Gamma_{c}(\mathfrak{S}_{g})$ is supported within
$V_{\rho}({\bm{\alpha}})$ with ${\bm{\alpha}}\neq\mathbf{0}$, then the block
form of $P$ implies

$\displaystyle[P,\phi_{i}^{\varepsilon}]u$
$\displaystyle=\left(P({\bm{\alpha}})+\sum_{j\in\mathrm{supp}\,{\bm{\alpha}}}e_{j}\cdot\partial_{j}\right)\phi_{i}^{\varepsilon}u-\phi_{i}^{\varepsilon}\cdot\left(P({\bm{\alpha}})+\sum_{j\in\mathrm{supp}\,{\bm{\alpha}}}e_{j}\cdot\partial_{j}\right)u$
$\displaystyle=\phi^{\varepsilon}_{i}(P({\bm{\alpha}})-P({\bm{\alpha}}))u+\sum_{j\in\mathrm{supp}\,{\bm{\alpha}}}[e_{j}\cdot\partial_{x_{j}},\phi_{i}^{\varepsilon}]u$
$\displaystyle=\mathrm{grad}(\phi_{i}^{\varepsilon})\cdot
u=\mathbf{c}(\mathrm{grad}(\phi_{i}^{\varepsilon}))(u).$

This shows property (ii). ∎

We use this partition of unity to derive the fundamental elliptic estimates
for block operators of Dirac type. The idea of the following two propositions
is borrowed from [ebert2017indexdiff]*Prop 3.3.

######  Proposition 4.3.4.

Let $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ be an elliptic
block operator of Dirac type. Then $P$ satisfies the fundamental elliptic
inequality, which means that there is a $C>0$ such that

$||u||_{1}\leq C\left(||u||_{0}+||Pu||_{0}\right)$

for all $u\in H^{1}(\mathfrak{S}_{g})$ .

###### Proof.

The proof will be carried out by induction over $n$, the dimension of the
Euclidean part of our block manifolds $M\times\mathord{\mathbb{R}}^{n}$. For
$n=0$ our base manifold is compact and the theorem follows from Lemma 4.3.2.

Assume the statement for $n-1$ and pick an elliptic block operator
$P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ over a block metric
$g\in\widetilde{\mathcal{R}_{n}}(M)$. Pick $R>r>0$ such that $g$ decomposes
outside $M\times(-r,r)^{n}$ and $M\times(-R,R)^{n}$ is a core for $P$. Since
$P$ is of block form the classical elliptic estimate gives us

$||u||_{1}\leq C_{0,R}\left(||u||_{0}+||Pu||_{0}\right)$

for some $C_{0,R}$ and all sections $u$ with support in $M\times(-R,R)^{n}$.
If $u$ is supported within $U_{r}(i,\varepsilon)=\\{\varepsilon x_{i}>r\\}$,
then we can identify $\mathfrak{S}_{g}|_{U_{r}(i,\varepsilon)}$ with
$\mathfrak{S}_{g(i,\varepsilon)}$ and $P$ with
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$.
Since $g$ is a product metric here, we can apply Fubini’s theorem to get:

$\displaystyle||u||_{1}^{2}$
$\displaystyle=\int_{\varepsilon\mathord{\mathbb{R}}_{>r}}||u(t)||_{1,g(i,\varepsilon)}^{2}\mathrm{d}t+\int_{\varepsilon\mathord{\mathbb{R}}_{>r}}||\partial_{t}u(t)||_{0,g(i,\varepsilon)}^{2}\mathrm{d}t$
$\displaystyle\leq\int
C^{2}\left(||u(t)||^{2}_{0,g(i,\varepsilon)}+||P(i,\varepsilon)u||^{2}_{0,g(i,\varepsilon)}\right)+||\partial_{t}u(t)||_{0,g(i,\varepsilon)}^{2}\mathrm{d}t$
$\displaystyle\leq
C^{2}\int||u(t)||^{2}_{0,g(i,\varepsilon)}+||P(i,\varepsilon)u||^{2}_{0,g(i,\varepsilon)}+||\partial_{t}u(t)||_{0,g(i,\varepsilon)}^{2}\mathrm{d}t$
$\displaystyle=C^{2}\int||u(t)||^{2}_{0,g(i,\varepsilon)}+\langle
P(i,\varepsilon)^{2}u,u\rangle_{0,g(i,\varepsilon)}-\langle\partial^{2}_{t}u(t),u(t)\rangle_{0,g(i,\varepsilon)}\mathrm{d}t$
$\displaystyle=C^{2}\int||u(t)||^{2}_{0,g(i,\varepsilon)}+\langle
P(i,\varepsilon)^{2}u-\partial^{2}_{t}u(t),u\rangle_{0,g(i,\varepsilon)}\mathrm{d}t$
$\displaystyle=C^{2}\int||u(t)||_{0,g(i,\varepsilon)}^{2}+\langle(P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})^{2}u(t),u(t)\rangle\mathrm{d}t$
$\displaystyle=C^{2}\int||u(t)||^{2}_{0,g(i,\varepsilon)}+||(Pu)(t)||_{0,g(i,\varepsilon)}^{2}\mathrm{d}t$
$\displaystyle=C^{2}(||u||_{0}^{2}+||Pu||_{0}^{2}),$

from which we immediately get the fundamental elliptic inequality for this
case.

Finally, we need to patch these inequalities together. We define the set
$L\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{0\\}\cup\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$ and let
$(\phi_{l})_{l\in L}$ be the partition of unity from Lemma 4.3.3 adapted to
$P$. We use this partition of unity to patch the inequalities together:

$\displaystyle||u||_{1}$
$\displaystyle=||\sum_{L}\phi_{l}u||_{1}\leq\sum_{L}||\phi_{l}u||_{1}$
$\displaystyle\leq
C_{0}(||\phi_{0}u||_{0}+||P\phi_{0}u||_{0})+\sum_{L\setminus\\{0\\}}C_{l}(||\phi_{l}u||_{0}+||P\phi_{l}u||_{0})$
$\displaystyle\leq(2n+1)\max\\{C_{l}\\}||u||_{0}+\sum_{L}C_{l}(||\phi_{l}Pu||_{0}+||[P,\phi_{l}\cdot]u||_{0})$
$\displaystyle\leq(2n+1)\max\\{C_{l},||[P,\phi_{l}\cdot]||_{0,0}\\}\left(||u||_{0}+||Pu||_{0}\right).$

This concludes the induction step. ∎

######  Definition 4.3.5.

A block operator $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ of
Dirac type is _invertible_ if its induced operator
$H^{1}(\mathfrak{S}_{g})\rightarrow H^{0}(\mathfrak{S}_{g})$ is an
isomorphism.

######  Remark 4.3.6.

By Proposition 4.3.4 the defining condition is equivalent to require that $P$
is an invertible unbounded operator $H^{0}(\mathfrak{S}_{g_{P}})$. Recall that
this means that the extension
$P\colon\overline{\Gamma_{c}(\mathfrak{S}_{g_{P}})}^{||\text{--}||_{P}}\rightarrow
H^{0}(\mathfrak{S}_{g})$ is invertible, where the domain is the completion of
$\Gamma_{c}(\mathfrak{S}_{g_{P}})$ with respect to the graph norm
$||\text{--}||_{P}=||\text{--}||+||P(\text{--})||$.

If the block operator $P$ has invertible faces, then the previous proposition
can be improved.

######  Proposition 4.3.7.

Let $P$ be a block operator of Dirac type whose faces $P(i,\varepsilon)$ are
invertible. Then $P$ satisfies the elliptic estimate

$||u||_{1}\leq
C\left(||Pu||_{0}+||\phi_{0}u||_{0}+\sum_{l=1}^{2n}||[P,\phi_{l}\cdot]u||_{0}\right),$

where $\\{\phi_{l}\\}$ is the partition of unity from Lemma 4.3.3.
Furthermore, $P$ has closed image and finite dimensional kernel.

If $P$ is self-adjoint, then $P$ is an (unbounded) Fredholm operator with
domain $H^{1}(M\times\mathord{\mathbb{R}}^{n};\mathfrak{S}_{g})$.

The proof of this proposition requires the main Fredholm Lemma.

######  Lemma 4.3.8 ([salamon2018functional]*Lemma 4.3.9).

Let $X,Y$ be Banach spaces and $Z$ be a normed vector space. Let $A\colon
X\rightarrow Y$ be a bounded linear map and let $K\colon X\rightarrow Z$ be a
compact operator. If there is a constant $C$ such that for all $x\in X$ the
inequality

$||x||_{X}\leq C(||Ax||_{Y}+||Kx||_{Z})$

holds, then $A$ has closed image and finite dimensional kernel.

###### Proof of Proposition 4.3.7.

Pick $R>r$ such that $M\times(-R,R)^{n}$ is a core of $P$ and that the block
operator $P$ decomposes on $U_{r}(i,\varepsilon)$ under the identification

$U_{r}(i,\varepsilon)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{(m,x)\in M\times\mathord{\mathbb{R}}^{n}\,:\,\varepsilon
x_{i}>r\\}\cong
M\times\mathord{\mathbb{R}}^{n}(i,\varepsilon)\times\\{\varepsilon x_{i}>r\\}$

into the operator
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$.

For each $u\in\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g})$
with support in $U_{r}(i,\varepsilon)$, we have

$\displaystyle||Pu||_{0}^{2}$
$\displaystyle=\int_{M\times\mathord{\mathbb{R}}^{n}}|Pu|^{2}=\int_{M\times\mathord{\mathbb{R}}^{n}}|P(i,\varepsilon)u+\partial_{i}\cdot\partial_{i}u|^{2}$
$\displaystyle=\int_{M\times\mathord{\mathbb{R}}^{n}}\langle
u,(P(i,\varepsilon)+\partial_{i}\cdot\partial_{i})^{2}u\rangle=\int_{M\times\mathord{\mathbb{R}}^{n}}\langle
u,P(i,\varepsilon)^{2}u-\partial_{i}^{2}u\rangle$
$\displaystyle=\int_{M\times\mathord{\mathbb{R}}^{n}}|P(i,\varepsilon)u|^{2}+|\partial_{i}u|^{2}$
$\displaystyle=\int_{\varepsilon\cdot\mathord{\mathbb{R}}_{\geq
r}}||(P(i,\varepsilon)u)(x)||^{2}_{L^{2}(M\times\mathord{\mathbb{R}}^{n}(i,\varepsilon),\mathfrak{S}_{g})}+\int_{M\times\mathord{\mathbb{R}}^{n}}|\partial_{i}u|^{2}$
$\displaystyle\geq\int_{\varepsilon\cdot\mathord{\mathbb{R}}_{\geq
r}}c^{-2}_{(i,\varepsilon)}||u(x)||^{2}_{H^{1}(M\times\mathord{\mathbb{R}}^{n}(i,\varepsilon),\mathfrak{S}_{g})}+\int_{M\times\mathord{\mathbb{R}}^{n}}|\nabla_{\partial_{i}}u|^{2}$
$\displaystyle\geq\mathrm{min}\\{c^{-2}_{(i,\varepsilon)},1\\}||u||^{2}_{H^{1}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g})}.$

In this chain of equations, we used in the second line that $P$ is symmetric
and that $P(i,\varepsilon)$ anti-commutes with $e_{i}$. In the fourth line, we
applied Fubini’s theorem, while in the fifth line, we used the invertibility
of $P(i,\varepsilon)$ and that $\nabla_{\partial_{i}}=\partial_{i}$. In the
last line we estimated against the minimum and identified the expression as
the Sobolev-1-norm.

On the other hand, if $u$ is supported within $M\times(-R,R)^{n}$, then $Pu$
is again supported within this relatively compact subset and the classical
elliptic estimate implies

$||u||_{1}\leq C(||u||_{0}+||Pu||_{0}).$

We now use the special adapted partition of unity from Lemma 4.3.3 with
$\rho=R$ and the notation from there to patch the inequalities together. Using
that $\phi_{l}$ is contained in some $U_{r}(i,\varepsilon)$ for $l\neq 0$, the
previous estimates yield

$\displaystyle||u||_{1}$
$\displaystyle=||\sum_{L}\phi_{l}u||_{1}\leq\sum_{L}||\phi_{l}u||_{1}$
$\displaystyle\leq
C_{0}(||\phi_{0}u||_{0}+||P\phi_{0}u||_{0})+\sum_{L\setminus\\{0\\}}C_{l}||P\phi_{l}u||_{0}$
$\displaystyle\leq
C_{0}||\phi_{0}u||_{0}+\sum_{L}C_{l}(||\phi_{l}Pu||_{0}+||[P,\phi_{l}\cdot]u||_{0})$
$\displaystyle\leq
C_{0}||\phi_{0}u||_{0}+\max\\{C_{l}\cdot||\phi_{l}||_{0,0},1\\}\cdot\left(\sum_{L}||Pu||_{0}+||[P,\phi_{l}\cdot]u||_{0}\right)$
$\displaystyle\leq
C\left(||Pu||_{0}+||\phi_{0}u||_{0}+\sum_{L}||[P,\phi_{l}\cdot]u||_{0}\right),$

where $C_{l}$ is the inverse of $\min\\{c_{(i,\varepsilon)^{-1}}\\}$ for the
pair $(i,\varepsilon)$ corresponding to $l$.

It turns out that $\phi_{0}$ and all $[P,\phi_{l}\cdot]$ are compact
operators. Indeed, $\phi_{0}$ is compactly supported, so $\phi_{0}\cdot$ is a
compact operator by the classical Rellich-Lemma.

To see that
$T_{l}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=[P,\phi_{l}\cdot]$ is compact, note first that it is an element of
$\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$. Symbol calculus shows that
its principal symbol vanishes. Pick a partition of unity as in Definition
4.2.5 such that $\chi^{\mathbf{0}}$ is identically $1$ on $M\times[-R,R]^{n}$.
We decompose it into
$T_{l}=\sum_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}}T_{l}\chi^{\bm{\alpha}}$.

By Lemma 4.2.6 the summand $T_{l}\chi^{\mathbf{0}}$ is supported within
$\mathrm{supp}\,\chi^{\mathbf{0}}$ in the sense that
$\mathrm{supp}\,T_{l}\chi^{\mathbf{0}}u\subseteq\mathrm{supp}\,\chi^{\mathbf{0}}$
and $T_{l}\chi^{\bm{\alpha}}u=0$ if
$\mathrm{supp}\,\chi^{\bm{\alpha}}\cap\mathrm{supp}\,u=\emptyset$. Since the
principal symbols of $T_{l}\chi^{\mathbf{0}}$ is zero, the Atiyah-Singer exact
sequence, see Lemma B.0.24, implies that $T_{l}\chi^{\mathbf{0}}$ is compact.

If ${\bm{\alpha}}\neq 0$, then
$\hat{{\bm{\alpha}}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}}$ is not the empty map and
$\chi^{\bm{\alpha}}u$ is supported within $U_{R}(\hat{{\bm{\alpha}}})$. Thus,
property (ii) of the partition of unity $\\{\phi_{l}\\}$ yields

$T_{l}\chi^{\bm{\alpha}}u=\mathrm{grad}\phi_{l}\cdot\chi^{\bm{\alpha}}u.$

By Lemma 4.3.3 (i) and the generalised Rellich lemma for complete manifolds,
see [bunke2009index]*p.50, we conclude that $T_{l}\chi_{\bm{\alpha}}$ is also
compact.

A finite sum of compact operators is still compact, so $T_{l}$ is a compact
operator. Lemma 4.3.8 now yields that $P$ has finite kernel and closed image.

If $P$ is a self-adjoint (unbounded) operator on
$L^{2}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g})$, then the maximal
domain agrees with the minimal domain. The elliptic estimate implies that the
graph norm of $P$ is equivalent to $||\text{--}||_{1}$, so the (minimal)
domain of $P$ agrees with
$H^{1}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g})$. The kernel and the
cokernel of a self-adjoint operator are isomorphic, so the cokernel is also
finite dimensional and $P$ is a Fredholm operator. ∎

Usually, it is not easy to show that a (pseudo) differential operator is self-
adjoint. On complete manifolds, there is, however, a general criterion, the
boundness of the propagation speed.

######  Proposition 4.3.9 ([higson2000analytic]*Prop. 10.2.11).

Let $N$ be a complete Riemannian manifold and let $D$ be a symmetric
differential operator on $M$. If the _propagation speed_

$c_{D}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\sup\\{||\mathrm{symb}_{1}(D)(x,\xi)||\,:\,x\in M,\xi\in
T^{\vee}_{x}M,||\xi||=1\\}$

is finite, then $D$ is (essentially) self-adjoint, i.e. its closure is a self-
adjoint operator.

######  Corollary 4.3.10.

Let $g\in\widetilde{\mathcal{R}_{n}}(M)$ be a block metric on
$M\times\mathord{\mathbb{R}}^{n}$. Then ${\not{\mathfrak{D}}}_{g}$ is a self-
adjoint operator.

More generally, if $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ is a
symmetric, block operator of Dirac type, such that
$P-{\not{\mathfrak{D}}}_{g}$ extends to a bounded operator on
$H^{0}(\mathfrak{S}_{g})=L^{2}(\mathfrak{S}_{g})$, then $P$ is a self-adjoint
operator.

###### Proof.

A block metric is complete by Proposition 3.1.5 and the propagation speed
obviously satisfies $c_{{\not{\mathfrak{D}}}_{g}}=1$. By the preceding
proposition, ${\not{\mathfrak{D}}}_{g}$ is therefore self-adjoint.

The second statement follows from Exercise 1.9.21 of [higson2000analytic]. ∎

A common strategy to verify that a self-adjoint operator is invertible is to
verify that it squares to a uniformly positive operator. We recall this
concept.

######  Definition 4.3.11.

An unbounded operator $T$ on a Hilbert space $H$ is called _non-negative_ , if
$\langle Tx,x\rangle\geq 0$ for all $x\in{\mathrm{dom}\,}T$. It is called
_uniformly positive_ , if there is a constant $c>0$ such that $\langle
Tx,x\rangle\geq c||x||^{2}$.

An unbounded operator is _bounded from below_ if there is a positive constant
$c>0$ such that $||Tx||\geq c||x||$ for all $x\in{\mathrm{dom}\,}T$.

Clearly, a closed unbounded operator $T$ is bounded from below if and only if
$T^{\ast}T$ is uniformly positive. The next theorem is an immediate
consequence from unbounded functional calculus, see
[werner2006funktionalanalysis].

######  Lemma 4.3.12.

Let $T$ be a self-adjoint unbounded operator on a separable Hilbert space.

  1. (i)

If $T$ is uniformly positive, then $T$ is invertible.

  2. (ii)

If $T$ is closed and invertible, then $T^{\ast}T$ is uniformly positive.

Being uniformly positive is an open condition for block operators of Dirac
type in the space $\mathrm{Hom}(H^{1},H^{0})$, where
$H^{j}=H^{j}(\mathfrak{S}_{g})$.

######  Lemma 4.3.13.

Let $P,Q\in\mathrm{Hom}(H^{1},H^{0})$ be bounded operators. Let $P$ be bounded
from below and satisfy a fundamental elliptic estimate. If $Q$ is sufficiently
close to $P$ in the operator norm of $\mathrm{Hom}(H^{1},H^{0})$, then $Q$ is
also bounded from below.

###### Proof.

Given $c\in(0,1)$ such that $||Px||_{0}>c||x||_{0}$ and a constant $C=C(P)$
making the fundamental elliptic estimate valid for $P$. We claim that all
operators $Q$ satisfying the inequality

$||P-Q||_{1,0}<\frac{c}{(c+1)C}$

are bounded from below.

Indeed, the fundamental elliptic estimate applied to $P$ and the assumed
inequality yield

$\displaystyle||Qu||_{0}$
$\displaystyle\geq\bigl{|}||Pu||_{0}-||(Q-P)u||_{0}\bigr{|}$
$\displaystyle\geq||Pu||_{0}-||P-Q||_{1,0}C(||u||_{0}+||Pu||_{0})$
$\displaystyle=(1-||P-Q||_{1,0}\cdot C)||Pu||_{0}-||P-Q||_{1,0}C||u||_{0}.$

Using uniform positivity and the assumed inequality a second time, we continue
the estimation with

$\displaystyle||Qu||_{0}$ $\displaystyle\geq
c(1-C)||P-Q||_{1,0}||u||_{0}-C||P-Q||_{1,0}||u||_{0}$
$\displaystyle=\underbrace{\bigl{(}c-(c+1)C||P-Q||_{1,0}\bigr{)}}_{>0}||u||_{0}$

This proves the claim. ∎

##### Gluing Techniques

In the forthcoming sections we will carry out several cut and paste arguments
for block operators. Although we will start and end with a block operator on a
complete manifold, during the procedures we will consider operators on non-
complete manifolds. Being self-adjoint or invertible are global properties of
an operator that are usually lost when we restrict the operator to an open
subset, so we cannot expect them to hold through the entire cut and paste
procedure. On the other hand, being symmetric and bounded from below are
properties that are preserved under such restrictions and the purpose of this
section is to show that they are also preserved under certain gluing
constructions.

The first gluing technique for block operator roughly says that being
symmetric is a sheaf-like property, provided the operators are suitably local.

######  Lemma 4.3.14.

Let $E\rightarrow M$ be a vector bundle over a not necessarily compact
manifold $M$. Let $\\{U_{\alpha}\\}_{\alpha\in A}$ be a locally finite open
cover and let $\\{P_{\alpha}\\}_{\alpha\in A}$ be a family of operators

$P_{\alpha}\colon\Gamma_{c}(U_{\alpha},E|_{U_{\alpha}})\rightarrow\Gamma_{c}(U_{\alpha},E|_{U_{\alpha}})\quad\text{
and }\quad P_{\alpha}\in\overline{\Psi\mathrm{DO}}^{1}(E|_{U_{\alpha}})$

such that $P_{\alpha}|_{U_{\alpha}\cap U_{\beta}}=P_{\beta}|_{U_{\alpha}\cap
U_{\beta}}\colon\Gamma_{c}(U_{\alpha\beta},E_{U_{\alpha\beta}})\rightarrow\Gamma_{c}(U_{\alpha\beta},E_{U_{\alpha\beta}})$
for all $\alpha,\beta\in A$. Assume further that there is a partition of unity
$\\{\chi_{\alpha}\\}_{\alpha\in A}$ that is subordinated to the cover
$\\{U_{\alpha}\\}_{\alpha\in A}$ such that the operators acts as derivation on
the partition of unity, that is

$[P_{\alpha},\chi_{\beta}\cdot]u=i^{-1}\mathrm{symb}_{1}(P)(\text{--},\mathrm{d}\chi_{\beta})\cdot
u.$

Then each $P_{\alpha}$ is symmetric if and only if the operator

$P\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\sum_{\alpha\in A}P_{\alpha}\circ\chi_{\alpha}$

is symmetric.

###### Proof.

Assume that $P$ is symmetric, then also $P_{a}$ is symmetric for all $a\in A$
because for all $u\in\Gamma_{c}(U_{a},E|_{U_{a}})$ we have

$Pu=\sum_{\alpha\in A}P_{\alpha}\chi_{\alpha}u=\sum_{\\{\alpha:U_{\alpha}\cap
U_{a}\neq\emptyset\\}}P_{a}\chi_{\alpha}u=P_{a}u.$

For the converse, let $u,v\in\Gamma_{c}(M,E)$ be two sections. Set
$u_{\alpha}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\chi_{\alpha}u$ and
$v_{\beta}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\chi_{\beta}v$. By bilinearity, it suffices to show that
$(P_{\alpha}u_{\alpha},v_{\beta})=(u_{\alpha},P_{\beta}v_{\beta})$ for all
$\alpha,\beta\in A$.

To verify this, we rely on the following two facts that are immediate
consequences from the derivation property:

  * (I)

$\mathrm{supp}\,\,P_{\beta}u_{\alpha}\subseteq\mathrm{supp}\,\,\chi_{\alpha}$
for all $\alpha,\beta\in A$.

  * (II)

If $u|_{\\{\chi_{\alpha}\neq 0\\}}=0$, then
$(P_{\beta}u)|_{\\{\chi_{\alpha}\neq 0\\}}=0$.

Indeed, property (I) follows from

$\displaystyle P_{\beta}u_{\alpha}$
$\displaystyle=P_{\beta}\chi_{\alpha}u=\chi_{\alpha}P_{\beta}u+[P_{\beta},\chi_{\alpha}]u=\chi_{\alpha}u+i^{-1}\mathrm{symb}_{1}(P_{\beta})(\text{--},\mathrm{d}\chi_{\alpha})u$

and that
$\mathrm{symb}_{1}(P_{\beta})(\text{--},\mathrm{d}\chi_{\alpha})=\mathrm{symb}_{1}(P_{\beta})(\text{--},0)=0$
on $\\{\chi_{\alpha}=0\\}$. Property (II) follows from

$\displaystyle(\chi_{\alpha}P_{\beta}u)|_{\chi_{\alpha}\neq
0}=P_{\beta}\underbrace{\chi_{\alpha}u}_{=0}-i^{-1}\mathrm{symb}_{1}(P_{\beta})(\text{--},\mathrm{d}\chi_{\alpha})\underbrace{u|_{\chi_{\alpha}\neq
0}}_{=0}.$

Now, for each $\alpha\in A$, we pick a smooth function $\psi_{\alpha}$ that is
identically $1$ near $\mathrm{supp}\,\,\chi_{\alpha}$ and supported within
$U_{\alpha}$. Symmetry of $P$ now follows from the calculation

$\displaystyle(P_{\alpha}u_{\alpha},v_{\beta})$
$\displaystyle\overset{(\mathrm{I})}{=}(\psi_{\alpha}P_{\alpha}u_{\alpha},v_{\beta})=(P_{\alpha}u_{\alpha},\psi_{\alpha}v_{\beta})$
$\displaystyle=(u_{\alpha},P_{\alpha}\psi_{\alpha}v_{\beta})$
$\displaystyle=(u_{\alpha},P_{\beta}\psi_{\alpha}v_{\beta})\qquad\text{because
}P_{\alpha}|_{U_{\alpha\beta}}=P_{\beta}|_{U_{\alpha\beta}}$
$\displaystyle=(u_{\alpha},(P_{\beta}\psi_{\alpha}v_{\beta})|_{\mathrm{supp}\,\,\chi_{\alpha}})$
$\displaystyle=(u_{\alpha},P_{\beta}v_{\beta})$

in which we use (II) and linearity of $P_{\beta}$ to deduce that
$(P_{\beta}\psi_{\alpha}v_{\beta})|_{\mathrm{supp}\,\,\chi_{\alpha}}$ only
depends on (the germ of)
$\psi_{\alpha}v_{\beta}|_{\mathrm{supp}\,\,\chi_{\alpha}}=v_{\beta}|_{\mathrm{supp}\,\,\chi_{\alpha}}$.
∎

The second gluing technique yields that two invertible block operators with
matching faces can be glued together to an invertible operator, provided the
“gluing strip” is sufficiently large. We carry out the gluing construction in
a more general setup as we allow our underlying Riemannian manifolds to be
non-complete. This forces us, to work with the following weaker notion that is
equivalent to invertibility if the operator in question is self-adjoint.

######  Definition 4.3.15.

Let $P$ be a densely defined operator on a Hilbert space $H$. We call $P$
_(uniformly) bounded from below_ , if there is a $c>0$ such that $||Pv||\geq
c||v||$ for all $v\in\mathrm{dom}(P)$. More generally, an operator
$P\in\overline{\Psi\mathrm{DO}}^{1}(E)$ is bounded from below, if
$P\colon\Gamma_{c}(M,E)\rightarrow\Gamma(M,E)\subset L^{2}_{loc}(M,E)$ is
bounded from below (meaning we also allow $||P\sigma||_{0}=\infty$).

We work in the following setup.

######  Setup 4.3.16.

Let $H\subseteq M$ be a separating hypersurface that is closed as a subset but
not necessarily compact. Set $M\setminus H=M^{-}\sqcup M^{+}$ and let
$\Phi\colon H\times(-a,a)\hookrightarrow M$ be an open embedding that
restricts to the inclusion on $H\times\\{0\\}=H$ and satisfies
$\Phi(H\times(-a,0))\subseteq M^{-}$ and $\Phi(H\times(0,a))\subseteq M^{+}$.
We further denote $M_{\pm
a}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=M^{\pm}\setminus\Phi(H\times(-a,a))$ and, for all
$t,t_{1},t_{2}\in(-a,a)$, we set

$\displaystyle
M_{<t}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=M_{-a}\,\cup\,\Phi(H\times(-a,t)),\qquad
M_{>t}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\Phi(H\times(t,a))\,\cup\,M_{a},\quad\text{and}$ $\displaystyle
M_{(t_{1},t_{2})}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\Phi(H\times(t_{1},t_{2})).$

######  Definition 4.3.17.

Let $\mathcal{A}\subseteq\mathcal{C}^{\infty}(M,\mathord{\mathbb{R}})$ be a
sub-algebra of smooth functions on $M$. An operator
$P\in\overline{\Psi\mathrm{DO}}^{1}(E)$ is an $\mathcal{A}$_-derivation_ , if

$[P,f\cdot]=i^{-1}\mathrm{symb}_{1}(P)(\text{--},\mathrm{d}f)$

for all $f\in\mathcal{A}$.

We are interested in the algebra
$\mathcal{A}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(\mathord{\mathrm{pr}}_{2}\circ\Phi^{-1})^{\ast}\mathcal{C}_{c}^{\infty}((-a,a),\mathord{\mathbb{R}}))$
considered as a sub-algebra of $\mathcal{C}^{\infty}(M,\mathord{\mathbb{R}})$
(via extension by zero).

######  Lemma 4.3.18.

Let $E\rightarrow M$ be a vector bundle over a manifold as in setup 4.3.16.
Assume that $P\in\overline{\Psi\mathrm{DO}}^{1}(E)$ restricts to $M_{>b}$ and
$M_{<-b}$ for some $0<b<a$ and that $P$ is an $\mathcal{A}$-derivation. If
$\sigma\in\Gamma_{c}(M,E)$ is supported within $M_{(t_{1},t_{2})}$, then
$P\sigma$ is also supported within $M_{(t_{1},t_{2})}$ for all
$-a<t_{1}<t_{2}<a$. In particular, $(P\sigma)|_{M_{(t_{1},t_{2})}}$ only
depends on the germ of $\sigma|_{\overline{M}_{(t_{1},t_{2})}}$.

Analogous statements hold for $M_{>t}$ and $M_{<t}$ for all $t\in(-b,b)$.

###### Proof.

We only prove the statement for $M_{(t_{1},t_{2})}$. The other proofs are
quite similar. Assume $\sigma$ is supported within this set. Since the support
is compact, there are $t_{1}<t_{1}^{\prime}<t_{2}^{\prime}<t_{2}$ such that
$\sigma$ is also supported within $M_{(t_{1}^{\prime}t_{2}^{\prime})}$. Let
$\chi\colon(t_{1},t_{2})\rightarrow[0,1]$ be identically $1$ on
$(t_{1}^{\prime},t_{2}^{\prime})$ and compactly supported. To ease the
notation, we denote $\chi\circ\mathord{\mathrm{pr}}_{2}\circ\Phi^{-1}$ (and
its extension by zero) again with $\chi$. We have

$\displaystyle P\sigma$ $\displaystyle=P\chi\sigma=\chi
P\sigma+[P,\chi\cdot]\sigma$ $\displaystyle=\chi
P\sigma+i^{-1}\mathrm{symb}_{1}(P)(\text{--},\chi^{\prime})\sigma$
$\displaystyle=\chi P\sigma,$

because $\chi\equiv 1$ on $\mathrm{supp}\,\sigma$. Thus, $P\sigma$ is
contained in $\mathrm{supp}\,\chi\subseteq M_{(t_{1},t_{2})}$.

To prove that $P\sigma|_{M_{(t_{1},t_{2})}}$ only depends on the germ of
$\sigma|_{\overline{M_{(t_{1},t_{2})}}}$, it suffices to show that if $\sigma$
is identically zero near $\overline{M_{(t_{1},t_{2})}}$, then $P\sigma$
vanishes identically on $M_{(t_{1},t_{2})}$. But if $\sigma$ vanishes near
$\overline{M_{(t_{1},t_{2})}}$, then, by compactness of
$\mathrm{supp}\,\sigma$, there are $t_{1}^{\prime}<t_{1}$ and
$t_{2}^{\prime}>t_{2}$ such that $\mathrm{supp}\,(\sigma)\cap
M_{(t_{1}^{\prime},t_{2}^{\prime})}=\emptyset$. Thus $\mathrm{supp}\,(\sigma)$
is supported within $M_{<t_{1}^{\prime}}$ and $M_{>t_{2}^{\prime}}$, so
$P\sigma$ must be supported there, too. ∎

######  Definition 4.3.19.

Let $R<b<a$ and let $P_{1}$, $P_{2}$ be operators on $M_{<a}$ and $M_{>-a}$,
respectively. Assume that $P_{1}$ and $P_{2}$ restrict to $M_{<-b}$ and
$M_{>b}$, respectively, and that they are $\mathcal{A}$-derivations. By the
previous Lemma, the two operators restrict to $M_{(-R,R)}$ and we assume that
the restrictions agree. This allows us to patch the two operators together:
Let $\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ is a smooth function that
is identically $1$ near $\mathord{\mathbb{R}}_{\leq-R}$ and identically $0$
near $\mathord{\mathbb{R}}_{\geq R}$. We extend he function via $0$ and $1$ to
$M$ and denote the result again with $\chi$. We define

$P_{1}\cup
P_{2}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=P_{1}(1-\chi)+P_{2}\chi.$

Since $P_{1}$ and $P_{2}$ agree on $M_{(-R,R)}$ and act as
$\mathcal{A}$-derivations, the result does not depend on the choice of $\chi$.
An application of Lemma 4.3.14 shows that $\cup$ preserves symmetry.

######  Corollary 4.3.20.

The operator $P_{1}\cup P_{2}$ is symmetric if and only if the operators
$P_{1}$ and $P_{2}$ are symmetric.

###### Proof.

The only if direction is easy because restrictions of symmetric operators are
symmetric and $P_{1}$ agrees with $P_{2}$ on $M_{(-R,R)}$.

For the other direction, we will prove that $M_{<R}$ and $M_{>-R}$ together
with $\chi$ and $1-\chi$ from Definition 4.3.19 satisfy the condition of Lemma
4.3.14 for $P_{1}$ and $P_{2}$. To this end, fix a function
$\eta\colon(-a,a)\rightarrow[0,1]$ with compact support and
$\eta|_{(-b,b)}\equiv 1$. Let $\eta^{-1}$, $\eta^{0}$, and $\eta^{1}$ be the
functions obtained from the construction in Definition 4.2.5 but considered as
function on $M$. They form a partition of unity.

For all compactly supported sections $\sigma\in\Gamma_{c}(M_{<R},E)$, the
section $\eta^{-1}\sigma$ is supported within $M_{<-b}$, so that
$P_{1}\eta^{-1}\sigma$ is also supported there by assumption. This implies

$\displaystyle P_{1}\chi\sigma$
$\displaystyle=P_{1}\chi\eta^{-1}\sigma+P_{1}\chi\eta^{0}\sigma=P_{1}\eta^{-1}+P_{1}\chi\eta^{0}\sigma$
$\displaystyle=\chi P_{1}\eta^{-1}\sigma+\chi
P_{1}\eta^{0}\sigma+[P_{1},\chi]\eta^{0}\sigma$ $\displaystyle=\chi
P_{1}(\eta^{-1}+\eta^{0})\sigma+i^{-1}\mathrm{symb}_{1}(P_{1})(\text{--},\chi^{\prime})\cdot\eta^{0}\sigma$
$\displaystyle=\chi
P_{1}\sigma+i^{-1}\mathrm{symb}_{1}(P_{1})(\text{--},\chi^{\prime})\sigma.$

Note that the last equation follows from $\chi^{\prime}\equiv 0$ on
$\mathrm{supp}\,\eta^{-1}$. Of course, this implies that $P_{1}$ acts on
$1-\chi$ as a derivation.

The corresponding statement for $P_{2}$ and $(1-\chi)$ can be derived in an
analogous manner. ∎

######  Lemma 4.3.21.

Let $P_{1}$ and $P_{2}$ be operators on $M_{<a}$ and $M_{>-a}$ as in setup
4.3.19. Assume that

$||\mathrm{symb}_{1}(P_{1})(\text{--},\mathrm{d}t)||_{\infty,M_{(-R,R)}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{||\mathrm{symb}_{1}(P_{1})(x,\mathrm{d}t)||_{\mathrm{op}}\,:\,x\in
M_{(-R,R)}\\}\leq C.$

If there is a section $u\in\Gamma_{c}(M,E)$ such that $||(P_{1}\cup
P_{2})u||_{0}<\varepsilon||u||_{0}$, then there is a $v\in\Gamma_{c}(M,E)$
that is supported within $M_{<R}$ that satisfies

$||P_{1}v||_{0}^{2}<4\left(\varepsilon^{2}+\frac{C^{2}}{R^{2}}\right)||v||_{0}^{2}$

or is supported within $M_{>-R}$ and satisfies an analogous inequality with
$P_{1}$ replaced by $P_{2}$.

###### Proof.

Abbreviate $P_{1}\cup P_{2}$ to $Q$. Let $u\in\Gamma_{c}(M,E)$ with
$||Qu||_{0}^{2}<\varepsilon^{2}||u||_{0}^{2}$. Assume without loss of
generality, that $||u||_{0,M_{\leq 0}}^{2}\geq 1/2||u||_{0}^{2}$. Let
$\eta\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth function that
satisfies $\eta\equiv 1$ on $\mathord{\mathbb{R}}_{\leq 0}$, $\eta\equiv 0$ on
$\mathord{\mathbb{R}}_{\geq 1}$, and $|\eta^{\prime}|\leq 1.2$. Set
$\eta_{R}(t)=\eta(R^{-1}t)$ and extend this function as described above to
$M$. We denote the result again with $\eta_{R}$.

The norm splits into

$\displaystyle||P_{1}\eta_{R}u||_{0}^{2}=||P_{1}\eta_{R}u||_{0,M_{\leq-R}}^{2}+||P_{1}u||_{0,M_{(-R,R)}}^{2}$

and, by Lemma 4.3.18, the summands only depend on the germs of the
restrictions to these domains. For example,

$||P_{1}\eta_{R}u||_{0,M_{\leq-R}}^{2}=||P_{1}\eta_{R}u|_{M_{\leq-R}}||_{0,M_{\leq-R}}^{2}=||P_{1}u||_{0,M_{\leq-R}}^{2}.$

Since also $Q$ satisfies the assumptions of Lemma 4.3.18, we have

$(P_{1}\eta_{R}u)|_{M\leq-R}=P_{1}(\eta_{R}u|_{M\leq-R})=(P_{1}u)|_{M\leq-R}=(Qu)|_{M\leq-R}$

and because $P_{1}$ and $P_{2}$ agree on $M_{(-R,R)}$ we have

$\displaystyle(Q\eta_{R}u)|_{M_{(-R,R)}}$
$\displaystyle=Q\eta_{R}(u|_{M_{(-R,R)}})=P_{1}\eta_{R}(u|_{M_{(-R,R)}})$
$\displaystyle=\eta_{R}P_{1}(u|_{M_{(-R,R)}})+[P,\eta_{R}\cdot](u|_{M_{(-R,R)}})$
$\displaystyle=(\eta_{R}Qu)|_{M_{(-R,R)}}+i^{-1}\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)|_{M_{(-R,R)}}.$

This implies

$\displaystyle||P_{1}\eta_{R}u||_{0}^{2}$
$\displaystyle=||Qu||^{2}_{0,M_{\leq-R}}+||\eta_{R}Qu+\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)||_{0,M_{(-R,R)}}^{2}$
$\displaystyle\leq||Qu||^{2}_{0,M_{\leq-R}}+2||\eta_{R}Qu||^{2}_{0,M_{(-R,R)}}+2||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)||_{0,M_{(-R,R)}}^{2}$
$\displaystyle\leq
2||Qu||^{2}_{0,M_{<R}}+2||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)||_{0,M_{(-R,R)}}^{2}$
$\displaystyle<2\varepsilon^{2}||u||^{2}_{0}+2\frac{C^{2}}{R^{2}}||u||_{0}^{2},$

where we used in the last inequality that
$||Qu||_{0}^{2}<\varepsilon^{2}||u||^{2}_{0}$ and that

$\displaystyle\quad\
||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)||_{0;M_{(-R,R)}}$
$\displaystyle\leq||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)||_{0,0;M_{(-R,R)}}||u||_{0,M_{(-R,R)}}$
$\displaystyle\leq||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)||_{\infty;M_{(-R,R)}}||u||_{0}$
$\displaystyle\leq\frac{|\eta^{\prime}|_{\infty}}{R}||\mathrm{symb}_{1}(P_{1})(\text{--},\mathrm{d}t)||_{\infty;M_{(-R,R)}}||u||_{0}$
$\displaystyle\leq\frac{C}{R}||u||_{0}$

From $1/2||u||_{0}^{2}\leq||u||_{0,M_{\leq 0}}^{2}$ it follows that
$||u||_{0}^{2}\leq 2||\eta_{R}u||_{0}^{2}$. Putting all inequalities together,
we get

$||P_{1}\eta_{R}u||_{0}^{2}<4\left(\varepsilon^{2}+C^{2}/R^{2}\right)||\eta_{R}u||_{0}^{2},$

so the claim follows for
$v\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\eta_{R}u$. ∎

The following implication is an indispensable tool in the following sections
because it allows us to glue invertible block operators of Dirac type
together. Note that, in this set up, we are allowed to choose the length of
the “gluing-strip” to be arbitrarily large without increasing the propagation
speed of the operator.

######  Corollary 4.3.22.

Let $P_{1}$ and $P_{2}$ be as in the previous lemma. If they are bounded from
below with lower bounds $c_{1}$ and $c_{2}$ respectively and if
$R>\max\\{C^{2}/c_{1}^{2},C^{2}/c_{2}^{2}\\}$, then $P_{1}\cup P_{2}$ is
bounded from below with lower bound

$c_{P_{1}\cup
P_{2}}^{2}=\frac{\min\left\\{{c_{1}^{2}-C^{2}/R^{2}},c_{2}^{2}-C^{2}/R^{2}\right\\}}{4}.$

###### Proof.

By assumption on $R$, the constant $c_{P_{1}\cup P_{2}}$ is positive. Assume
that there is a section $u\in\Gamma_{c}(M,E)$ such that $||(P_{1}\cup
P_{2})(u)||_{0}<c_{P_{1}\cup P_{2}}||u||_{0}$, then the previous Lemma implies
the existence of $v\in\Gamma_{c}(M,E)$ that is, without loss of generality,
supported within $M_{<R}$ and satisfies

$\displaystyle||P_{1}v||_{0}^{2}$ $\displaystyle<4\left(c_{P_{1}\cup
P_{2}}^{2}+C^{2}/R^{2}\right)||v||_{0}^{2}<c_{1}^{2}||v||_{0}^{2},$

which contradicts that $P_{1}$ is bounded from below by $c_{1}$. ∎

###  4.4 Foundations of the Operator Concordance Set

With the theory developed in the previous section we are finally in the
position to define the operator concordance set
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ as the cubical subset of
all block Dirac operators.

But, first we give a combinatorial model for the classifying space for real
$K$-theory, by considering the cubical set of block maps into
$\Psi\mathrm{Dir}(M)$ and $\Psi\mathrm{Dir}^{\times}(M)$ instead of the spaces
themselves.

######  Definition 4.4.1.

Let $\Psi\mathrm{Dir}_{\bullet}(M)$ be the cubical set whose $n$-cubes
consists of smooth block maps
$P\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{Dir}(M)$. The
connecting maps are given by

$\partial^{\varepsilon}_{i}P=\lim_{R\to\infty}P\circ{\delta^{R\varepsilon}_{i}}\qquad\text{
and }\qquad\sigma_{i}P=P\circ p_{i}.$

Let $\Psi\mathrm{Dir}^{\times}_{\bullet}(M)$ be the cubical subset consisting
of all smooth block maps with values in $\Psi\mathrm{Dir}^{\times}(M)$.

Recall that $\Psi\mathrm{Dir}(M)$ is an affine bundle over $\mathrm{Riem}(M)$
with fibre $\Psi\mathrm{Dir}(M)_{g}$, the set of all self-adjoint, odd,
Clifford-linear pseudo differential operators of order $1$ whose principal
symbol agree with the principal symbol of the Dirac operator
${\not{\mathfrak{D}}}_{g}$. The map
$\Psi\mathrm{Dir}(M)\rightarrow\mathrm{Riem}(M)$ assigns to an operator $P$
its underlying metric $g$. This map induces a map of cubical sets
$\Psi\mathrm{Dir}_{\bullet}(M)\rightarrow\mathcal{R}^{+}_{\bullet}(M)$.

Now, we are going to construct the combinatorial reference space.

######  Definition 4.4.2.

Let $g$ be a block metric on $M\times\mathord{\mathbb{R}}^{n}$. A block
operator $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ of Dirac type
is called a _block Dirac operator_ if it is symmetric and
$P-{\not{\mathfrak{D}}}_{g}$ extends to a bounded endomorphism on
$H^{0}(\mathfrak{S}_{g})=L^{2}(\mathfrak{S}_{g})$.

An immediate consequence of Corollary 4.3.10 is that a block Dirac operator is
automatically essentially self-adjoint.

For the definition of the operator concordance set it will be important to
trace permutation of coordinates. A permutation
$\sigma\in\mathrm{Aut}(\\{1,\dots,n\\})=S_{n}$ induces a linear map

$\sigma\colon(x_{1},\dots,x_{n})\mapsto(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(n)}).$

If $g$ is a block metric on $M\times\mathord{\mathbb{R}}^{n}$, then
$\sigma^{\ast}g$ is also a block metric on $M\times\mathord{\mathbb{R}}^{n}$.
Since $\mathord{\mathbb{R}}^{n}$ carries up to equivalence only one
$\mathrm{Pin}^{-}$-structure, permutations are Pin structure preserving.
Functoriality of the spinor bundle construction yields a map, which we denote
with

$\mathfrak{S}(\sigma)\colon\mathfrak{S}_{\sigma^{\ast}g}\rightarrow\mathfrak{S}_{g}\quad\text{
or equivalently
}\quad\mathfrak{S}(\sigma)\colon\mathfrak{S}_{g}\rightarrow\mathfrak{S}_{\sigma_{\ast}g}.$

We are particularly interested in cyclic permutations.

######  Definition 4.4.3.

For positive integers $a<b$ let $\mathrm{cycl}(a,b)$ be the cyclic permutation
given by $a\mapsto a+1$ and $b\mapsto a$ and let
$\mathrm{cycl}(b,a)=\mathrm{cycl}(a,b)^{-1}$. We denote the induced maps on
spinor bundles with $\mathrm{Cycl}(a,b)$ and $\mathrm{Cycl}(b,a)$,
respectively.

######  Definition 4.4.4.

Let $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ be the cubical set whose set
of $n$-cubes is given by

$\widetilde{\Psi\mathrm{Dir}}_{n}(M)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})\,:\,g\in\widetilde{\mathcal{R}_{n}}(M),\,P\text{
block Dirac operator}\\}$

and whose connecting maps are given by

$\displaystyle\partial^{\varepsilon}_{i}P$
$\displaystyle\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\lim_{R\to\infty}{{\delta^{R\varepsilon}_{i}}}^{\ast}P(i,\varepsilon)\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{\partial^{\varepsilon}_{i}g}),$
$\displaystyle\sigma_{i}P$
$\displaystyle\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{Cycl}(i,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{\sigma_{i}g})$

######  Remark 4.4.5.

A comment to the (subtle) notation. Recall that if a block operator $P$ with
underlying block metric $g$ decomposes on $M\times
U_{r}(i,\varepsilon)=\\{(m,x)\,:\,\varepsilon x_{i}>r\\}$ into
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$
under the isometry $U_{r}(i,\varepsilon)\cong M\times\\{\varepsilon
x_{i}=r\\}\times\varepsilon\mathord{\mathbb{R}}_{>r}$ given by permutation,
then the operator $P(i,\varepsilon)$ restricts to a block operator on
$\mathfrak{S}_{g(i,\varepsilon)}\rightarrow M\times\\{\varepsilon x_{i}=R\\}$
for all sufficiently large $R$ and is independent of $R$. The map

${\delta^{R\varepsilon}_{i}}\colon(M\times\mathord{\mathbb{R}}^{n-1},\partial^{\varepsilon}_{i}g)\rightarrow\bigl{(}M\times\\{\varepsilon
x_{i}=R\\},g(i,\varepsilon)\bigr{)}$

is a Pin-structure preserving isometry and thus provides a bundle isometry
$\mathfrak{S}({\delta^{R\varepsilon}_{i}})$ between the corresponding spinor
bundles that induces an isomorphism between the corresponding sections with
compact support. We use this isomorphism to pull $P(i,\varepsilon)$, an
operator on $\Gamma_{c}(\mathfrak{S}_{g(i,\varepsilon)})$, back to
${{\delta^{R\varepsilon}_{i}}}^{\ast}P(i,\varepsilon)$, an operator on
$\Gamma_{c}(\mathfrak{S}_{\partial^{\varepsilon}_{i}g})$.

For the degeneracy map, we use that the cyclic permutation

$\mathrm{cycl}(i,n+1)\colon\left(M\times\\{\varepsilon
x_{i}>r\\},\sigma_{i}g\right)\rightarrow\left(M\times\mathord{\mathbb{R}}^{n}\times\varepsilon\mathord{\mathbb{R}}_{>r},g\oplus\mathrm{d}x_{n+1}^{2}\right)$

is an isometry and the natural isomorphism $\mathfrak{S}_{g}\boxtimes
Cl_{1,0}\cong\mathfrak{S}_{g\oplus\mathrm{d}x_{n+1}^{2}}$.

######  Lemma 4.4.6.

The connecting maps of $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ are well-
defined and satisfy the cubical identities.

The proof requires the following simple functional analytic result.

######  Lemma 4.4.7.

Let $H_{1}$, $H_{2}$ be Hilbert spaces and $A\colon H_{1}\rightarrow H_{1}$ be
an unbounded operator. Then $A\otimes\mathrm{id}$ is a bounded operator on
$H_{1}\otimes H_{2}$ if and only if $A$ is bounded on $H_{1}$. In this case,
the operator norms satisfy
$||A\otimes\mathrm{id}||_{\mathrm{op}}=||A||_{\mathrm{op}}$.

###### Proof.

Since the algebraic tensor product $H_{1}\odot H_{2}$ lies dense in
$H_{1}\otimes H_{2}$, so does
${\mathrm{dom}\,}(A\otimes\mathrm{id})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={\mathrm{dom}\,}A\odot H_{2}$. If $A$ is bounded, then we can
continuously extend it to an operator on $H_{1}\otimes H_{2}$ and the operator
norm of this extension necessarily agrees with $||A||_{\mathrm{op}}$.

Conversely, assume that $A\otimes\mathrm{id}$ is bounded. Let $v\in H_{2}$ be
a unit-length vector. The map $\iota_{v}\colon H_{1}\rightarrow H_{1}\otimes
H_{2}$ given by $x\mapsto x\otimes v$ is linear and bounded. The map $x\otimes
y\mapsto\langle y,v\rangle\cdot x$ extends from $H_{1}\odot H_{2}$ to a
bounded linear map $p_{v}\colon H_{1}\otimes H_{2}\rightarrow H_{1}$. From
$A=p_{v}\circ A\otimes\mathrm{id}\circ\iota_{v}$ we deduce that $A$ is a
bounded operator. ∎

###### Proof of Lemma 4.4.6.

By Corollary 4.2.3, $\partial^{\varepsilon}_{i}P$ is a again a block operator.
Symbol calculus shows that the principal symbol of
$\partial^{\varepsilon}_{i}P$ agrees with the principal symbol of
${\not{\mathfrak{D}}}_{\partial^{\varepsilon}_{i}g}$, so
$\partial^{\varepsilon}_{i}P$ is a block operator of Dirac type. In a similar
fashion, but using Lemma 4.2.4 instead, we see that $\sigma_{i}P$ is a block
operator of Dirac type.

We need to show that $\partial^{\varepsilon}_{i}P$ and $\sigma_{i}P$ are
symmetric operators if $P$ is symmetric. We start with
$\sigma_{i}P=\mathrm{Cycl}(i,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})$.
Since $\mathrm{Cycl}(i,n+1)$ induces an isometry between the corresponding
Hilbert spaces of square integrable sections, the push-forward
$\mathrm{Cycl}(i,n+1)_{\ast}$ maps symmetric operators to symmetric operators.
Thus, it suffices to prove that
$P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$ is a symmetric
operator on

$L^{2}(M\times\mathord{\mathbb{R}}^{n+1};\mathfrak{S}_{g\oplus\mathrm{d}x_{n+1}^{2}})\cong
L^{2}(M\times\mathord{\mathbb{R}}^{n};\mathfrak{S}_{g})\otimes
L^{2}(\mathord{\mathbb{R}},Cl_{1,0}),$

which is true because the tensor product of two symmetric operators is again
symmetric and all four operators in question $P$, $\mathrm{id}$,
$\mathrm{e/o}$ and ${\not{\mathit{D}}}$ are symmetric.

To argue that $\partial^{\varepsilon}_{i}P$ is symmetric, we proceed as
follows: On
$U(i,\varepsilon)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=U_{R}(i,\varepsilon)$, for some sufficiently large $R>0$, we can
identify the symmetric operator $P|_{U(i,\varepsilon)}$ with
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}$
via an isometric bundle morphism. The second summand is symmetric, so
$P(i,\varepsilon)\boxtimes\mathrm{id}$ must be a symmetric operator on
$\mathfrak{S}_{g(i,\varepsilon)\oplus\mathrm{d}x_{n}^{2}}\rightarrow(M\times\mathord{\mathbb{R}}^{n})(i,\varepsilon)\times\varepsilon\mathord{\mathbb{R}}_{>R}$.
This implies that $P(i,\varepsilon)$ is symmetric on
$\mathfrak{S}_{g(i,\varepsilon)}$ and consequently that
$\partial^{\varepsilon}_{i}P={\delta^{R\varepsilon}_{i}}^{\ast}P(i,\varepsilon)$
is symmetric because
${\delta^{R\varepsilon}_{i}}\colon(M\times\mathord{\mathbb{R}}^{n-1},\partial^{\varepsilon}_{i}g)\rightarrow(M\times\mathord{\mathbb{R}}^{n}(i,\varepsilon),g(i,\varepsilon))$
is an isometry.

From

$\displaystyle\quad\ \sigma_{i}P-{\not{\mathfrak{D}}}_{\sigma_{i}g}$
$\displaystyle=\mathrm{Cycl}(i,n+1)_{\ast}\left(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)-\mathrm{Cycl}(i,n+1)_{\ast}\left({\not{\mathfrak{D}}}_{g}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)$
$\displaystyle=\mathrm{Cycl}(i,n+1)_{\ast}((P-{\not{\mathfrak{D}}}_{g})\boxtimes\mathrm{id})$

and Lemma 4.4.7 it follows that the difference
$\sigma_{i}P-{\not{\mathfrak{D}}}_{\sigma_{i}g}$ extends to a bounded operator
on $H^{0}(\mathfrak{S}_{\sigma_{i}g})$ with operator norm
$||P-{\not{\mathfrak{D}}}_{g}||_{0,0}$.

To see that
$\partial^{\varepsilon}_{i}P-{\not{\mathfrak{D}}}_{\partial^{\varepsilon}_{i}g}$
extends to a bounded operator on
$H^{0}(\mathfrak{S}_{\partial^{\varepsilon}_{i}g})$ we argue as follows. For
$R>0$ sufficiently large the operator $P-{\not{\mathfrak{D}}}_{g}$ restricts
to $\Gamma_{c}(U_{R}(i,\varepsilon),\mathfrak{S}_{g})$. Since
$P-{\not{\mathfrak{D}}}_{g}$ extends to a bounded operator on
$H^{0}(\mathfrak{S}_{g})$, the restriction extends to a bounded operator on
$H^{0}(U_{R}(i,\varepsilon),\mathfrak{S}_{g})$, which is isometric equivalent
to $H^{0}(M\times\\{\varepsilon
x_{i}=R\\},\mathfrak{S}_{g(i,\varepsilon)})\otimes
L^{2}(\varepsilon\mathord{\mathbb{R}}_{>R},Cl_{1,0})$. Under this
identification, the restriction of $P-{\not{\mathfrak{D}}}_{g}$ agrees with
$\left(P(i,\varepsilon)-{\not{\mathfrak{D}}}_{g(i,\varepsilon)}\right)\boxtimes\mathrm{id}$.
By Lemma 4.4.7, this implies that
$P(i,\varepsilon)-{\not{\mathfrak{D}}}_{g(i,\varepsilon)}$ is bounded. Thus,
$\partial^{\varepsilon}_{i}P-{\not{\mathfrak{D}}}_{\partial^{\varepsilon}_{i}g}={\delta^{R\varepsilon\,\ast}_{i}}\left(P(i,\varepsilon)-{\not{\mathfrak{D}}}_{g(i,\varepsilon)}\right)$
is a bounded operator on $H^{0}(\mathfrak{S}_{\partial^{\varepsilon}_{i}g})$.

Lastly, we need to show, that the connecting maps satisfy the cubical
identities. For $i<j$ and $R>0$ sufficiently large, we have

$\displaystyle\partial^{\varepsilon}_{i}\partial^{\omega}_{j}P$
$\displaystyle=\partial^{\varepsilon}_{i}\left({\delta^{\omega
R}_{j}}^{\ast}P(j,\omega)\right)$ $\displaystyle={\delta^{\varepsilon
R}_{i}}^{\ast}\left({\delta^{\omega
R}_{j}}^{\ast}\left(P(j,\omega)\right)(i,\varepsilon)\right)$
$\displaystyle=\left({\delta^{\omega R}_{j}}{\delta^{\varepsilon
R}_{i}}\right)^{\ast}P(i,j,\varepsilon,\omega)$
$\displaystyle=\left({\delta^{\varepsilon R}_{i}}{\delta^{\omega
R}_{j-1}}\right)^{\ast}P(i,j,\varepsilon,\omega)$
$\displaystyle={\delta^{\omega R}_{j-1}}^{\ast}\left({\delta^{\varepsilon
R}_{i}}^{\ast}\left(P(i,\varepsilon)\right)(j-1,\omega)\right)$
$\displaystyle=\partial^{\omega}_{j-1}\partial^{\varepsilon}_{i}P.$

For $i<j$ and a sufficiently large $R>0$, we calculate

$\displaystyle\partial^{\varepsilon}_{i}\sigma_{j}(P)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}(\sigma_{j}(P)(i,\varepsilon))$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}\mathrm{Cycl}(j,n+1)_{\ast}\left(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)(\mathrm{cycl}(j,n+1)^{-1}(i),\varepsilon)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}\mathrm{Cycl}(j,n+1)_{\ast}(P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})\quad\quad\quad\text{
for }i<j$
$\displaystyle=\left({\delta^{R\varepsilon}_{i}}^{-1}\right)_{\ast}\mathrm{Cycl}(j,n+1)_{\ast}(P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})$
$\displaystyle=\mathrm{Cycl}(j-1,n)_{\ast}\left({\delta^{R\varepsilon}_{i}}^{-1}\right)_{\ast}\left(P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g(i,\varepsilon)}}\boxtimes{\not{\mathit{D}}}\right)$
$\displaystyle=\mathrm{Cycl}(j-1,n)_{\ast}(\partial^{\varepsilon}_{i}P\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{\partial^{\varepsilon}_{i}g}}\boxtimes{\not{\mathit{D}}})$
$\displaystyle=\sigma_{n-1}\partial^{\varepsilon}_{i}P,$

and analogously
$\partial^{\varepsilon}_{i}\sigma_{j}P=\sigma_{j}\partial^{\varepsilon}_{i-1}P$
for $i>j$. Finally, with a slight abuse of notation in the last lines, we
calculate

$\displaystyle\partial^{\varepsilon}_{i}\sigma_{i}P$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}\left((\sigma_{i}P)(i,\varepsilon)\right)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}\left(\mathrm{Cycl}(i,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})(i,\varepsilon)\right)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{-1}_{\ast}\mathrm{Cycl}(i,n+1)_{\ast}\left((P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})(\mathrm{cycl}(i,n+1)^{-1}(i),\varepsilon)\right)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{-1}_{\ast}\mathrm{Cycl}(i,n+1)_{\ast}\left((P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})(n+1,\varepsilon)\right)$
$\displaystyle={\delta^{R\varepsilon}_{n+1}}^{-1}_{\ast}\left((P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})(n+1,\varepsilon)\right)=P.$

For $i\leq j$, we use the identity

$\mathrm{cycl}(i,n+2)\circ\mathrm{cycl}(j,n+1)=\mathrm{cycl}(j+1,n+2)\circ\mathrm{cycl}(i,n+1)\circ\tau_{n+1,n+2}$

in which $\tau_{n+1,n+2}$ denotes the transposition that interchanges $n+1$
and $n+2$, to conclude

$\displaystyle\sigma_{i}\sigma_{j}(P)$
$\displaystyle=\sigma_{i}(\mathrm{Cycl}(j,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}))$
$\displaystyle\begin{split}&=\mathrm{Cycl}(i,n+2)_{\ast}\left(\mathrm{Cycl}(j,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}})\boxtimes\mathrm{id}\right.\\\
&\quad\left.+\mathrm{e/o}_{\mathfrak{S}_{\sigma_{j}(g)}}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle\begin{split}&=\mathrm{Cycl}(i,n+2)_{\ast}\circ\mathrm{Cycl}(j,n+1)_{\ast}\biggl{(}P\boxtimes\mathrm{id}\boxtimes\mathrm{id}\\\
&\quad+\left.\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{\sigma_{n+1}(g)}}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle\begin{split}&=\mathrm{Cycl}(i,n+2)_{\ast}\circ\mathrm{Cycl}(j,n+1)_{\ast}\biggl{(}P\boxtimes\mathrm{id}\boxtimes\mathrm{id}\\\
&\quad\left.+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle\begin{split}&=\mathrm{Cycl}(j+1,n+2)_{\ast}\circ\mathrm{Cycl}(i,n+1)_{\ast}\circ{\tau_{n+1,n+2}}_{\ast}\biggl{(}P\boxtimes\mathrm{id}\boxtimes\mathrm{id}\\\
&\quad\left.+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle\begin{split}&=\mathrm{Cycl}(j+1,n+2)_{\ast}\circ\mathrm{Cycl}(i,n+1)_{\ast}\biggl{(}P\boxtimes\mathrm{id}\boxtimes\mathrm{id}\\\
&\quad\left.+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle=...=\sigma_{j+1}\sigma_{i}(P).$

In the firth line, we can drop ${\tau_{n+1,n+2}}_{\ast}$ in the second factor
to go to the sixth line because

${\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}=\partial_{x_{n+1}}\cdot\frac{\partial}{\partial{x_{n+1}}}+\partial_{x_{n+2}}\cdot\frac{\partial}{\partial{x_{n+2}}}$

is symmetric in the $(n+1)$-th and $(n+2)$-th coordinate. ∎

We come now to one of the most essential definitions of this thesis.

######  Definition 4.4.8.

Let $\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ be the sequence of
subsets of $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$, where
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(M)$ consists of all block Dirac
operator $P\in\widetilde{\Psi\mathrm{Dir}}_{n}(M)$ that extend to an
invertible operator $P\colon H^{1}(\mathfrak{S}_{g_{P}})\rightarrow
H^{0}(\mathfrak{S}_{g_{P}})$.

The proof that the face maps of $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$
restricts to $\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ requires the
following lemma that roughly says that positivity descends to hypersurfaces.

######  Lemma 4.4.9.

Let $\mathfrak{S}_{g}\rightarrow M$ be a spinor bundle over a Riemannian
manifold $(M,g)$. For $Q\in\overline{\Psi\mathrm{DO}}^{1}(E)$ symmetric and
odd, define
$P\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=Q\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$ on
$\mathfrak{S}\boxtimes Cl_{1,0}\rightarrow M\times(-a,a)$ for all
$0<a\leq\infty$.

If $||Pu||_{0}\geq c||u||_{0}$ for all
$u\in\Gamma_{c}(M\times(-a,a),\mathfrak{S}_{g}\boxtimes Cl_{1,0})$, then

$||Qv||_{0}^{2}\geq\max\\{0,c-4/a^{2}\\}||v||_{0}^{2}$

for all $v\in\Gamma_{c}(M,\mathfrak{S}_{g})$.

###### Proof.

Let $v\in\Gamma_{c}(M,\mathfrak{S}_{g})$ and
$f\in\Gamma_{c}((-a,a),Cl_{1,0})$. The operator $P$ is again symmetric by the
proof of Lemma 4.4.6 and odd. Using Fubini’s theorem and that $P$ is bounded
from below, we get

$\displaystyle||Pv\otimes f||_{0}^{2}$ $\displaystyle=\langle P^{2}(v\otimes
f),v\otimes f\rangle$ $\displaystyle=\langle Q^{2}(v)\otimes
f-v\otimes\Delta_{\mathord{\mathbb{R}}}(f),v\otimes f\rangle$
$\displaystyle=||Q(v)\otimes f||_{0}^{2}+||v\otimes f^{\prime}||_{0}^{2}$
$\displaystyle=||Q(v)||_{0}^{2}||f||_{0}^{2}+||v||_{0}^{2}||f^{\prime}||_{0}^{2}$
$\displaystyle\geq c||v||_{0}^{2}||f||_{0}^{2}=c||v\otimes f||_{0}^{2}.$

The function

$\varphi\colon(-1,1)\rightarrow\mathord{\mathbb{R}}\quad\text{ given by
}\quad\varphi(t)=\begin{cases}1+t,&\text{if }t\leq 0,\\\ 1-t,&\text{if }t\geq
0,\end{cases}$

satisfies $||\varphi||_{0}^{2}=2/3$ while $||\varphi^{\prime}||_{0}^{2}=2$.
Pick a compactly supported function $f$ that is sufficiently
$||\text{--}||_{1}$-close to $\varphi$ such that
$||f^{\prime}||_{0}^{2}/||f||_{0}^{2}\leq 4$.

By the transformation formula, the assignment $f\mapsto
a^{-1/2}f(a^{-1}\cdot)$ is an isometry $L^{2}(-1,1)\rightarrow L^{2}(-a,a)$.
Thus,
$||f_{a}^{\prime}||_{0}^{2}/||f_{a}||_{0}^{2}=a^{-2}||f^{\prime}||_{0}^{2}/||f||_{0}^{2}\leq
4/a^{2}$.

In conclusion, if we plug into $f_{a}$ in the previous inequality, we obtain

$\displaystyle||Qv||_{0}^{2}$
$\displaystyle\geq\left(c-||f_{a}^{\prime}||_{0}^{2}/||f_{a}||_{0}^{2}\right)||v||_{0}^{2}$
$\displaystyle\geq\left(c-4/a^{2}\right)||v||_{0}^{2},$

which is positive if $c>4/a^{2}$. ∎

######  Lemma 4.4.10.

The connecting maps of $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ send
invertible elements to invertible elements. Thus,
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ is a cubical subset.

###### Proof.

Fix a sufficiently large $R>0$ such that $g=g_{P}$ decomposes into
$g(i,\varepsilon)\oplus\mathrm{d}x_{i}^{2}$ on $U_{R}{(i,\varepsilon)}$ and,
by abusing notation, $P$ decomposes into
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}$.
Since $P$ is bounded from below, its restriction $P|_{U_{R}(i,\varepsilon)}$
is also bounded from below. Lemma 4.4.9 implies that $P(i,\varepsilon)$, and
hence $\partial^{\varepsilon}_{i}P$ too, is bounded from below. Since
$\partial^{\varepsilon}_{i}P$ is self-adjoint, it must be invertible.

To see that $\sigma_{i}(P)$ is invertible, it suffices to check that
$P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$ is invertible.
Since $P\boxtimes\mathrm{id}$ and $\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$
anti-commute we have

$\left(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)^{2}=P^{2}\boxtimes\mathrm{id}-\mathrm{id}\boxtimes\Delta.$

Both summands are non-negative operators. Since $P$ is invertible, $P^{2}$ is
positive and therefore the sum is positive and therefore invertible. ∎

We would like to know whether $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ and
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ are Kan sets. For
$\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ even more is true.

######  Proposition 4.4.11.

The cubical set $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ is combinatorially
contractible.

The proposition follows immediately from the following elementary lemma.

######  Lemma 4.4.12.

Let $P_{(j,\omega)}$ and $P_{(k,\eta)}$ be two block Dirac operators on
$M\times\mathord{\mathbb{R}}^{n-1}$. Assume that $j\leq k$, that
$\partial^{\omega}_{j}P_{(k,\eta)}=\partial^{\eta}_{k-1}P_{(j,\omega)}$ and
that the two operators decompose outside of $M\times\rho I^{n}$. Then the
operators $\sigma_{j}P_{(j,\omega)}$ and $\sigma_{k}P_{(k,\eta)}$ agree on the
set $\\{\omega x_{j}>\rho,\eta x_{k}>\rho\\}$.

###### Proof.

Abbreviate $\\{\omega x_{j}>\rho,\eta x_{k}>\rho\\}$ to $U$. If $j=k$ there is
nothing to prove for $U$ is either empty or $(j,\omega)=(k,\eta)$. In the
other cases, it follows from Lemma 3.1.10 that the underlying metrics agree on
$U$. The proof of Lemma 4.2.4 shows that $\sigma_{j}P_{(j,\omega)}$ restricts
to $U$. Furthermore, for $j<k$, the calculation

$\displaystyle\sigma_{j}P_{(j,\omega)}|_{U}$