# On the $L^{p}$ Spectrum of the Dirac operator

Nelia Charalambous  Department of Mathematics and Statistics, University of
Cyprus, Nicosia, 1678, Cyprus<EMAIL_ADDRESS>and  Nadine Große
Mathematisches Institut, Universität Freiburg, 79100 Freiburg, Germany
<EMAIL_ADDRESS>
###### Abstract.

Our main goal in the present paper is to expand the known class of open
manifolds over which the $L^{2}$-spectrum of a general Dirac operator and its
square is maximal. To achieve this, we first find sufficient conditions on the
manifold so that the $L^{p}$-spectrum of the Dirac operator and its square is
independent of $p$ for $p\geq 1$. Using the $L^{1}$-spectrum, which is simpler
to compute, we generalize the class of manifolds over which the
$L^{p}$-spectrum of the Dirac operator is the real line for all $p$. We also
show that by applying the generalized Weyl criterion, we can find large
classes of manifolds with asymptotically nonnegative Ricci curvature, or which
are asymptotically flat, such that the $L^{2}$-spectrum of a general Dirac
operator and its square is maximal.

###### Key words and phrases:

Dirac operator, independence of $L^{p}$-spectrum

###### 2010 Mathematics Subject Classification:

Primary: 58J50; Secondary: 58J35, 53C27

The first author is partially supported by a University of Cyprus Internal
Grant. The second author is partially supported by DFG SPP2026.

##  1\. Introduction

The spectrum of geometric operators like the Laplacian or the Dirac operator
plays a fundamental role in several analytic and geometric problems. However,
computing the spectrum is in general a difficult task without any additional
assumptions on the symmetries and structure of the manifold. Depending on the
manifold there are several methods at hand. For example, it is in general
easier to compute the $L^{1}$-spectrum of an operator over a noncompact
manifold (instead of its $L^{2}$-spectrum) due to the fact that
$L^{1}$-estimates for approximate eigenfunctions are easier to obtain in
comparison to $L^{2}$-estimates. This is also reflected in the fact that the
$L^{2}$-spectrum is in general a subset of the $L^{1}$-spectrum. As a result,
whenever there exists a $p$-independence result for the $L^{p}$-spectrum, one
can determine the $L^{2}$-spectrum by computing instead the $L^{1}$-spectrum
of the operator. Wang in [Wang97]*Thm. 3 was the first one to utilize such a
technique to prove that the $L^{2}$-spectrum of the Laplacian acting on
functions is $[0,\infty)$ (in other words maximal) on a much larger class of
manifolds than was previously known; he took advantage of the $p$-independence
result of Sturm [sturm].

Our first main goal is to find sufficient conditions on the manifold such that
a $p$-independence result holds for a general Dirac operator and its square
for $p\geq 1$. Most of the $p$-independence results known up to now are for
Laplace-type operators. For closed manifolds it is easy to see that the
spectrum is independent of $p$. So the interesting cases are noncompact
manifolds. Hempel and Voigt [HemVo, HemVo2] proved $p$-independence of the
spectrum for Schrödinger operators with potentials admitting certain
singularities on $\mathbb{R}^{n}$. Later Kordyukov [Kordu] obtained such a
$p$-independence result for uniformly elliptic operators with uniformly
bounded smooth coefficients on manifolds of bounded geometry having
subexponential volume growth. Independently, Sturm [sturm] showed the
independence of the $L^{p}$-spectrum for a class of uniformly elliptic
operators in divergence form on manifolds with uniformly subexponential volume
growth and Ricci curvature bounded from below. Both results include the
Laplacian acting on functions. The condition on subexponential volume growth
is an essential one since the $L^{p}$-spectrum of the Laplacian on the
hyperbolic space does depend on $p$ [Davies]*Thm. 5.7.1 and is the convex hull
of a $p$-dependent parabola in the complex plane. Later the first author
proved in [Char1]*Prop. 9 that under the assumptions of the result by Sturm
and whenever the curvature tensor on $k$-forms in bounded below, then the
Hodge-Laplacian acting on $k$-forms has a $p$-independent spectrum. The case
of the drifting Laplacian over a weighted manifold was studied in [ChLu1].

For the $p$-independence of the Dirac operator much less is known. The
classical Dirac operator on hyperbolic spaces also has $p$-dependent spectrum
as was calculated in [AG16]. But a $p$-independence result for the spectrum of
the Dirac operator and its square on nice enough noncompact manifolds was not
previously known. In this paper we will prove a $p$-independence result for
the square Dirac operator on $L^{p}$, which extends to the Dirac operator on
$L^{p}$ whenever the manifold has the right dimension or certain symmetric
structures. In particular, we will prove the following result.

######  Theorem 1.1.

Let $M^{n}$ be a complete Riemannian manifold with Clifford bundle $S$ and
associated Dirac operator $D$. Suppose that the Ricci curvature of $M$ and the
Clifford contraction $\mathcal{R}$ in (1) are both bounded below, and the
volume of $M$ grows uniformly subexponentially. Then the $L^{p}$-spectrum of
the square Dirac operator acting on $L^{p}$, $D_{p}^{2}$, is independent of
$p$ for all $p\in[1,\infty]$. Moreover, the set of isolated eigenvalues of
finite multiplicity and their algebraic multiplicity is independent of $p$ for
all $p\in[1,\infty]$.

If additionally $\sigma(D_{2})$ is symmetric and
$\sigma(D_{1})\neq\mathbb{C}$, then the $L^{p}$-spectrum of $D_{p}$ is
independent of $p$ for all $p\in[1,\infty)$.

Here the lower index on the operator refers to the $L^{p}$-spaces the operator
acts on, see Section 2 for the precise definitions.

We make the assumption $\sigma(D_{1})\neq\mathbb{C}$ so that we can compare
the $L^{p}$-spectrum of $D_{p}$ with that of $D_{p}^{2}$ (see Section 5 for
further details). We believe that this assumption is not necessary in general
or it can be replaced by more geometric assumptions, but we hope to address
this issue in future work. For the particular case of the classical Dirac
operator we will show:

######  Corollary 1.2.

Let $M^{n}$ be a spin Riemannian manifold with classical Dirac operator
$\not{D}$. Suppose that the Ricci curvature of $M$ is bounded below, and the
volume of $M$ grows uniformly subexponentially. Then the $L^{p}$-spectrum of
$\not{D}_{p}^{2}$ is independent of $p$ for all $p\in[1,\infty]$.

If additionally $\sigma(\not{D}_{1})\neq\mathbb{C}$ and if $n\neq 3$ modulo
$4$ or $M$ has an orientation-reserving isometry that lifts to spin structure,
then the $L^{p}$-spectrum of $\not{D}_{p}$ is independent of $p$ for all
$p\in[1,\infty)$.

Part of the machinery used for the $p$-independence result by Sturm or the
first author are upper estimates for the heat kernel as in [saco]. In order to
prove Theorem 1.1 we adapt those semigroup methods in order to be able to
apply them to the square of the Dirac operator $D^{2}$. One difference is that
the semigroup to $D^{2}$ is no longer a contraction. However, under our
curvature assumption it will still be dominated by the semigroup associated to
the Laplacian plus a certain constant. This allows us to use techniques
developed in [HSU] and make the semigroup method work. The
$L^{p}$-independence of the Dirac operator is also of significance. We prove
that it can occur over manifolds with certain symmetries, and relies on
proving that the generator of the semigroup coincides with the square of the
Dirac operator on $L^{p}$. The proofs of Theorem 1.1 and Corollary 1.2 are
found in Sections 2-5.

Our second main goal is to find large classes of manifolds where the
$L^{2}$-spectrum of the Dirac operator and its square is maximal, but also
where the $L^{p}$-spectrum is contained in the real line. Previously, not many
examples where the $L^{2}$ spectrum of a Dirac operator is maximal were known,
and they were mainly in settings with very particular requirements on the
metric, e.g. by Kawai in [Kaw1, Kaw2] and Bär in [Barhyp], and only for the
classical Dirac operator. Kawai considered spin manifolds with an end, whose
metric can be written in radial polar coordinates outside a compact set such
that the sectional curvature decays in absolute value at a rate $cr^{-2},$
where $r$ is the radial distance function to a fixed point. By constructing
approximate eigenspinors he proved that $\sigma(\not{D}_{2})=\mathbb{R}$ on
these manifolds. Bär considered hyperbolic manifolds with finite volume. With
our the $L^{p}$ independence result, but also via the generalized Weyl
criterion of the first author and Lu, we are able to expand the collection of
manifolds where one can prove that the $L^{2}$-spectrum of a general Dirac
operator and its square is maximal, and coincides with the $L^{p}$-spectrum
for all $p\in[1,\infty)$.

In Section 6 we find two very general classes of manifolds where the
$L^{p}$-spectrum of the square Dirac operator is the nonegative real line for
all $p\in[1,\infty]$, by using the $L^{p}$-independence result of the spectrum
and computing instead the $L^{1}$-spectrum. One example treats the classical
Dirac operator over manifolds with an end in a special warped product form,
and the other manifolds with asymptotically nonnegative Ricci curvature that
admit asymptotically $D^{2}$-harmonic spinors. Using Theorem 1.1 we are also
able to find large classes of manifolds where the $L^{2}$ spectrum of the
Dirac operator is maximal, in other words it is $\mathbb{R}$, but also where
$\sigma(D_{p})=\mathbb{R}$ for all $p\in[1,\infty)$. Our examples provide
broad classes of manifolds where the $L^{2}$-spectrum of Dirac operators and
their square is maximal.

In Section 7 we use the generalized Weyl criterion by the first author and Lu
[ChLu2] to find another large class of manifolds where the $L^{2}$ spectrum of
the Dirac operator and its square is maximal. We consider manifolds with
asymptotically nonnegative Ricci curvature in the radial direction, which
admit an asymptotically $D^{2}$-harmonic spinor. We also consider
asymptotically flat manifolds in the sense of Bartnik, since they admit a
Witten spinor that can be used to produce test eigenspinors for the classical
square Dirac operator. Finally in Appendix A we prove the technical result
that the generator of the heat semigroup of the Dirac operator on $L^{p}$
coincides with the square of the Dirac operator acting on $L^{p}$.

##  2\. Preliminaries

###  2.1. Spinorial Preliminaries

Let $(M^{n},g)$ be a complete Riemannian manifold $(M^{n},g)$. Let $S\to M$ be
a _Clifford bundle_ , in other words it is a bundle of Clifford modules over
$M$ and equipped with a fiberwise Hermitian metric $\langle.,.\rangle$, a
metric connection $\nabla$ on $S$ such that the Clifford action is skew-
adjoint with respect to $\langle.,.\rangle$, and $\nabla$ is compatible with
the Levi-Civita connection on $M$, cf. [Roe]*Def. 3.4. The sections of $S$
will be called _spinors_.

The Riemannian metric induces an $L^{2}$-inner product on spinors over $M^{n}$
by

$(\varphi,\eta)\mathrm{:=}\int_{M}\langle\varphi,\eta\rangle dv$

where $dv$ is the Riemannian measure.

Let $D_{c}$ be the associated Dirac operator defined on compactly supported
smooth spinors $C^{\infty}_{c}(S)$, see [Roe]*Chap. 3. When $M$ is spin and
$S$ is the associated spinor bundle, we will refer to the associated Dirac
operator as the _classical Dirac operator_ denoted by $\not{D}$. Note that if
$M$ is spin, we always assume that a spin structure is fixed and already
chosen. Many of the interesting properties that Dirac operators exhibit follow
from the study of the closely related square Dirac operator, $D^{2}$. By
[Roe]*(3.8) there exists an operator $\mathcal{R}\in\mathrm{End}(S)$ such that
the square Dirac operator satisfies the following Weitzenböck formula

(1) $\displaystyle D^{2}=\nabla^{*}\nabla+\mathcal{R}.$

The operator $\mathcal{R}$ is known as the _Clifford contraction_ and acts as
a tensor on spinors. This decomposition allows us to consider the square Dirac
operator as a Schrödinger type operator, and hence obtain useful heat kernel
bounds depending on the behavior of the potential-like term $\mathcal{R}$. For
the classical Dirac operator $\not{D}$ the above Weitzenböck formula is also
referred to as a Schrödinger-Lichnerowicz formula.

######  Example 2.1.

  1. (i)

In the case that $M$ is spin, $S$ is the associated spinor bundle and
$\not{D}$ is the classical Dirac operator, then $\mathcal{R}=\mathrm{scal}/4$
where $\mathrm{scal}$ is the scalar curvature of $M^{n}$. Recall that the
scalar curvature of $M^{n}$ is the trace of the Ricci tensor.

  2. (ii)

The complexified exterior bundle of an oriented Riemannian manifold equipped
with its natural metric and connection admits the structure of a Clifford
bundle, see [Roe]*Ex. 3.19. In this setting the associated Dirac operator is
the de Rham operator $d^{*}+d$. Then $\mathcal{R}$ is determined by the
Riemannian curvature tensor of the underlying manifold, which in the case of
one-forms simplifies to the Ricci curvature tensor [BGV]*p. 130.

  3. (iii)

For complex manifolds $M$ the exterior bundle of the complexified tangent
bundle carries the structure of a Clifford bundle, see [BGV]*p. 135. In this
setting smooth spinors are the smooth differential forms on $M$, and the
associated Dirac operator is the Dolbeaut operator
$\sqrt{2}(\bar{\partial}+\bar{\partial}^{*})$ plus an endomorophism. In the
case of Kähler manifolds this endomorophism vanishes and then the Dirac
operator coincides with the Dolbeaut operator.

###  2.2. Spectral preliminaries

For an operator $H$ on a Banach space $\mathcal{B}$, we say that a point
$\lambda\in\mathbb{C}$ is a point in the _resolvent set_ of $H$, $\rho(H)$, if
$H-\lambda I$ has a bounded inverse on $\mathcal{B}$. If $\lambda\in\rho(H)$,
then $(H-\lambda I)^{-1}$ is the _resolvent operator_. The complement of the
resolvent set in $\mathbb{C}$ is the _spectrum_ of the operator, $\sigma(H)$.
In the general case the domain of an operator $H$ will be denoted
$\text{dom}(H)$.

######  Definition 2.2.

For $p\in[1,\infty)$ we denote by $D_{p}$ the completion of $D_{c}\colon
C_{c}^{\infty}(S)\subset L^{p}(S)\to C_{c}^{\infty}(S)\subset L^{p}(S)$ with
respect to the graph norm as an operator from $L^{p}$ to $L^{p}$. To simplify
notation, we will often write $D_{p}\colon L^{p}\to L^{p}$.

We will also refer to the Dirac operator as $D$ when the domain and context
are clear.

##  3\. Semigroup Properties

In this section we collect properties of the $L^{p}$-semigroup associated to a
general square Dirac operator $D^{2}$. Throughout this section $S$ is a
Clifford bundle over a complete Riemannian manifold $(M,g)$ and $D$ is the
associated Dirac operator. From now on we denote the bound of an operator
$A\colon L^{p}\to L^{q}$ by $\|A\|_{p\to q}$.

Denote by $\Delta_{2}$ the nonnegative definite Laplacian on the space of
$L^{2}$-integrable functions. The $L^{p}$-independence of the spectrum of
$D^{2}_{{p}}$ relies on specific estimates for its heat kernel. The following
generalized Kato inequality between $D^{2}$ and the operator $\Delta-K_{1}$
over $C_{c}^{\infty}(S)$, whenever the Clifford contraction is bounded below,
is a key ingredient for eventually proving the above estimates.

######  Lemma 3.1.

Let $D$ be a Dirac operator over a Clifford bundle $S$. Suppose that the
Clifford contraction $\mathcal{R}$ in (1) satisfies $\mathcal{R}\geq-K_{1}$
for some $K_{1}\geq 0$. Then for any smooth compactly supported section of the
Clifford bundle $\varphi\in C_{c}^{\infty}(S)$ we have the pointwise estimate

$\displaystyle{\rm Re}\langle D^{2}\varphi,\varphi\rangle\geq|\varphi|(\Delta-
K_{1})|\varphi|.$

###### Proof.

By the Weitzenböck formula we have $D^{2}=\nabla^{*}\nabla+\mathcal{R}$. Using

$\displaystyle\langle\nabla^{*}\nabla\varphi,\varphi\rangle=$
$\displaystyle\frac{1}{2}\Delta|\varphi|^{2}+|\nabla\varphi|^{2}=|\varphi|\Delta|\varphi|-|d|\varphi||^{2}+|\nabla\varphi|^{2}$

and the Kato inequality $|d|\varphi||\leq|\nabla\varphi|$ , we obtain

$\displaystyle\rm{Re}\langle D^{2}\varphi,\varphi\rangle$
$\displaystyle=\rm{Re}\langle\nabla^{*}\nabla\varphi,\varphi\rangle+\langle\mathcal{R}(\varphi),\varphi\rangle$
$\displaystyle=|\varphi|\Delta|\varphi|-|d|\varphi||^{2}+|\nabla\varphi|^{2}+\langle\mathcal{R}(\varphi),\varphi\rangle\geq|\varphi|(\Delta-
K_{1})|\varphi|$

given our lower bound on the Clifford contraction $\mathcal{R}$. ∎

We denote by $D_{2}^{2}$ the unique self-adjoint extension of the square Dirac
operator on $L^{2}(S)$. Lemma 3.1 together with the same calculation as in the
proof of [HSU]*Thm. 3.3 shows that $D_{2}^{2}$ and $\Delta_{2}-K_{1}$ satisfy
Kato’s inequality in the sense of condition $(e)$ of Theorem 2.15 in [HSU],
where in our case $H=D_{2}^{2},$ $\ \mathcal{H}=L^{2}(S)$, $\
K=\Delta_{2}-K_{1}$, $\ \mathcal{K}=L^{2}(M)$ and $\
\mathcal{K}^{+}=\overline{C_{c}^{\infty}(M,\mathbb{R}_{\geq 0})}^{L^{2}}$. The
following result is then immediate from $(a)$ in the same theorem.

######  Proposition 3.2.

Suppose that $\mathcal{R}\geq-K_{1}$ for some $K_{1}\geq 0$. Then the
semigroup $e^{-t(\Delta_{2}-K_{1})}$ on $L^{2}(M)$ dominates the semigroup
$e^{-tD_{2}^{2}}$, in other words for any
$\varphi\in\mathrm{dom}(D_{2}^{2})\subset L^{2}(S)$

$\left|e^{-tD_{2}^{2}}\varphi\right|\leq
e^{-t(\Delta_{2}-K_{1})}\left|\varphi\right|.$

Proposition 3.2 is essential for the remaining results of this section. First
of all, given that we have a complete Riemannian manifold with Ricci curvature
bounded below, the heat operator $e^{-t\Delta}$ is a Markov semigroup which is
a contraction on $L^{p}$ for all $1\leq p\leq\infty$ [Dav2]. Combining this
fact with Proposition 3.2 we immediately get the following lemma.

######  Lemma 3.3.

Under the assumptions of Proposition 3.2 the heat operator

$e^{-tD^{2}}\colon L^{2}(S)\cap L^{p}(S)\to L^{2}(S)\cap L^{p}(S)$

can be extended to a bounded operator on $L^{p}\to L^{p}$ for all
$p\in[1,\infty)$. In the case $p=\infty$, $e^{-tD^{2}}$ is defined as the dual
of the operator on $L^{1}$. Even though it is not necessarily a contraction on
$L^{p}$, for all $t\geq 0$ and $p\in[1,\infty]$ we have the upper bound

$\|e^{-tD^{2}}\|_{p\to p}\leq e^{K_{1}t}.$

We would like to remark that this result is a consequence of Kato’s inequality
for $D^{2}$ which essentially implies that whenever $\mathcal{R}\geq-K_{1}$,

$\int_{M}\langle d|\varphi|,d|\varphi|\rangle\,dv\leq\int_{M}\langle
D\varphi,D\varphi\rangle\,dv+K_{1}\int_{M}|\varphi|^{2}dv.$

for all $\varphi\in C_{c}^{\infty}(S)$. For the Laplacian on functions, one
can prove Lemma 3.3 with $K_{1}=0$ in a more direct way, using Davies
[Davies]*Theorems 1.3.2, 1.3.3 because it is a real-valued operator and
satisfies the pointwise Kato inequality $|\,d|u|\,|\leq|\nabla u|(=|du|)$.

Whenever the Clifford contraction is bounded below, Lemma 3.3 allows us to
define the infinitesimal generator, $H_{p}$, of the semigroup $e^{-tD^{2}}$ on
$L^{p}$ for all $1\leq p\leq\infty$. In Appendix A we prove that for $1\leq
p<\infty$ the infinitesimal generator $H_{p}$ coincides—as one expects—with
the square of the Dirac operator $D_{p}^{2}$ on $L^{p}$. Note that it is still
unclear whether $H_{\infty}$ coincides with $(D_{1}^{*})^{2}$ in general. To
simplify notation we will denote $H_{p}$ by $D_{p}^{2}$ for all $1\leq
p\leq\infty$ from now on.

Proposition 3.2 also allows us to prove the existence of an integral kernel
for the operator $e^{-tD_{2}^{2}}$ and obtain a Gaussian-type upper bound for
it, whenever such an upper bound holds for the Laplacian on functions. We can
prove the following.

######  Proposition 3.4.

Consider a complete Riemannian manifold $M^{n}$ with Ricci curvature bounded
below $\rm{Ric}\geq-K_{0}$ for some $K_{0}\geq 0$. Let $S\to M$ be a Clifford
bundle with a Dirac operator $D$ that satisfies (1) and assume that the
Clifford contraction is bounded below, namely $\mathcal{R}\geq-K_{1}$ for some
$K_{1}\geq 0$. Then the heat kernel, $\vec{H}(x,y,t)$, corresponding to the
square Dirac operator $D_{2}^{2}$ exists, and for any $x,y\in M$, $t>0$ and
$\delta\in(0,1)$ it satisfies the upper bound

(2)
$\begin{split}\left|\vec{H}(x,y,t)\right|\leq\;C_{3}(\delta,n)&\;V^{-1/2}(x,\sqrt{t})\,V^{-1/2}(y,\sqrt{t})\\\
&\cdot\exp\left[-\,\frac{d^{2}(x,y)}{C_{4}(\delta,n)\,t}+C_{5}(n)\sqrt{K_{0}\,t}+K_{1}t\,\right]\end{split}$

for some positive constants $C_{3}(\delta,n),C_{4}(\delta,n)$ and $C_{5}(n).$

In the above estimate $V(x,r)$ denotes the Riemannian volume of the geodesic
ball of radius $r$ centered at $x$, and $d(x,y)$ is the distance between $x,y$
induced by the Riemannian metric.

###### Proof.

Our lower bound on Ricci curvature implies that there exists a unique heat
kernel $h(x,y,t)$ for the Laplacian on functions, [Chavel]*p. 188/189 with a
Gaussian type upper bound given in [saco]*Theorem 6.1. The pointwise
domination property of Lemma 3.3 also implies the existence of a heat kernel
for $D_{2}^{2}$, $\vec{H}(x,y,t)$. By following the proof of Theorem 3.5 in
[Ros], this domination property can be used to show

$\left|\vec{H}(x,y,t)\right|\leq\left|h(x,y,t)\,e^{K_{1}t}\right|.\qed$

We refer the interested reader to the book [BGV]*Chapter 2 for a formal
definition of the heat semigroup for operators such as $D_{2}^{2}$ over
Clifford bundles, as well as a detailed construction of its heat kernel in the
case the manifold is compact.

Note that through the same argument one also has the following.

######  Corollary 3.5.

Suppose that $\not{D}$ is the classical Dirac operator over a complete
Riemannian spin manifold. Suppose that the scalar curvature of $M$ is bounded
below by $\mathrm{scal}\geq-4K_{1}$ for some $K_{1}\geq 0$. Then the self-
adjoint extension of $\not{D}^{2}$ on $L^{2}(S)$ satisfies the domination
property of Proposition 3.2 and the upper bound of Lemma 3.3 for its
semigroup. Moreover, whenever the Ricci curvature of $M$ is bounded below
$\mathrm{Ric}\geq-K_{0}$ for some $K_{0}\geq 0$, then the heat kernel of the
semigroup $e^{-t\not{D}_{2}^{2}}$ exists and satisfies the Gaussian upper
bound of Theorem 3.4 with $K_{1}=\frac{n}{4}K_{0}$.

##  4\. $L^{p}$-Independence of the Spectrum of $D_{p}^{2}$

In this section we prove the main part of Theorem 1.1 – the $p$-independence
of the spectrum of $D^{2}_{p}$. We will prove the second part of the theorem
concerning the spectrum of $D_{p}$ in the next section.

We begin with some preliminary definitions and notations. Recall that
$\rho(D^{2}_{p})$ denotes the resolvent set of $D^{2}_{p}$, and
$\sigma(D^{2}_{p})=\mathbb{C}\setminus\rho(D^{2}_{p})$ is its spectrum. We
will show that under the assumptions of Theorem 1.1 the resolvent set is
independent of $p$. For this, it suffices to show that
$\rho(D^{2}_{p})\subset\rho(D^{2}_{2})$ and
$\rho(D^{2}_{2})\subset\rho(D^{2}_{p})$ for all $p$. The first containment is
the one that holds under more restrictive geometric assumptions, whereas the
other one holds in a more general setting as we will see.

We note that in general $\sigma(D^{2}_{p})=\sigma(D^{2}_{q})$ whenever
$1/p+1/q=1$ by duality even in the case $p=1$, since
$D^{2}_{\infty}=H_{\infty}$ is by definition the dual operator to $D^{2}_{1}$.

On $M^{n}$ we denote as $B(x,r)$ the geodesic ball of radius $r$ centered at
$x$, and the Riemannian volume of $B(x,r)$ is again denoted as $V(x,r)$. The
following property is key when proving an $L^{p}$-independence result for the
spectrum.

######  Definition 4.1.

A Riemannian manifold is said to have _uniformly subexponential volume growth_
, if for any $\varepsilon>0$ there exists a constant $C{(\varepsilon)}$,
independent of $x$, such that for any $r>0$ and $x\in M$

$V(x,r)\leq C{(\varepsilon)}\,V(x,1)\,e^{\varepsilon\,r}.$

Define $\mu(x)\mathrm{:=}V(x,1)^{-1/2}$. The assumption on the uniformly
subexponential volume growth implies in particular the following estimate:

######  Lemma 4.2.

[sturm]*Proposition 1 Let $M^{n}$ be a Riemannian manifold whose volume grows
uniformly subexponentially. Then for any $\beta>0$

$\sup_{x\in M}\int_{M}\mu(x)\,\mu(y)\,e^{-\beta\,d(x,y)}\,dv(y)<\infty,$

where $d$ is the distance function on $(M,g)$.

To prove the $L^{p}$-independence of the spectrum of $D_{p}^{2}$, we will use
the heat kernel estimate technique that was introduced by [sturm], although we
need to be a bit more careful with our arguments as our operator is defined on
spinors. A set of auxiliary results is necessary, which we outline below.

We begin by defining the set of functions

$\Psi_{\varepsilon}\mathrm{:=}\\{\psi\in{\mathcal{C}}_{0}^{1}(M)\,|\,|\nabla\psi|\leq\varepsilon\\}$

where ${\mathcal{C}}_{0}^{1}(M)$ are compactly supported $C^{1}$-functions on
$M$. The set $\Psi_{\varepsilon}$ will be used to define perturbations of the
operator. Note that for any pair of points $x,y\in M$,

$\displaystyle\sup\\{|\psi(x)-\psi(y)|\,|\,\psi\in\Psi_{\varepsilon}\\}\leq\varepsilon\,d(x,y).$

We consider perturbations of the operator $D^{2}_{2}$ of the form
$e^{\psi}D^{2}_{2}e^{-\psi}$. The next lemma is an analytic result that is
independent of the geometric conditions on the manifold.

######  Lemma 4.3.

For any compact subset $K$ of the resolvent set of $\rho(D^{2}_{2})$ there
exist $\varepsilon>0$ and $C<\infty$ such for all $\xi\in K$ and
$\psi\in\Psi_{\varepsilon}$, $\xi$ belongs to the resolvent set of the
operator $e^{\psi}D^{2}_{2}e^{-\psi}$ and

$\|(e^{\psi}D^{2}_{2}e^{-\psi}-\xi)^{-1}\|_{2\to 2}\leq C.$

###### Proof.

The proof follows as in [sturm]*Lem. 1 since the computation does not change
when taking sections in a Clifford bundle and $H_{2}=D_{2}^{2}$. In
particular, the proof uses Theorem VI.3.9 in [Kato] to get a uniform bound for
the resolvent operator of these perturbations over compact sets of the
resolvent set of $D^{2}_{2}$. ∎

Our next goal is to prove the $L^{p}$-boundedness of some power of the
resolvent operator, $(D^{2}_{2}-\xi)^{-m}$, whenever $\xi$ belongs to
$\rho(D^{2}_{2})$. This requires the geometric assumptions on the manifold as
it relies on upper Gaussian estimates for the heat kernel of $D^{2}_{2}$ and
volume comparison estimates. These lead to the following strong result about
the heat kernel.

######  Lemma 4.4.

Consider a complete Riemannian manifold $M^{n}$ and a Dirac operator $D$ on a
Clifford bundle over $M$ which satisfies (1). Suppose that the Clifford
contraction and Ricci curvature are bounded below such that $\mathcal{R}\geq-
K_{1}$ and $\mathrm{Ric}\geq-K_{0}$, for some $K_{1},K_{0}\geq 0$. Then for
any $\beta>0$ there exists a negative number $\alpha$ and constant
$C(n,K_{0},\beta)$ such that the heat kernel of $D^{2}_{2}$ satisfies the
upper bound

(3) $\left|\vec{H}(x,y,t)\right|\leq
C\,\mu(x)^{2}\,\sup\\{t^{-n/2},1\\}\,e^{-\beta\,d(x,y)}\,e^{-(\alpha+1)t}.$

###### Proof.

The proof of the lemma uses the Gaussian upper bound that we proved in
Proposition 3.4, and the assumption on Ricci curvature (see for example
[Char1, sturm, ChLu1] for similar estimates).

By the Bishop volume comparison Theorem, whenever $\rm{Ric}\geq-K_{0}$, there
exists $C=C(n)$ such that

(4) $\frac{V(x,R)}{V(x,r)}\leq\,\frac{R^{n}}{r^{n}}\,e^{C\,\sqrt{K_{0}}\,R}$

for any $0<r\leq R$.

Taking $r<1=R$, we have

$V(x,r)^{-1}\leq\mu(x)^{2}\,r^{-n}\,C(n,K_{0}),$

whereas

$V(x,r)^{-1}\leq\mu(x)^{2}$

holds for any $r\geq 1$ by monotonicity of the volume. Therefore, for any
$r>0$

(5) $V(x,r)^{-1}\leq\mu(x)^{2}\sup\\{r^{-n},1\\}\,C(n,K_{0}),$

The volume comparison (4) also yields

(6) $\mu(y)^{2}\leq C\,\mu(x)^{2}\,e^{\bar{C}\,(\sqrt{K_{0}}+1)\,d(x,y)}$

for all $x,y\in M$ and constants $C,\bar{C}$ that only depend on $n$.

Moreover, for any $\gamma\in\mathbb{R}$, the exponential function satisfies
the inequality

(7) $e^{-d^{2}/4ct}\leq e^{-\gamma\,d}\,e^{c\,\gamma^{2}t}.$

Using (5), (6) and (7) we can estimate the right side of (2) to get

$\begin{split}\left|\vec{H}(x,y,t)\right|&\leq\;C\,\mu(x)^{2}\,\sup\\{t^{-n/2},1\\}\;\\\
&\cdot\,\exp\left[\left(\bar{C}\,(\sqrt{K_{0}}+1)-\gamma\right)d(x,y)+C_{5}\sqrt{K_{0}\,t}+K_{1}\,t+\frac{1}{4}\gamma^{2}C_{4}\,t\,\right].\end{split}$

Choosing $\gamma$ such that $\beta=-\bar{C}\,(\sqrt{K_{0}}+1)+\gamma$ is any
positive number, we get the desired estimate with
$\alpha=-1-C_{5}\sqrt{K_{0}}-K_{1}-\frac{1}{4}\gamma^{2}C_{4}.$ ∎

We are now ready to state and prove the $L^{p}$-boundedness of the resolvent
operator. The proof is a standard analytical argument which we include for
completion.

######  Proposition 4.5.

Consider a complete Riemannian manifold $M^{n}$ and $D$ a Dirac operator on a
Clifford bundle over $M$ which satisfies (1). Suppose that the Clifford
contraction and Ricci curvature are bounded below such that $\mathcal{R}\geq-
K_{1}$ and $\rm{Ric}\geq-K_{0}$, for some $K_{1},K_{0}\geq 0$. Assume moreover
that the volume of $M$ grows uniformly subexponentially.

Let $K$ be a compact subset of $\rho(D^{2}_{2})$. Then for any $\xi\in K$, and
for any $m>n+2$, $(D^{2}_{2}-\xi)^{-m}$ has a smooth kernel function
$\vec{G}_{\xi}(x,y)$ satisfying

(8) $|\vec{G}_{\xi}(x,y)|\leq C\mu(x)\mu(y)e^{-\varepsilon d(x,y)}$

for some $\varepsilon>0$ and $C>1$.

In addition, the operator $(D^{2}_{2}-\xi)^{-m}$ is bounded from $L^{p}$ to
$L^{p}$ for all $1\leq p\leq\infty$.

Dunford and Pettis proved that an operator on functions which is bounded from
$L^{1}$ to $L^{\infty}$ has an integral kernel. In [Char1]*Lem. 11 the first
author showed that an integral kernel exists for an operator on the $k$-form
bundle $\Lambda^{k}(M)$ which is bounded from $L^{1}$ to $L^{\infty}$. By
following the same argument, but for a Clifford bundle, we can also show that
an operator has an integral kernel whenever it is bounded from $L^{1}$ to
$L^{\infty}$:

###### Proof.

In order to prove (8) it suffices to show that the perturbed resolvent
operator $\mu^{-1}\,e^{\psi}(D_{2}^{2}-\xi)^{-m}e^{-\psi}\mu^{-1}$ is bounded
from $L^{1}$ to $L^{\infty}$ for all $\psi\in\Psi_{\varepsilon}$ for some
$\varepsilon>0$. To this end, we use powers of the resolvent equation
[Kato]*5.5 on p. 36 to get

(9)
$\begin{split}\mu^{-1}\,e^{\psi}(D^{2}_{2}-\xi)^{-m}e^{-\psi}\mu^{-1}&=\sum_{j=0}^{m}\left(\begin{array}[]{c}m\\\
j\\\
\end{array}\right)(\xi-\alpha)^{j}[\mu^{-1}\,e^{\psi}(D^{2}_{2}-\alpha)^{-m/2}e^{-\psi}]\\\
&\qquad\cdot[e^{\psi}(D^{2}_{2}-\xi)^{-1}e^{-\psi}]^{j}\;[e^{\psi}(D^{2}_{2}-\alpha)^{-m/2}e^{-\psi}\mu^{-1}]\end{split}$

where $\alpha$ is a negative constant. The resolvents on the right side of the
equations commute. But this splitting is convenient in order to show that this
actually is a bounded operator from $L^{1}$ to $L^{\infty}$, see below.

On the right side, we have in essence three operators. First note that by
Lemma 4.3, the middle operator is bounded on $L^{2}$. In other words there
exists an $\varepsilon,C>0$, depending on $\xi$, such that for all
$\psi\in\Psi_{\varepsilon}$

(10) $\|e^{\psi}(D^{2}_{2}-\xi)^{-1}e^{-\psi}\|_{2\to 2}\leq C.$

Since $\alpha<0$, the operator $(D^{2}_{2}-\alpha)^{-m/2}$ satisfies the
identity (see [EN]*p.57)

$(D^{2}_{2}-\alpha)^{-m/2}=C(m)\int_{0}^{\infty}e^{-t\,D^{2}_{2}}\,t^{\frac{m}{2}-1}\,e^{\alpha
t}\,dt.$

By Proposition 3.4 the heat operator has a kernel. Hence, the integrability of
the right hand side of the last equation guarantees the existence of a kernel
for the resolvent operator $(D^{2}_{2}-\alpha)^{-m/2}$ as well. We denote this
kernel by $\vec{G}^{m/2}_{\alpha}(x,y)$. By Lemma 4.4 we know that right side
is integrable whenever $m>n+2$ and for $\alpha$ sufficiently negative. More
specifically, for any $\beta>0$ and $m>n+2$ we can find an $\alpha$
sufficiently negative such that

$|\vec{G}^{m/2}_{\alpha}(x,y)|\leq
C(n,K_{0},\beta)\,\mu(x)^{2}e^{-\beta\,d(x,y)}.$

For any $\omega\in L^{2}$, if we use the estimate (6), Lemma 4.2 and choose
$\beta>0$ large enough, we can see that

$\begin{split}\|\mu^{-1}\,e^{\psi}(D_{2}^{2}-\alpha)^{-m/2}&e^{-\psi}\omega\|_{L^{\infty}}=\sup_{y}\mu(y)^{-1}\,e^{\psi(y)}\bigl{|}\int_{M}\vec{G}^{m/2}_{\alpha}(x,y)e^{-\psi(x)}\,\omega(x)\,dv(x)\bigr{|}\\\
&\leq
C\sup_{y}\bigl{|}\int_{M}\mu(y)^{-2}\,\mu(x)^{2}e^{2(\varepsilon-\beta)d(x,y)}\,dv(x)\bigr{|}^{1/2}\,\|\omega(x)\|_{L^{2}}\\\
&\leq C\|\omega(x)\|_{L^{2}}.\end{split}$

In other words,

$\|\mu^{-1}\,e^{\psi}(D^{2}_{2}-\alpha)^{-m/2}e^{-\psi}\|_{2\to\infty}\leq C.$

By taking adjoints, we get that the third operator in the right side of (9) is
bounded from $L^{1}$ to $L^{2}$,

$\|e^{\psi}(D^{2}_{2}-\alpha)^{-m/2}e^{-\psi}\mu^{-1}\|_{1\to 2}\leq C.$

Combing the three estimates for the operators on the right side of (9), we get
that $\mu^{-1}\,e^{\psi}(D^{2}_{2}-\xi)^{-m}e^{-\psi}\mu^{-1}$ is bounded from
$L^{1}$ to $L^{\infty}$. Hence it has an integral kernel. The pointwise bound
on $\vec{G}^{m}_{\xi}(x,y)$ follows.

Lemma 4.2 now implies that $(D^{2}_{2}-\xi)^{-m}$ is a bounded operator on
$L^{1}$. By the definition of $\xi$ it is also bounded on $L^{2}$. Using a
standard interpolation argument, it follows that it is bounded on $L^{p}$ for
any $1\leq p\leq 2$. Taking adjoints we get that it is bounded for any $1\leq
p\leq\infty$. ∎

We are now ready to prove the $L^{p}$-independence for $\sigma(D_{p}^{2})$:

###### Proof of the $p$-independence of $\sigma(D^{2}_{p})$ in Theorem 1.1.

We will first prove the inclusion $\sigma(D^{2}_{p})\subset\sigma(D^{2}_{2})$
which is the most difficult one to hold.

By Proposition 4.5, with $m=n+3$, the operator $(D^{2}_{2}-\xi)^{-m}$ is
bounded on $L^{p}$ for all $p\in[1,\infty]$ and for all
$\xi\in\rho(D^{2}_{2})$. Given that $\sigma(D^{2}_{2})$ is contained in the
nonnegative real line, any $\xi\in\rho(D^{2}_{2})$ can be path connected to a
negative number. By uniqueness of extension we conclude that
$(D^{2}_{p}-\xi)^{-m}=(D^{2}_{2}-\xi)^{-m}$ for all $\xi\in\rho(D^{2}_{2})$
(see also [HemVo, HemVo2, sturm, Char1, ChLu1] for further details).

Recall that the spectrum $D^{2}_{p}$ is closed for all $p$. If
$\sigma(D^{2}_{p})$ is not contained in $\sigma(D^{2}_{2})$, then we can find
a point $\xi_{o}$ on the boundary of $\sigma(D^{2}_{p})$ which does not belong
to $\sigma(D^{2}_{2})$. Moreover, for any $\varepsilon>0$, we can find $m-1$
distinct points $\xi_{j}\in\rho(D^{2}_{p})$ for $1\leq j\leq m-1$ with
$|\xi_{j}-\xi_{0}|<\epsilon$. Using the resolvent equation [Kato]*Equation
(5.6) p. 37, we get

$\begin{split}&\left((D^{2}_{p}-\xi_{o})(D^{2}_{p}-\xi_{1})\cdots(D^{2}_{p}-\xi_{m-1})\right)^{-1}\\\
&\quad=\sum_{l_{1},\cdots,l_{m-1}=0}^{\infty}(D^{2}_{p}-\xi_{o})^{-m-l_{1}-\cdots-
l_{m-1}}(\xi_{1}-\xi_{o})^{l_{1}}\cdots(\xi_{m-1}-\xi_{o})^{l_{m-1}}.\end{split}$

Recall that $(D^{2}_{2}-\xi_{o})^{-m}$ is bounded on $L^{p}$ by Proposition
4.5. Since $\|(D^{2}_{p}-\xi_{o})^{-m}\|_{p\to
p}=\|(D^{2}_{2}-\xi_{o})^{-m}\|_{p\to p}=C$, then

$\|(D^{2}_{p}-\xi_{o})^{-l}\|_{p\to p}\leq C^{l/m}$

for any $l\geq m$. Choosing $\varepsilon$ small enough, the resolvent
expansion above gives us that
$\left((D^{2}_{p}-\xi_{o})(D^{2}_{p}-\xi_{1})\cdots(D^{2}_{p}-\xi_{m-1})\right)^{-1}$
is bounded on $L^{p}$.

At the same time, by [Kato]*(5.5) on p. 36

$\left((D^{2}_{p}-\xi_{o})(D^{2}_{p}-\xi_{1})\cdots(D^{2}_{p}-\xi_{m-1})\right)^{-1}=\alpha_{o}(D^{2}_{p}-\xi_{o})^{-1}+\sum_{j=1}^{m-1}\alpha_{j}(D^{2}_{p}-\xi_{j})^{-1}$

where $\alpha_{o}\ldots\alpha_{m-1}$ are constants (depending on $\xi_{j}$).
As a result, $(D^{2}_{p}-\xi_{o})^{-1}$ is bounded on $L^{p}$, since every
other operator in the above equation is bounded on $L^{p}$. In other words,
$\xi_{o}$ is in the resolvent set, which is a contradiction.

The other containment, $\sigma(D^{2}_{p})\supset\sigma(D^{2}_{2})$, holds in a
very general setting and only requires an interpolation argument. The idea is
to consider $D^{2}_{p}$ and $D^{2}_{p^{*}}$ where $p,p^{*}$ are dual such that
$\frac{1}{p}+\frac{1}{p^{*}}=1$. Then $\rho(D^{2}_{p})=\rho(D^{2}_{p^{*}})$
and the resolvent operators $(D^{2}_{p}-\xi)^{-1}$, $(D^{2}_{p^{*}}-\xi)^{-1}$
are bounded on $L^{p}$ and $L^{p^{*}}$ respectively for all
$\xi\in\rho(D^{2}_{p})$. Using the Calderón Lions Interpolation Theorem
[ReSiII]*Theorem IX.20 and the argument by Hempel and Voigt [HemVo2], we get
that the interpolated bounded operator on $L^{2}$ is in fact
$(D^{2}_{2}-\xi)^{-1}$. Therefore $\xi\in\rho(D^{2}_{2})$.

The proof of the $L^{p}$-independence of isolated eigenvalues of finite
multiplicity follows similarly as in [HemVo]*Proposition 3.1 (see also [Aut,
Char1]). The main idea is to consider the Laurent series expansion for the
resolvent operator $(D^{2}_{p}-\xi)^{-1}$ on a small disc around the
eigenvalue $\lambda$ of finite multiplicity, which contains no other points in
the spectrum other than $\lambda$. The order of the pole at $\lambda$
corresponds to the multiplicity of the eigenvalue. The $L^{p}$-independence of
the spectrum allows us to show that the Laurent series expansions are the same
for all $p$, and in consequence the order of the pole is also the same. ∎

##  5\. The $L^{p}$ spectrum of $D_{p}$

Up to now we examined when the $L^{p}$-spectrum of $D_{p}^{2}$ is independent
on $p$. Of course, it is interesting to know whether the $L^{p}$-spectrum of
$D_{p}$ itself is also independent of $p$.

First by the Riesz-Thorin interpolation theorem
$\sigma(D_{2})\subset\sigma(D_{q})\subset\sigma(D_{p})$ for all $\infty>p\geq
q\geq 2$ or $1\leq p\leq q\leq 2$. Moreover, we note that the proof of Lemma
B.8 in [AG16] stays exactly the same for Dirac operators on arbitrary Clifford
bundles over complete Riemannian manifolds. Hence, we know that if the
spectrum $\sigma(D_{p})$ is not the entire complex plane, then
$\lambda^{2}\in\sigma(D^{2}_{p})$ if and only if $\lambda$ or $-\lambda$
belongs to $\sigma(D_{p})$. This is the only statement where
$\sigma(D_{p})\neq\mathbb{C}$ is used. We think that for the manifolds
considered in Theorem 1.1 this is actually automatically the case but up to
now we were not able to prove this in full generality. We hope to address such
questions in future work. If $(M,g)$ and the Clifford bundle are of bounded
geometry, the assumption $\sigma(D_{1})\neq\mathbb{C}$ is automatically
fulfilled, see [AG16]*Rem. B9 and cp. Lemma B.1.

###### Proof of Theorem 1.1 – the last statement concerning $\sigma(D_{p})$.

As proved in the last section $\sigma(D_{p}^{2})$ is independent of $p$ – in
particular $\sigma(D_{p}^{2})=\sigma(D_{2}^{2})$. From the last paragraph
above we obtain that whenever $\sigma(D_{1})\neq\mathbb{C}$ then also
$\sigma(D_{p})\neq\mathbb{C}$ for $p<\infty$ and hence
$\lambda^{2}\in\sigma(D^{2}_{p})$ if and only if $\lambda$ or $-\lambda$
belongs to $\sigma(D_{p})$. Now consider a point $\lambda\in\sigma(D_{p})$.
Then, $\lambda^{2}\in\sigma(D_{p}^{2})=\sigma(D_{2}^{2})$. Since
$\sigma(D_{2})$ is assumped to be symmetric, $\lambda\in\sigma(D_{2})$. Hence,
$\sigma(D_{p})\subset\sigma(D_{2})$. Together with the converse inclusion, we
conclude that $\sigma(D_{p})$ is independent of $p$. ∎

###### Proof of Corollary 1.2.

For the special case of the classical Dirac operator [AG16]*Lemmas B.10 and
B.11 gives that for $\text{dim}\,M\neq 3$ modulo $4$, or if $M$ admits an
orientation reversing isometry that lifts to the spin structure, the
$L^{p}$-spectrum of $\not{D}_{p}$ is always symmetric. Hence, we can use the
last theorem which completes the proof of Corollary 1.2. ∎

##  6\. Computation of the spectrum of $D_{p}$ and $D_{p}^{2}$ via computation
of $L^{1}$ spectrum.

In this section, we give two examples (one very concrete and a more abstract
one) where we prove that the $L^{p}$-spectrum of the square Dirac operator is
the nonnegative real line for all $p\in[1,\infty)$. We achieve this by using
the $p$-independence result of Theorem 1.1, and showing that the $L^{1}$\-
spectrum contains the nonnegative real line. This implies that
$\sigma(D^{2}_{2})$ is maximal in these cases. Using the same theorem, we can
also find large classes of manifolds where $\sigma(D_{2})$ is maximal (in
other words equal to $\mathbb{R}$) and where $\sigma(D_{p})=\mathbb{R}$ for
all $p\in[1,\infty)$.

Our first example treats the classical Dirac operator over manifolds with ends
in special warped product form. The second example considers the classical
Dirac operator over manifolds that admit a bounded, asymptotically harmonic
spinor. In both cases the geometric conditions on the structure of the
manifold are such that the $L^{p}$ spectrum of the square Dirac operator is
$p$-independent.

###  6.1. Some manifolds with warped product ends

In this section we want to apply our results to a class of manifolds with
warped product ends.

######  Theorem 6.1.

Let $(\tilde{M},\tilde{g})$ be a complete spin Riemannian manifold of
dimension $n$ which has uniformly subexponential volume growth such that there
exists an isometric embedding
$(M=N\times[0,\infty),g=f(t)^{2}g_{N}+dt^{2})\hookrightarrow(\tilde{M},\tilde{g})$
with $\text{ker}\,\not{D}^{N}\neq\varnothing$. Assume that $f$ is positive,
monotonically increasing with

(11) $\lim_{t\to\infty}\tfrac{f^{n-1}(2t)}{tf^{n-1}(t)}\to 0.$

Then, the $L^{p}$-spectrum of the square of the classical Dirac operator on
$(\tilde{M},\tilde{g})$ is $[0,\infty)$.

If additionally $\mathrm{dim}\,\tilde{M}\neq 3\ \text{mod}\ 4$ or $M$ admits
an orientation preserving isometry that lifts to the spin structure, then, the
$L^{2}$-spectrum of the classical Dirac operator on $(\tilde{M},\tilde{g})$ is
$\mathbb{R}$, it is in other words maximal. If we also know that
$\sigma(\not{D}_{1})\neq\mathbb{C},$ then $\sigma(\not{D}_{p})=\mathbb{R}$ for
all $p\in[1,\infty)$.

Admissible functions $f$ include some functions with quasi-polynomial growth
such as $f^{n-1}(t)=e^{\frac{1}{4}(\ln t)^{2}}$. Also functions with
polynomial growth are admissible in the above theorem. Note that for
polynomial growth functions one could directly compute the $L^{2}$-spectrum,
but not for the general class of $f$ that satisfies (11).

###### Proof.

By [BGM]*(3.6) the Dirac operator on the end $M$ is given by

$\not{D}^{M}=\nu\cdot\left(\tilde{\not{D}}^{N}+\frac{n-1}{2}\frac{f^{\prime}(t)}{f(t)}+\nabla^{M}_{\nu}\right),$

where $\widetilde{\not{D}}^{N}=\not{D}^{N}$ when the dimension of $N$ is even,
$\tilde{\not{D}}^{N}=\text{diag}(\not{D}^{N},-\not{D}^{N})$ otherwise, and
$\nu\mathrm{:=}\partial_{t}$. Let $\phi\in\mathrm{ker}\,\tilde{\not{D}}^{N}$
be such that $\nu\cdot\phi=\mathrm{i}\phi$. This is possible since
$\nu^{2}=-\text{Id}$ and
$\tilde{\not{D}}^{N}(\nu\cdot.)=-\nu\cdot\tilde{\not{D}}^{N}(.)$.

Let $\eta_{T}\colon\mathbb{R}\to[0,1]$ be a smooth cut-off function with
$\text{supp}\,\eta_{T}=[T,4T]$, $\eta_{T}|_{[2T,3T]}=1$ and
$|\eta_{T}^{\prime}|\leq 2/T$. We define spinots $\psi_{T}$ with support in
$M$ by setting
$\psi_{T}(x,t)=\eta_{T}(t)\text{exp}(-\lambda\mathrm{i}t)\phi(x,t)$ for some
$\lambda\in\mathbb{R}$ and all $(x,t)\in M=N\times[0,\infty)$. Then

$\displaystyle(\not{D}^{M}-\lambda)\psi_{T}=$
$\displaystyle\left(\frac{n-1}{2}\frac{f^{\prime}}{f}\eta_{T}+(\eta_{T}^{\prime}-\lambda\mathrm{i}\eta_{T})\right)\text{exp}(-\lambda\mathrm{i}t)\nu\cdot\phi-\lambda\eta_{T}\text{exp}(-\lambda\mathrm{i}t)\phi$
$\displaystyle=$
$\displaystyle\left(\frac{n-1}{2}\frac{f^{\prime}}{f}\eta_{T}+\eta_{T}^{\prime}\right)\text{exp}(-\lambda\mathrm{i}t)\nu\cdot\phi$

which implies

$\displaystyle\frac{\|(\not{D}^{M}-\lambda)\psi_{T}\|_{L^{p}}}{\|\psi_{T}\|_{L^{p}}}\leq\frac{\|\frac{n-1}{2}\frac{f^{\prime}}{f}\eta_{T}\phi\|_{L^{p}}+\|\eta_{T}^{\prime}\phi\|_{L^{p}}}{\|\psi_{T}\|_{L^{p}}}$
$\displaystyle\leq$
$\displaystyle\frac{\left(\int_{N\times[T,4T]}\left(\frac{n-1}{2}\left|\frac{f^{\prime}(t)}{f(t)}\right|\,|\phi|\right)^{p}f^{n-1}(t)\,dt\,dv_{N}\right)^{\frac{1}{p}}+\frac{2}{T}\left(\int_{N\times[T,4T]}|\phi|^{p}\,f^{n-1}(t)dt\,dv_{N}\right)^{\frac{1}{p}}}{\|\phi\|_{L^{p}(N)}\left(\int_{[2T,3T]}f^{n-1}(t)\,dt\right)^{\frac{1}{p}}}$
$\displaystyle\leq$
$\displaystyle\frac{\left(\int_{[T,4T]}|\frac{n-1}{2}|^{p}\left|\frac{f^{\prime}(t)}{f(t)}\right|^{p}f^{n-1}(t)\,dt\right)^{\frac{1}{p}}}{\left(\int_{[2T,3T]}f^{n-1}(t)\,dt\right)^{\frac{1}{p}}}+\frac{2\left(\int_{[T,4T]}f^{n-1}(t)\,dt\right)^{\frac{1}{p}}}{T\left(\int_{[2T,3T]}f^{n-1}(t)\,dt\right)^{\frac{1}{p}}}.$

where we used that
$\int_{N\times[a,b]}|\phi|^{p}F(t)\,dt\,dv_{N}=\|\phi\|_{L^{p}(N)}^{p}\int_{a}^{b}F(t)\,dt$.

For $p=1$ and using that $f$ is nondecreasing and positive this gives

$\displaystyle\frac{\|(\not{D}^{M}-\lambda)\psi_{T}\|_{L^{1}}}{\|\psi_{T}\|_{L^{1}}}\leq\frac{f^{n-1}(4T)-f^{n-1}(T)}{2Tf^{n-1}(2T)}+\frac{6f^{n-1}(4T)}{Tf^{n-1}(2T)}\leq\frac{7f^{n-1}(4T)}{Tf^{n-1}(2T)}.$

Then, (11) implies that $\mathbb{R}\subset\sigma(\not{D}_{1}^{M})$ and hence,
$[0,\infty)\subset\sigma((\not{D}_{1}^{M})^{2})$. Since the spectrum of
$\not{D}^{2}_{p}$ is $p$-independent for $1\leq p<\infty$ by Theorem 1.1, we
have $[0,\infty)\subset\sigma(\not{D}_{2}^{2})$. Since $[0,\infty)$ is the
maximal possible spectrum of the nonnegative selfadjoint operator
$\not{D}_{2}^{2}$, we have equality.

By Lemma B.10 and B.11 of [AG16] if $n$ is not congruent $3$ modulo $4$ or $M$
admits an isometry preserving orientation that lifts to the spin structure,
the $L^{2}$-spectrum of $\not{D}$ is automatically symmetric, which implies
$\sigma(\not{D}_{2})=\mathbb{R}$. Moreover, if we also know that
$\sigma(\not{D}_{1})\neq\mathbb{C},$ then $\sigma(\not{D}_{p})=\mathbb{R}$ for
all $p\in[1,\infty)$, by Corollary 1.2. ∎

###  6.2. Manifolds with asymptotically $D^{2}$-harmonic spinors

As another application we consider manifolds with asymptotically
$D^{2}$-harmonic spinors. This assumption is a relatively general one. In the
following section we will consider the particular case of asymptotically flat
manifolds which admit a Witten spinor.

######  Theorem 6.2.

Let $(M,g)$ be a complete Riemannian manifold with Clifford bundle $S$ and
associated Dirac operator $D$. Suppose that the Clifford contraction
$\mathcal{R}$ in (1) is bounded below and that the Ricci curvature of $M$
satisfies

$\text{Ric}(x)\geq-\delta(n)r^{-2}(x)$

where $r(x)=d(x_{o},x)$ is the distance from $x$ to a fixed point $x_{o}$ and
$\delta(n)$ is a small constant that depends only on the dimension of the
manifold.

Moreover, assume that for all $R>0$ large enough there is a smooth section
$\phi_{R}$ of $S$ restricted to $M\setminus B_{x_{o}}(R)$ with
$0<c_{1}<|\phi_{R}|<c_{2}$ on $M\setminus B_{x_{o}}(R)$, $|\nabla\phi_{R}|\leq
C$, and $\|D^{2}\phi_{R}\|_{L^{\infty}}\to 0$ as $R\to\infty$.

Then, the $L^{p}$-spectrum of $D^{2}_{p}$ is $[0,\infty)$ for all
$p\in[1,\infty]$.

If additionally, $\sigma(D_{1})\neq\mathbb{C}$ and $\sigma(D_{2})$ is
symmetric, then $\sigma(D_{p})=\mathbb{R}$ for all $p\in[1,\infty)$.

###### Proof.

As Wang proves in [Wang97], under the above assumption on the Ricci curvature,
the manifold has uniformly subexponential volume growth, and as a result the
spectrum of $D_{p}^{2}$ is $L^{p}$ independent by Theorem 1.1. It therefore
suffices to prove that the spectrum of $D_{1}^{2}$ contains $[0,\infty)$.

Let $\lambda\in[0,\infty)$. By the estimates of [Wang97]*p. 478 it follows
that for any $R$ large enough there is a function $\eta_{R}\in C^{\infty}(M)$
with support on $M\setminus B_{x_{o}}(R)$, such that

$\|\Delta\eta_{R}-\lambda\eta_{R}\|_{L^{1}}\leq(c/R)\;\|\eta_{R}\|_{L^{1}}\quad\text{and}\quad\|\nabla\eta_{R}\|_{L^{1}}\leq(c/R)\;\|\eta_{R}\|_{L^{1}}.$

where $c$ is a uniform positive constant. Since

(12)
$\begin{split}(D^{2}-\lambda)(\eta_{R}\phi_{R})=&D(\eta_{R}D\phi_{R}+\nabla\eta_{R}\cdot\phi_{R})-\lambda\eta_{R}\phi_{R}\\\
=&\eta_{R}D^{2}\phi_{R}-2\nabla_{\nabla\eta_{R}}\phi_{R}+(\Delta\eta_{R}-\lambda\eta_{R})\phi_{R}\end{split}$

we have the estimate

$\displaystyle\frac{\|(D^{2}-\lambda)(\eta_{R}\phi_{R})\|_{L^{1}}}{\|\eta_{R}\phi_{R}\|_{L^{1}}}\leq$
$\displaystyle\frac{\|\eta_{R}D^{2}\phi_{R}\|_{L^{1}}+2\|\nabla\phi_{R}\|_{L^{\infty}}\,\|\nabla\eta_{R}\|_{L^{1}}+c_{2}\|(\Delta\eta_{R}-\lambda\eta_{R})\|_{L^{1}}}{c_{1}\|\eta_{R}\|_{L^{1}}}$
$\displaystyle\leq$
$\displaystyle\frac{C\left(1/R+\|D^{2}\phi_{R}\|_{L^{\infty}}\right)\|\eta_{R}\|_{L^{1}}}{\|\eta_{R}\|_{L^{1}}}\to
0\ \ \text{ as }\ \ R\to\infty.$

Under our additional assumption that $\sigma(D_{1})\neq\mathbb{C}$ and
$\sigma(D_{2})$ is symmetric, we have that $\sigma(D_{p})$ is independent of
$p$, see Theorem 1.1. Since $\sigma(D_{2})$ is symmetric, then
$\sigma(D_{2})=\mathbb{R}$. Hence $\sigma(D_{p})=\mathbb{R}$ for all $p$. ∎

######  Corollary 6.3.

Let $(M^{n},g)$ be a complete Riemannian spin manifold with associated
classical Dirac operator $\not{D}$. Suppose that the Ricci curvature of $M$ is
asymptotically nonnegative as in Theorem 6.2 and for all $R>0$ large enough
there is a spinor $\phi_{R}$ on $M\setminus B_{p}(R)$ with
$0<c_{1}<|\phi_{R}|<c_{2}$ on $M\setminus B_{p}(R)$, $|\nabla\phi_{R}|\leq C$,
and $\|\not{D}^{2}\phi_{R}\|_{L^{\infty}}\to 0$ as $R\to\infty$.

Then, the $L^{p}$-spectrum of the classical square Dirac operator
$\not{D}^{2}_{p}$ is $[0,\infty)$ for all $p\in[1,\infty]$.

If in addition $n\neq 3$ modulo $4$ or $M$ has an orientation-reserving
isometry that lifts to spin structure, then $\sigma(D_{2})=\mathbb{R}$. If we
also know that $\sigma(\not{D}_{1})\neq\mathbb{C},$ then
$\sigma(\not{D}_{p})=\mathbb{R}$ for all $p\in[1,\infty)$.

###### Proof.

The proof of the Corollary is immediately given by the last Theorem since in
this case the scalar curvature is also asymptotically nonnegative and hence
bounded from below.

The remaining statement for $\sigma(D_{p})$ follows from Corollary 1.2. ∎

##  7\. The $L^{2}$-spectrum of the square Dirac operator

In [ChLu2]*Theorem 2.3 the first author and Lu, proved a generalized Weyl
criterion for the $L^{2}$-spectrum of a densely defined self-adjoint operator.
This criterion can also be applied to the square of the Dirac operator over a
Clifford bundle, since it is a self-adjoint and nonnegative operator. In fact,
given that the resolvent operator $(D^{2}+1)^{-1}$ of $D^{2}$ is bounded on
$L^{\infty}$, the proof of Theorem 1.1 in [ChLu2] which was stated for the
particular case of the Laplacian on functions directly generalizes to our
setting, giving us the following result.

######  Theorem 7.1.

Let $(M,g)$ be a complete Riemannian manifold with Clifford bundle $S$ and
associated Dirac operator $D$. Assume that for $\lambda>0$ there exists a
sequence of sections of the Clifford bundle $\psi_{i}\in L^{1}\cap L^{\infty}$
in the domain of $D_{1}^{2}$ such that

$\frac{\|\psi_{i}\|_{L^{\infty}}\|(D^{2}-\lambda)\psi_{i}\|_{L^{1}}}{\|\psi_{i}\|_{L^{2}}^{2}}\to
0\ \ \text{as}\ \ i\to\infty.$

Then $\lambda\in\sigma(D_{2}^{2})$.

In addition, if for any compact subset $K$ of $M$ the support of the
$\psi_{i}$ lies outside $K$ for $i$ large enough, then $\lambda$ belongs to
the essential spectrum of $D_{2}^{2}$.

In Section 6 of [ChLu2] the first author and Lu proved the following

######  Lemma 7.2.

Let $r(x)$ be the radial function with respect to $p$ on the manifold $M$.
Suppose that the Ricci curvature in the radial direction away from $p$
satisfies

${\rm Ric}(\partial_{r},\partial_{r})\geq-(n-1)\delta(r),$

where $\delta(r)$ is a continuous function on $\mathbb{R}^{+}$ such that
${\displaystyle\lim_{r\to\infty}\delta(r)=0}$, and $\partial_{r}$ is the unit
vector field in the radial direction. In the case that the volume of $M$ is
finite, we make the further assumption that its volume does not decay
exponentially at $p$. Then there exists a sequence of uniformly bounded test
functions $\eta_{i}$ with disjoint supports such that

$\frac{\|(\Delta-\lambda)\eta_{i}\|_{L^{1}}}{\|\eta_{i}\|_{L^{2}}^{2}}\to 0\ \
\text{as}\ \ i\to\infty$

The functions $\eta_{i}$ are defined as

(13)
$\eta_{i}(x)=\chi_{i}({\tilde{r}}/{R_{i}})\,e^{\sqrt{-1}\sqrt{\lambda}\,\tilde{r}}.$

where $\tilde{r}(x)$ is a smoothing of the radial function $r(x)$, with
$\nabla\tilde{r}$ uniformly bounded, and
$\chi_{i}\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$ are smooth cut-off functions
with support on $[x_{i}/R_{i}-1,y_{i}/R_{i}+1]$, where $x_{i},y_{i},R_{i}$ are
large positive numbers such that $x_{i}>2R_{i}$ and $y_{i}>x_{i}+2R_{i}$.
Moreover, $0\leq\chi_{i}\leq 1$ on $M$, $\chi_{i}=1$ on
$[x_{i}/R_{i},y_{i}/R_{i}]$, and
$|\chi^{\prime}_{i}|,|\chi^{\prime\prime}_{i}|$ are bounded uniformly.

Take a family of sections of the Clifford bundle $\phi_{i}$ such that for all
$i$ large enough $0<c_{1}<|\phi_{i}|<c_{2}$, and set
$\psi_{i}=\eta_{i}\phi_{i}$. Then, by using (12),

$\displaystyle\|(D^{2}-\lambda)\psi_{i}\|_{L^{1}}\leq$ $\displaystyle
C\|D^{2}\phi_{i}\|_{L^{1}(B_{p}(y_{i}+R_{i})\setminus B_{p}(x_{i}-R_{i}))}$
$\displaystyle\quad+\frac{C}{R_{i}}\|\nabla\phi_{i}\|_{L^{1}(B_{p}(y_{i}+R_{i})\setminus
B_{p}(x_{i}-R_{i}))}+\|(\Delta-\lambda)\eta_{i}\|_{L^{1}}$

Note that for manifolds as in Lemma 7.2 it was proved in [ChLu2] that
$\text{Vol}(B_{p}(y_{i}+R_{i}))\leq C\|\eta_{i}\|_{L^{2}}^{2}$, therefore if
we have sections with the property that $D^{2}\phi_{i}\to 0$ as $i\to\infty$
and $|\nabla\phi_{i}|\leq C$, then

$\displaystyle\frac{\|\psi_{i}\|_{L^{\infty}}\|(D^{2}-\lambda)\psi_{i}\|_{L^{1}}}{\|\psi_{i}\|_{L^{2}}^{2}}\leq
C\|D^{2}\phi_{i}\|_{L^{\infty}(B_{p}(y_{i}+R_{i})\setminus
B_{p}(x_{i}-R_{i}))}+\frac{C}{R_{i}}+\frac{\|(\Delta-\lambda)\eta_{i}\|_{L^{1}}}{\|\eta_{i}\|_{L^{2}}^{2}}.$

This observation together with Theorem 7.1 proves the following result.

######  Theorem 7.3.

Suppose that $M$ is a complete Riemannian manifold with Ricci curvature
asymptotically nonnegative in the radial direction and satisfying the
assumptions of Lemma 7.2. In addition suppose that there exists a family of
sections of the Clifford bundle $\phi_{i}$ such that (except possibly on a
compact set $K$) for all $i$ large enough $0<c_{1}<|\phi_{i}|<c_{2}$,
$|\nabla\phi_{i}|\leq C$, and $\|D^{2}\phi_{i}\|_{L^{\infty}}\to 0$ as
$i\to\infty$. Then $\sigma(D_{2}^{2})=[0,\infty)$.

The above result allows us to find a large class of manifolds over which the
$L^{2}$ spectrum of $D^{2}$ is maximal. For the particular case of the square
of the classical Dirac operator over an asymptotically Euclidean spin manifold
we can use the Witten spinor constructed by Bartnik to compute the
$L^{2}$-spectrum of $\not{D}^{2}$ and $\not{D}$ [Bart, Witten81].

######  Definition 7.4.

A manifold $(M^{n},g)$ with $n\geq 3$ is asymptotically flat if its scalar
curvature is in $L^{1}(M)$ and there is a compact set $K\subset M$ and a
diffeomorphism $\Phi\colon M\setminus K\to\mathbb{R}^{n}\setminus
B_{r_{o}}(0)$ such that

$(\Phi_{*}g)_{ij}=\delta_{ij}+O(r^{2-n}),\quad\partial_{k}(\Phi_{*}g)_{ij}=O(r^{1-n}),\quad\partial_{kl}(\Phi_{*}g)_{ij}=O(r^{-n}).$

If in addition to being asymptotically flat the manifold has positive scalar
curvature and is spin, then Bartnik (see also Finster and Kath [FK]) proves
the following result: We consider the boundary problem

(14) $\not{D}\psi=0,\ \ \lim_{|x|\to\infty}\psi=\psi_{0}$

where $\psi_{0}$ is a constant spinor (that is viewed as a spinor on
$M\setminus K$ using $\Phi$) . As Bartnik proves, this problem has a unique
solution, called a Witten spinor, which is smooth and satisfies the decay
estimates

$\psi=\psi_{0}+O(r^{2-n}),\quad\partial_{k}\psi=O(r^{1-n}),\quad\partial_{kl}\psi=O(r^{-n}).$

By Theorem 7.3 the following result is immediate.

######  Proposition 7.5.

Suppose that $M^{n}$ with $n\geq 3$ is an asymptotically flat spin manifold
with positive scalar curvature. Then the spectrum of $\not{D}_{2}^{2}$ is
$[0,\infty)$. If additionally $n\neq 3$ modulo $4$ or $M$ has an orientation-
reserving isometry that lifts to spin structure, then
$\sigma(\not{D}_{2})=\mathbb{R}$.

Note that a Witten spinor $\psi$ which satisfies (14) has the property that
$\not{D}^{2}\psi=0$. Hence, Theorem 1.1 leads to the following immediate
result.

######  Proposition 7.6.

Let $M^{n}$ with $n\geq 3$ be a spin manifold whose Ricci curvature satisfies

$\text{Ric}(x)\geq-\delta(n)r^{-2}(x)$

where $r(x)=d(x_{o},x)$ is the distance from $x$ to a fixed point $x_{o}$ and
$\delta(n)$ is a small constant that depends only on the dimension of the
manifold.

Suppose that $M$ admits a Witten spinor which is asymptotically constant and
whose gradient vanishes asymptotically.

Then, the $\sigma(\not{D}_{p}^{2})=[0,\infty)$ for all $p\in[1,\infty]$.

If additionally $n\neq 3$ modulo $4$ or $M$ has an orientation-reserving
isometry that lifts to spin structure, then $\sigma(\not{D}_{2})=\mathbb{R}$.
If we also know that $\sigma(\not{D}_{1})\neq\mathbb{C},$ then
$\sigma(\not{D}_{p})=\mathbb{R}$ for all $p\in[1,\infty)$.

##  Appendix A On the definition of the Dirac operator on $L^{p}$

In [AG16] the classical Dirac operator on $L^{p}$, $\not{D}_{p}$, for $1\leq
p<\infty$ is defined as the completion of $\not{D}\colon
C_{c}^{\infty}(S)\subset L^{p}(S)\to C_{c}^{\infty}(S)\subset L^{p}(S)$ with
respect to the graph norm. Analogously
$\not{D}_{p}^{2}\left(=(\not{D}_{p})^{2}\right)$ is defined and we use this
definition as well for general Dirac-type operators as in Section 2. However,
the $L^{p}$-independence result we proved in Theorem 1.1 is a statement on the
generator, $H_{p}$, of the semigroup $e^{-tD^{2}}$ on $L^{p}$ as defined in
Lemma 3.3.

We show next that for a Riemannian manifold with Ricci curvature bounded from
below this generator does in fact coincide with the operator $D^{2}$ defined
above. We begin with the following result.

######  Lemma A.1.

Let $S_{t}\colon L^{2}\to L^{2}$ be a strongly continuous semigroup, and let
$\mathfrak{D}\subset L^{2}\cap L^{p}$ be a core of $S_{t}$. For each $1\leq
p<\infty$ we assume that $\mathfrak{D}$ is dense in $L^{p}$. Define
$S_{t}\colon L^{p}\to L^{p}$ to be the completion of $S_{t}|_{\mathfrak{D}}$
in the graph norm. Let $H_{p}\colon L^{p}\to L^{p}$ be the corresponding
generator, in other words its domain is given by
$\text{dom}(H_{p})\mathrm{:=}\\{\psi\in L^{p}\ |\ \exists\phi\in
L^{p}\colon\|t^{-1}(\text{Id}-S_{t})\psi-\phi\|_{L^{p}}\to 0\text{ as }t\to
0\\}$ and $H_{p}\psi\mathrm{:=}\phi$. It follows that $S_{t}=e^{-H_{p}t}$.

Assume that $S_{t}(\mathfrak{D})\cup\mathfrak{D}\subset\text{dom}(H_{p})$,
that $S_{t}H_{p}=H_{p}S_{t}$ on $S_{t}(\mathfrak{D})\cup\mathfrak{D}$ and that
there is a positive constant $K>0$ such that $\|S_{t}\|_{p\to p}\leq e^{Kt}$
for all $t>0$. Then $\mathfrak{D}$ is a core for $H_{p}\colon L^{p}\to L^{p}$
and $H_{p}$ is the closure of $H_{p}|_{\mathfrak{D}}$ with respect to the
graph norm.

###### Proof.

The proof goes along the lines of [Rob]*pp. 14–17. However, although Robinson
assumes that $S_{t}$ is a contraction semigroup and that
$S_{t}(\mathfrak{D})\subset\mathfrak{D}$, we will prove that the condition
$\|S_{t}\|_{p\to p}\leq e^{Kt}$ is enough to obtain the statement, and that we
can replace the assumption $S_{t}(\mathfrak{D})\subset\mathfrak{D}$ by
$S_{t}(\mathfrak{D})\cup\mathfrak{D}\subset\text{dom}(H_{p})$.

Fix a constant $\lambda<-K<0$ and define

$R_{\lambda}(H_{p})\phi\mathrm{:=}-\int_{0}^{\infty}ds\;e^{\lambda
s}S_{s}\phi$

This operator is bounded on $L^{p}$ since for $\phi\in L^{p}$

$\|R_{\lambda}(H_{p})\phi\|_{L^{p}}\leq\int_{0}^{\infty}ds\;e^{(\lambda+K)s}\|\phi\|_{L^{p}}=-(\lambda+K)^{-1}\|\phi\|_{L^{p}}.$

Moreover,

$\displaystyle\|t^{-1}(\text{Id}-S_{t})$ $\displaystyle
R_{\lambda}(H_{p})\phi+\phi-\lambda R_{\lambda}(H_{p})\phi\|_{L^{p}}$
$\displaystyle\leq$ $\displaystyle\;\|t^{-1}\int_{0}^{\infty}ds\;e^{\lambda
s}(S_{s+t}-S_{s})\phi+\phi+\lambda\int_{0}^{\infty}ds\;e^{\lambda
s}S_{s}\phi\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\;\|\left(t^{-1}(e^{-\lambda
t}-1)+\lambda\right)\int_{0}^{\infty}ds\;e^{\lambda
s}S_{s}\phi+\phi-t^{-1}\int_{0}^{t}ds\;e^{\lambda(s-t)}S_{s}\phi\|_{L^{p}}$
$\displaystyle\leq$ $\displaystyle\;|t^{-1}(e^{-\lambda
t}-1)+\lambda|\;\|R_{\lambda}(H_{p})\phi\|_{L^{p}}+\|\phi-t^{-1}\int_{0}^{t}ds\;e^{\lambda(s-t)}S_{s}\phi\|_{L^{p}}$
$\displaystyle\ \to 0\ \ \text{as}\ \ t\to 0$

This implies $R_{\lambda}(H_{p})(L^{p})\subset\text{dom}(H_{p})$ and
$(\lambda\,\text{Id}-H_{p})R_{\lambda}(H_{p})\phi=\phi$. In particular, we
have that $\text{Image}(\lambda\,\text{Id}-{H_{p}})=L^{p}$ and that
$(\lambda\text{Id}-{H_{p}})$ is injective.

Let $\hat{H_{p}}$ be the closure of $H_{p}|_{\mathfrak{D}}\colon L^{p}\to
L^{p}$ in the graph norm. If we can prove that $H_{p}=\hat{H}_{p}$, then
$\mathfrak{D}$ would be a core of $H_{p}$. Given that
$\text{Image}(\lambda\text{Id}-{H_{p}})=L^{p}$ and the map
$(\lambda\text{Id}-{H_{p}})$ is injective for some $\lambda<-K<0$, if we can
also show that $\text{Image}(\lambda\text{Id}-\hat{H_{p}})=L^{p}$, then we
would immediately get that $H_{p}=\hat{H}_{p}$.

Hence, it remains to prove that
$\text{Image}(\lambda\text{Id}-\hat{H}_{p})=L^{p}$ for some $\lambda<-K$. To
this end we define

$\psi_{n,k}\mathrm{:=}-\frac{k}{n}{\sum_{i=1}^{n}e^{\lambda\frac{ki}{n}}S_{\frac{ki}{n}}}\psi$

for any $\psi\in L^{2}\cap L^{p}$. Then
$\psi_{k}\mathrm{:=}\lim_{n\to\infty}\psi_{n,k}=-\int_{0}^{k}ds\;e^{\lambda
s}S_{s}\psi$ and $\lim_{k\to\infty}\psi_{k}=R_{\lambda}(H_{p})\psi$ in
$L^{p}$. Moreover, since $S_{t}H_{p}=H_{p}S_{t}$ on
$S_{t}\mathfrak{D}\cup\mathfrak{D}$ we have
$\lim_{k\to\infty}\lim_{n\to\infty}(\lambda-H_{p})\phi_{n,k}=\phi$ in $L^{p}$
for all $\phi\in S_{t}(\mathfrak{D})\cup\mathfrak{D}$.

Now let $\phi\in\mathfrak{D}$. Then $\phi_{n,k}\in
S_{t}(\mathfrak{D})\cup\mathfrak{D}$ and as a result, $(\lambda-
H_{p})\phi_{n,k}=((\lambda-H_{p})\phi)_{n,k}$. Hence
$\lim_{k\to\infty}\lim_{n\to\infty}((\lambda-H_{p})\phi)_{n,k}=\phi$ in
$L^{p}$ which implies
$\mathfrak{D}\subset\text{Image}(\lambda-\hat{H}_{p})\subset\text{Image}(\lambda-{H}_{p})=L^{p}$.

Therefore, it remains to show that $\text{Image}(\lambda-\hat{H}_{p})$ is norm
closed, which then implies $\text{Image}(\lambda\text{Id}-\hat{H}_{p})=L^{p}$:
Since $(\lambda\text{Id}-\hat{H}_{p})^{-1}$ is bounded, then
$(\lambda\text{Id}-\hat{H}_{p})$ is bounded below. To show that
$\text{Image}(\lambda\text{Id}-\hat{H}_{p})=L^{p}$, let
$\phi_{i}\in\text{Image}(\lambda-\hat{H}_{p})$ and $\phi_{i}\to\phi$ in
$L^{p}$. Since $(\lambda-H_{p})^{-1}$ is bounded and
$\text{dom}(H_{p})\subset\text{dom}(\hat{H}_{p})$,
$\psi_{i}\mathrm{:=}(\lambda-\hat{H}_{p})^{-1}\phi_{i}=(\lambda-{H}_{p})^{-1}\phi_{i}\in\text{dom}(\hat{H}_{p})$
converges to $\psi\mathrm{:=}(\lambda-H_{p})^{-1}\phi$ in $L^{p}$. Thus,
$\psi_{i}\to\psi$ and $\hat{H}_{p}\psi_{i}\to\lambda\psi-\phi$ in $L^{p}$.
Since $\hat{H}_{p}$ is closed in graph norm $\psi\in\text{dom}(\hat{H}_{p})$
and hence $\text{Image}(\lambda-\hat{H}_{p})$ is norm closed ∎

######  Corollary A.2.

Let $(M,g)$ be a complete Riemannian manifold with Clifford bunle $S$ and
associated Dirac operator $D$. Suppose that the Ricci curvature of $(M,g)$ and
the respective Clifford contraction are bounded from below. Then, for $1\leq
p<\infty$ the generator of the semigroup $e^{-tD^{2}}\colon L^{p}\to L^{p}$ is
the square of the completion of $D_{c}\colon C_{c}^{\infty}(S)\subset
L^{p}(S)\to C_{c}^{\infty}(S)\subset L^{p}(S)$ with respect to the graph norm.

###### Proof.

Set $S_{t}\mathrm{:=}e^{-tD^{2}}\colon L^{2}\to L^{2}$. Then $S_{t}$ is a
strongly continuous semigroup with $|S_{t}|\leq e^{K_{1}t}$ by Lemma 3.3. Let
$\mathfrak{D}=C_{c}^{\infty}(S)$. Then, $\mathfrak{D}$ is a core of $S_{t}$
and dense in $L^{p}$ for $1\leq p<\infty$. Let $S_{t}\colon L^{p}\to L^{p}$ be
the completion of $S_{t}|_{\mathfrak{D}}$ in the graph norm. The corresponding
generator of this semigroup will be denoted by $H_{p}$. Clearly,
$\mathfrak{D}\subset\text{dom}(H_{p})$.

Let $\psi\in\mathfrak{D}$ and $H_{p}\psi=\phi$. Then,

$\|t^{-1}(\text{Id}-S_{t})S_{t}\psi-S_{t}\phi\|_{L^{p}}\leq
e^{K_{1}t}\|t^{-1}(\text{Id}-S_{t})\psi-\phi\|_{L^{p}}\to 0\quad\text{as}\quad
t\to 0.$

Hence $S_{t}(\mathfrak{D})\subset\text{dom}(H)$. Moreover, by construction
$H_{2}$ is the generator $D^{2}$ of the semigroup $S_{t}$ on $L^{2}$, and
$S_{t}$ and $H_{2}$ commute on $\text{dom}(H_{2})\subset L^{2}$ by functional
calculus. Since all $H_{p}$ coincide on
$\text{dom}(H_{2})\cap\text{dom}(H_{p})$ (and the same for $S_{t}$ for the
different $p$) we have that $S_{t}$ and $H_{p}$ commute on
$S_{t}(\mathfrak{D})\cup\mathfrak{D}$.

Using Lemma A.1 and the fact that $H|_{\mathfrak{D}}=D_{c}^{2}$ we get that
$H_{p}$ is the completion of $D_{c}^{2}\colon C_{c}^{\infty}(S)\subset
L^{p}(S)\to C_{c}^{\infty}(S)\subset L^{p}(S)$ with respect to the graph norm.
∎

##  Appendix B Spectrum for bounded geometry

In [AG16]*Lem B.8 the implication: $\lambda^{2}\in\sigma(D_{p}^{2})$ if and
only if $\lambda$ or $-\lambda$ belongs to $\sigma(D_{p})$ is proved for the
spin Dirac operator under the additional assumption that
$\sigma(D_{p})\neq\mathbb{C}$. The proof immediately carries over to manifolds
with Clifford bundles and associated Dirac operator $D$. Under additional
geometric assumptions on the manifold and the Clifford bundle, the assumption
$\sigma(D_{p})\neq\mathbb{C}$ is automatic. We add here a corresponding
statement for completeness.

######  Lemma B.1.

Let $(M,g)$ be a complete Riemannian manifold with Clifford bundle $S$ and
associated Dirac operator $D$. Assume that $(M,g)$ and $S$ are of bounded
geometry. Let $p\in[1,\infty]$. Then $\lambda^{2}\in\sigma(D_{p}^{2})$ if and
only if $\lambda$ or $-\lambda$ belongs to $\sigma(D_{p})$.

###### Proof.

The ‘only if’-direction holds without the assumption of bounded geometry as
shown in [AG16]*Lem B.8. The proof of the ‘if’-direction is also the same as
therein as soon as we have: If $\psi\in L^{p}$ and $D^{2}\psi\in L^{p}$, then
$D\psi\in L^{p}$. In case of bounded geometry this follows from elliptic
regularity theory. We have $\|D\phi\|_{L^{p}}\leq
c\left(\|\phi\|_{L^{p}}+\|D^{2}\phi\|_{L^{p}}\right)$ for some $c>0$ and hence
$D\psi\in L^{p}$. ∎

We note that in the case of a spin manifold $(M,g)$ with $S$ the corresponding
spinor bundle, the assumption that $(M,g)$ is of bounded geometry
automatically implies that $S$ is of bounded geometry.

## References