# Kippenhahn Varieties and the Weyl Calculus for Several Matrices I

Brian Jefferies  School of Mathematics
NRISAWD
NSW 2484 AUSTRALIA<EMAIL_ADDRESS>
###### Abstract.

The paper reviews properties of the Weyl functional calculus for several
operators and its relation to the generalised numerical range of n hermitian
matrices. The support and singular support of the Weyl functional calculus for
$n$ hermitian matrices are determined by Kippenhahn varieties in algebraic
geometry.

Dedicated to the memory of Alan McIntosh.

##  1\. Introduction

Let $n$ be an integer and ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ an $n$-tuple
of linear operators densely defined in a complex Banach space $X$ whose joint
spectra satisfy the spectral reality condition

(1)
${\sigma}({\langle{{\boldsymbol{A}}},\xi\rangle})\subset{\mathbb{R}},\qquad\xi\in{\mathbb{R}}^{n},$

with ${\langle{{\boldsymbol{A}}},\xi\rangle}=A_{1}\xi_{1}+\dots+A_{n}\xi_{n}$.
Such a system ${\boldsymbol{A}}$ is called hyperbolic.

The term comes from the case when ${{\boldsymbol{A}}}$ consists of $N\times N$
matrices with $N=2,3,\dots$. With the characteristic polynomial of a matrix
$B$ defined by $p_{B}(z)=\det(B-zI)$ for $z\in{\mathbb{C}}$, the $n$-tuple
${{\boldsymbol{A}}}$ is hyperbolic exactly when the only solutions $z$ of the
equation $p_{{\langle{{\boldsymbol{A}}},\xi\rangle}}(z)=0$ are real for any
$\xi\in{\mathbb{R}}^{n}$, $\xi\neq 0$, or that
$P^{{\boldsymbol{A}}}:\xi\longmapsto\det(\xi_{0}I+{\langle{{\boldsymbol{A}}},{\boldsymbol{\xi}}\rangle})$,
$\xi\in{\mathbb{R}}^{n+1}$, is a homogeneous hyperbolic polynomial with
respect to $e_{0}=(1,0,\dots,0)\in{\mathbb{R}}^{n+1}$. Here the vector
$\xi=(\xi_{0},\dots,\xi_{n})\in{\mathbb{R}}^{n+1}$ has been written as
$\xi=\xi_{0}e_{0}+{\boldsymbol{\xi}}$ with
${\boldsymbol{\xi}}=\sum_{j=1}^{n}\xi_{j}e_{j}$ for the standard basis
$(e_{0},\dots,e_{n})$ of ${\mathbb{R}}^{n+1}$ and the same notation is used
for ${\zeta}\in{\mathbb{C}}^{n+1}$.

General hyperbolic polynomials $p:{\mathbb{R}}^{n+1}\to{\mathbb{R}}$ were
studied by L. Gårding [16] in relation to hyperbolic partial differential
equations in order that the equation $p(\tau,{\boldsymbol{D}})={\delta}_{0}$
be well-posed in the sense of distributions with respect to the differential
operators

$\tau=\frac{1}{i}\frac{\partial}{\partial
t},\qquad{\boldsymbol{D}}=\bigg{(}\frac{1}{i}\frac{\partial}{\partial
x_{1}},\dots,\frac{1}{i}\frac{\partial}{\partial x_{n}}\bigg{)}.$

The unique distributional solution $F_{p}$ of
$p(\tau,{\boldsymbol{D}})={\delta}_{0}$ [20, Theorem 12.5.1] may then be
expressed by a formula of Herglotz-Petrovsky-Leray systematically studied in
[3],[4] and [20, Chapter 12]. The associated hyperbolic system

(2) $I\frac{\partial}{\partial
t}+{\langle{{\boldsymbol{A}}},\nabla\rangle}=i\delta_{0}I$

then has a unique solution
$F_{{\boldsymbol{A}}}=\Phi({{\boldsymbol{A}}})F_{P^{{\boldsymbol{A}}}}$ for a
matrix differential operator $\Phi({{\boldsymbol{A}}})$ of order $N-1$ in
$(\tau,{\boldsymbol{D}})$ formed by multiplying the Fourier transform of
$F_{P^{{\boldsymbol{A}}}}$ by the $(N-1)$ matrix minors of
$\xi_{0}I+{\langle{{\boldsymbol{A}}},{\boldsymbol{\xi}}\rangle}$,
$\xi=\xi_{0}e_{0}+{\boldsymbol{\xi}}\in{\mathbb{R}}^{n+1}$.

The Fourier transform of the Schwartz function
$f\in{\mathcal{S}}({\mathbb{R}}^{n})$ is taken to be

$\hat{f}(\xi)=\int_{{\mathbb{R}}^{n}}e^{-i{\langle
x,\xi\rangle}}f(x)\,dx,\quad\xi\in{\mathbb{R}}^{n}.$

On the other hand, taking the Fourier transform of equation (2) in the space
variables only for $t>0$ and solving for the initial value problem, we get the
distribution ${\mathcal{W}}_{t{{\boldsymbol{A}}}}$, $t\in{\mathbb{R}}$. At
$t=1$, the operator valued distribution ${\mathcal{W}}_{{{\boldsymbol{A}}}}$
given by

(3) ${\langle{\mathcal{W}}_{{{\boldsymbol{A}}}},f\rangle}\
=\frac{1}{(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}e^{i{\langle{{\boldsymbol{A}}},\xi\rangle}}\hat{f}(\xi)\,d\xi,\quad
f\in{\mathcal{S}}({\mathbb{R}}^{n}),$

is the Weyl functional calculus for ${{\boldsymbol{A}}}$ studied since the
late 60’s by E. Nelson [28], M. Taylor [35] and R.F.V. Anderson [1, 2]. This
expression has the advantage that it does not involve determinants so
${\mathcal{W}}_{{{\boldsymbol{A}}}}$ makes sense for any system
${{\boldsymbol{A}}}$ of collectively densely defined linear operators which
satisfy an exponential estimate of the form

(4) $\|e^{i{\langle{{\boldsymbol{A}}},\xi\rangle}}\|\leq
C(1+|\xi|)^{s},\quad\xi\in{\mathbb{R}}^{n},$

for some $C,s>0$, for example, selfadjoint operators defined in a Hilbert
space with $s=0$. A single bounded linear operator satisfying the estimate
$(\ref{eq:type})$ is a generalised scalar operator [13, Theorem 5.4.5]. More
precisely, if ${{\boldsymbol{A}}}=(A_{1},\dots,A_{n})$ are densely defined
selfadjoint operators such that every real linear combination has a
selfadjoint closure ${\langle{{\boldsymbol{A}}},\xi\rangle}$ on the
intersection of the relevant domains, then (4) follows with $C=1$ and $s=0$
from successive applications of the Lie-Kato-Trotter product formula (see [14,
Chapter III, Corollary 5.8])

The prime candidate for the Weyl functional calculus is the system
$({\boldsymbol{X}},{\boldsymbol{D}})$ of densely defined operators. Here
${\boldsymbol{X}}=(X_{1},\dots,X_{n})$ with $X_{j}$ the operator of
multiplication by the variable $x_{j}$, $j=1,\dots,n$, so that
${\mathcal{W}}_{({\boldsymbol{X}},{\boldsymbol{D}})}$ is a
${\mathcal{L}}(L^{2}({\mathbb{R}}^{n}))$-valued Schwartz distribution on
${\mathbb{R}}^{2n}$. This is the original Weyl functional calculus and much
studied as a pseudodifferental operator [18],[36, VII §14]. The more general
definition of ${\mathcal{W}}_{{{\boldsymbol{A}}}}$ for selfadjoint operators
${{\boldsymbol{A}}}$ formulated by E. Nelson [28] was inspired by ideas of R.
Feynman on a general operator calculus and “disentangling” procedure
[15],[24]. In the case that ${{\boldsymbol{A}}}$ are selfadjoint elements of a
von Neumann algebra with a given trace $\rho$, then the scalar distribution
$\rho\circ{\mathcal{W}}_{{\boldsymbol{A}}}$ is the Wigner transform [12, 33].

In a similar fashion, the Kohn-Nirenberg functional calculus
${\sigma}\mapsto{\sigma}({\boldsymbol{X}},{\boldsymbol{D}})_{\text{KN}}$
defined for symbols ${\sigma}\in{\mathcal{S}}({\mathbb{R}}^{2n})$ by

${\sigma}({\boldsymbol{X}},{\boldsymbol{D}})_{\text{KN}}=\frac{1}{(2\pi)^{2n}}\int_{{\mathbb{R}}^{2n}}e^{i{\langle{{\boldsymbol{X}}},q\rangle}}e^{i{\langle{{\boldsymbol{D}}},p\rangle}}\hat{\sigma}(q,p)\,dq\,dp$

is an important tool in the analysis of partial differential equations [36,
VII (1.10)].

If ${{\boldsymbol{A}}}$ are bounded linear operators on a Banach space $X$
satisfying the exponential bound (4) then by the Paley-Wiener theorem
${\mathcal{W}}_{{{\boldsymbol{A}}}}$ has compact support and there exist $r>0$
such that

(5) $\|e^{i{\langle{{\boldsymbol{A}}},{\zeta}\rangle}}\|\leq
C^{\prime}(1+|\Re{\zeta}|)^{s^{\prime}}e^{r|\Im{\zeta}|},\quad{\zeta}\in{\mathbb{C}}^{n},$

for some $C^{\prime},s^{\prime}>0$. The operator valued distribution
${\mathcal{W}}_{{\boldsymbol{A}}}$ has support in the ball $B_{r}$ of radius
$r$ centred at zero. Any number $r$ strictly greater than
$\|{{\boldsymbol{A}}}\|=(\|A_{1}\|^{2}+\dots+\|A_{n}\|^{2})^{\frac{1}{2}}$
will do. All such linear operators satisfy the spectral reality condition (1).

Back to hyperbolic $(N\times N)$ matrices ${{\boldsymbol{A}}}$, the solution
of (2) has an expression by the Herglotz-Petrovsky-Leray formula so the bound
(5) is satisfied with $s^{\prime}$ at most $N-1$. In the infinite dimensional
situation this may no longer be the case. The paper of Nelson [28] also
contains a formula for hermitian matrices ${\boldsymbol{A}}$ acting on
${\mathbb{C}}^{N}$ expressing ${\mathcal{W}}_{{\boldsymbol{A}}}$ as a
differential operator of order $(N-1)$ acting on the measure $\mu\circ
n_{{{\boldsymbol{A}}}}^{-1}$. Here $\mu$ is the unitarily invariant
probability measure on the unit sphere $S({\mathbb{C}}^{N})$ in
${\mathbb{C}}^{N}$ centred at zero and

(6) $n_{{\boldsymbol{A}}}:h\longmapsto(\langle A_{1}h,h\rangle,\dots,\langle
A_{n}h,h\rangle),\qquad h\in S({\mathbb{C}}^{N}),$

is the joint numerical range map. The Payley-Wiener theorem shows that
${\mathcal{W}}_{{\boldsymbol{A}}}$ is supported by the convex hull of the
joint numerical range
$N_{{\boldsymbol{A}}}=n_{{\boldsymbol{A}}}(S({\mathbb{C}}^{N}))$, even for
unbounded selfadjoint operators (with a common dense domain so that
${\langle{{\boldsymbol{A}}},\xi\rangle}$ has a selfadjoint closure for all
$\xi\in{\mathbb{R}}^{n}$).

The joint numerical range $N_{{\boldsymbol{A}}}$ of an $n$-tuple
${{\boldsymbol{A}}}$ of hermitian matrices has attracted recent interest
because it arises in many areas such as optimisation theory, quantum optics,
quantum statistical mechanics, Wigner transforms and quantum error correction,
see [30] for an overview. In the Herglotz-Petrovsky-Leray formula the joint
numerical range $N_{{\boldsymbol{A}}}$ manifests as the trace of the
propagation cone $K(P^{\boldsymbol{A}})$ at time $t=1$ and the fundamental
formula

(7) $\mu\circ
n_{{{\boldsymbol{A}}}}^{-1}=(-i)^{N}(N-1)!F_{P^{{\boldsymbol{A}}}}(1,{\,\cdot\,})$

holds as distributions linking the Nelson and Herglotz-Petrovsky-Leray
representations for systems of hermitian matrices. It is not immediately
obvious why $F_{p}(1,{\,\cdot\,})$ should be a distribution of order zero in
the case that $p=P^{{\boldsymbol{A}}}$ is a hyperbolic determinantal
polynomial for a system ${{\boldsymbol{A}}}$ of hermitian matrices, see
Theorem 5.2 below.

For the case $n=2$ of two hermitian matrices
${{\boldsymbol{A}}}=(A_{1},A_{2})$, the set $N_{{\boldsymbol{A}}}$ may be
identified with the usual numerical range in ${\mathbb{C}}$ of the single
matrix $A=A_{1}+iA_{2}$ under the identification $j:(x,y)\mapsto x+iy$,
$x,y\in{\mathbb{R}}$. Then $jN_{{\boldsymbol{A}}}$ is the convex hull of an
algebraic curve $C(A)=jC({{\boldsymbol{A}}})$ studied by R. Kippenhahn [26].
In this case M. Atiyah, R. Bott and L. Gårding show that the singular support
$\operatorname{ss}({\mathcal{W}}_{{\boldsymbol{A}}})$ of the Weyl calculus for
two hermitian matrices ${{\boldsymbol{A}}}$ is exactly $C({{\boldsymbol{A}}})$
[4, Theorem 14.20]. That
$\operatorname{ss}({\mathcal{W}}_{{\boldsymbol{A}}})\subset
C({{\boldsymbol{A}}})$ was shown by J. Bazer and D. Yen [6]. None of these
authors reference the numerical range studies of Kippenhahn [26]. Bazer and
Yen manage to avoid discussing hyperbolic polynomials by invoking a plane wave
decomposition of the distribution ${\mathcal{W}}_{t{{\boldsymbol{A}}}}$ whose
density with respect to Lebesgue measure is the “Riemann matrix” for equation
(2).

At this stage it is prudent to introduce some modern developments. The Cauchy
transform of a Schwartz distribution
$T\in{\mathcal{S}}^{\prime}({\mathbb{R}})$ is the function
$\tilde{T}:z\mapsto{\langle T,g_{z}\rangle}$ with $g_{z}(x)=\frac{1}{2\pi
i}\frac{1}{x-z}$, $z\in{\mathbb{C}}\setminus{\mathbb{R}}$, $x\in{\mathbb{R}}$
[10, §5.1]. Then

(8) $T=\lim_{{\epsilon}\to
0+}\tilde{T}(x+i{\epsilon})-\tilde{T}(x+i{\epsilon})$

in the sense of distribution [10, §5.6]. If we want a similar formula for
distributions on ${\mathbb{R}}^{n}$ like the Weyl functional calculus
${\mathcal{W}}_{{\boldsymbol{A}}}$, then we need an analogue of the normalised
Cauchy kernel $\frac{1}{2\pi i}\frac{1}{z}$, $z\in{\mathbb{C}}$, $z\neq 0$, in
higher dimensions. Then we may obtain a representation for for the density of
${\mathcal{W}}_{{\boldsymbol{A}}}$ with respect to Lebesgue measure like the
Riemann matrix of Bazer and Yen [5, 6]. This is the point where Clifford
analysis enters as a functional calculus technique.

The Clifford algebra ${\mathbb{C}}_{(n)}$ over the field ${\mathbb{C}}$ is
generated by standard basis vectors
$(e_{0},e_{1},\dots,e_{n})$ in ${\mathbb{R}}^{n+1}$ with multiplication so
that $e_{0}$ is the unit and the formula $x^{2}=-|x|^{2}$ holds for
$x=\sum_{j=1}^{n}x_{j}e_{j}$, $x_{j}\in{\mathbb{R}}$. Then we obtain
$e_{j}e_{k}=-e_{k}e_{j}$ for $j\neq k$, $j,k=1,\dots,n$ and $e_{j}^{2}=-1$ for
$j=1,\dots,n$. The Kelvin inverse of $x\in{\mathbb{R}}^{n+1}$ is the vector
$x^{-1}={\overline{x}}/|x|^{2}$, $x\neq 0$.

The Cauchy-Riemann operator is $D=\sum_{j=0}^{n}e_{j}\frac{\partial}{\partial
x_{j}}$ and for
$\Sigma_{n}=\frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)}$ the
function

$E(x)=\frac{1}{\Sigma_{n}}\frac{{\overline{x}}}{|x|^{n+1}},\quad
x\in{\mathbb{R}}^{n+1},\ x\neq 0,$

is the corresponding Cauchy kernel with $DE=0$ on
${\mathbb{R}}^{n+1}\setminus\\{0\\}$. The involution is given by
${\overline{x}}=x_{0}-{\boldsymbol{x}}$ for $x=x_{0}e_{0}+{\boldsymbol{x}}$
with $x_{0}\in{\mathbb{R}}$ and
${\boldsymbol{x}}\in{\mathbb{R}}^{n}\equiv\\{0\\}\times{\mathbb{R}}^{n}\hookrightarrow{\mathbb{R}}^{n+1}$.
For product vectors
${\overline{e_{j}e_{k}}}={\overline{e}}_{k}\,{\overline{e}}_{j}$, $j\neq k$,
$j,k\neq 0$, and so on. It is easily checked that
$E={\overline{D}}{\Gamma}_{n+1}$ for the fundamental solution ${\Gamma}_{n+1}$
of the Laplacian operator $\Delta e_{0}=D{\overline{D}}$ in
${\mathbb{R}}^{n+1}$ so that
${\Gamma}_{n+1}=\frac{1}{\Sigma_{n}}\frac{1}{|x|^{n-1}}$ and
$\Delta{\Gamma}_{n+1}={\delta}_{0}$ for $n\geq 2$. Setting
$G_{\omega}(x)=E({\omega}-x)$, $x\in{\mathbb{R}}^{n}$, for any Schwartz
distribution $T\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{n})$ we have

(9) $T=\lim_{{\epsilon}\to
0+}\tilde{T}(x+{\epsilon}e_{0})-\tilde{T}(x-{\epsilon}e_{0})$

in the sense of distributions for the Cauchy transform
$\tilde{T}({\omega})={\langle T,G_{\omega}\rangle}$,
${\omega}\in{\mathbb{R}}^{n+1}$, ${\omega}_{0}\neq 0$ [9, Theorem 27.7] . When
the Weyl functional calculus ${\mathcal{W}}_{{\boldsymbol{A}}}$ exists for
operators ${{\boldsymbol{A}}}$, the operator valued function ${\omega}\mapsto
G_{\omega}({\boldsymbol{A}})=\widetilde{{\mathcal{W}}_{{\boldsymbol{A}}}}({\omega})$
defined for all ${\omega}\in{\mathbb{R}}^{n+1}$ with ${\omega}_{0}\neq 0$ is
called the Cauchy kernel for ${{\boldsymbol{A}}}$. For any bounded linear
operators ${{\boldsymbol{A}}}$ on a Banach space $X$, the Cauchy kernel
$G_{\omega}({\boldsymbol{A}})$ can be defined by a series expansion for
${\omega}\in{\mathbb{R}}^{n+1}$ with $|{\omega}|$ large enough but this is not
very useful. For $n=1$, $G_{\omega}(A)=\frac{1}{2\pi}(j{\omega}-A)^{-1}$ in
${\mathcal{L}}(X)$ if $j{\omega}\notin{\sigma}(A)$.

Finally, if ${\boldsymbol{A}}$ is a hyperbolic system of bounded linear
operators (or unbounded with a common dense domain and uniform resolvent
bounds) then $G_{\omega}({\boldsymbol{A}})$ may be defined by a plane wave
decomposition which agrees with the definition in case ${{\boldsymbol{A}}}$
satisfies the exponential bounds (5) and so ${\mathcal{W}}_{{\boldsymbol{A}}}$
exists. This central idea is due to Alan McIntosh in 1988 out of which the
present investigation grew.

When $n$ is even, the plane wave formula and equation (9) tell us that the
distribution ${\mathcal{W}}_{{\boldsymbol{A}}}$ is a constant times the limit

(10) $\lim_{{\epsilon}\to
0+}\int\limits_{S^{n-1}}\left(\langle{\boldsymbol{x}}I-{\boldsymbol{A}},s\rangle-{\epsilon}sI\right)^{-n}+\left(\langle{\boldsymbol{x}}I-{\boldsymbol{A}},s\rangle+{\epsilon}sI\right)^{-n}\/ds$

in the sense of distributions for ${\boldsymbol{x}}\in{\mathbb{R}}^{n}$. Here
$S^{n-1}$ is the unit sphere in ${\mathbb{R}}^{n}$. Even for matrices this is
a useful observation because (10) is similar to the plane wave formula
employed by Bazer and Yen [5] in their study of the Riemann matrix but with
the pleasing advantage that it also works for $n\geq 2$. The formula for odd
$n$ is

(11) $\lim_{{\epsilon}\to
0+}\int\limits_{S^{n-1}}s\left(\langle{\boldsymbol{x}}I-{\boldsymbol{A}},s\rangle+{\epsilon}sI\right)^{-n}-s\left(\langle{\boldsymbol{x}}I-{\boldsymbol{A}},s\rangle-{\epsilon}sI\right)^{-n}\/ds.$

By perturbing the integrals over $S^{n-1}$ into the complex domain and
observing that we have a closed matrix valued differential form by
homogeneity, the limits can be evaluated pointwise and converge to the density
of the distribution ${\mathcal{W}}_{{\boldsymbol{A}}}$ with respect to
Lebesgue measure. This is essentially the argument of Bazer and Yen in the
case $n=2$ for two hermitian matrices. The method just described is limited to
finite dimensional operators but works for all hyperbolic $n$-tuples
${{\boldsymbol{A}}}$ of matrices and uses the full force of the Herglotz-
Petrovsky-Leray representation devised by Atiyah, Bott and Gårding.

In addition, we need the analogue $C({\boldsymbol{A}})$ of the Kippenhahn
curves in dimensions $n=3,4,\dots$ for which
$N_{{\boldsymbol{A}}}=\operatorname{co}(C({\boldsymbol{A}}))$. These semi-
algebraic sets have only recently been determined by Plaumann, Sinn and Weis
[30] and promise to have applications to the many areas where joint numerical
range is a central concept. After settling the many issues involved, the
position of the singular support
$\operatorname{ss}({\mathcal{W}}_{{\boldsymbol{A}}})$ of the Weyl functional
calculus with respect to the boundary generating set $C({\boldsymbol{A}})$ or
Kippenhahn variety in ${\mathbb{R}}^{n}$ is examined.

##  2\. The Kippenhahn Curves

Let ${\boldsymbol{A}}=(A_{1},A_{2})$ be a pair of $(N\times N)$ hermitian
matrices. Set $A=A_{1}+iA_{2}$. As mentioned in the Introduction, an
application of the Paley-Wiener Theorem yields that the convex hull of the
support $\hbox{\rm supp}({\mathcal{W}}_{{{\boldsymbol{A}}}})$ of the
associated Weyl distribution ${\mathcal{W}}_{{{\boldsymbol{A}}}}$ coincides
with the numerical range

$\displaystyle K(A)$ $\displaystyle:=jN_{{\boldsymbol{A}}}$
$\displaystyle=\bigl{\\{}\langle Ax,x\rangle\,\bigm{|}\,x\in{\mathbb{C}}^{N},\
|x|=1\bigr{\\}}$

of the matrix $A$. For more precise information on the location of $\hbox{\rm
supp}({\mathcal{W}}_{{{\boldsymbol{A}}}})$ within the numerical range of $A$,
we need to have a closer look at the fine structure of $K(A)$.

Of particular interest are certain plane algebraic curves associated with $A$
that were first investigated by R. Kippenhahn [26] in 1952. We briefly recall
the concepts involved.

Let ${\mathbb{F}}={\mathbb{R}}$ or ${\mathbb{C}}$. For $0\leq k\leq 3$, the
Grassmannian $G_{3,k}{\mathbb{F}}$, defined as the set of all $k$-dimensional
${\mathbb{F}}$-subspaces of ${\mathbb{F}}^{3}$, is a compact analytic
${\mathbb{F}}$-manifold of dimension $k(3-k)$. It has a natural topology,
induced by the differential structure of the manifold, which is determined,
for example, by the metric $h$ on $G_{3,k}{\mathbb{F}}$ with

$h(U,V)\ =\ \sup_{v\in V,\ |v|=1}\inf_{\phantom{\mbox{l}}u\in U,\
|u|=1\phantom{\mbox{l}}}|\\!|u-v|\\!|\qquad\mbox{ for all }U,V\in
G_{3,k}{\mathbb{F}}\,.$

The projective plane $\hbox{PG}({\mathbb{F}}^{3})$ over ${\mathbb{F}}$ is
given by

$\hbox{PG}({\mathbb{F}}^{3})\ =\ \bigcup_{0\leq k\leq
3}G_{3,k}{\mathbb{F}}\,.$

The one and two dimensional subspaces of ${\mathbb{F}}^{3}$ are usually called
the points and lines in $\hbox{PG}({\mathbb{F}}^{3})$, respectively.

By common abuse of notation we introduce homogeneous coordinates for the
points in $\hbox{PG}({\mathbb{F}}^{3})$ as
$(u_{1}:u_{2}:u_{3})={\mathbb{F}}(u_{1},u_{2},u_{3})$. The coordinates of a
vector in ${\mathbb{F}}^{3}$ are expressed with respect to the standard basis
for ${\mathbb{F}}^{3}$.

A polarity of $\hbox{PG}({\mathbb{F}}^{3})$ is a bijection on
$\hbox{PG}({\mathbb{F}}^{3})$ which reverses the inclusion of subspaces and
the square of which equals the identity mapping. The standard polarity $\pi$
is characterised by

$u^{\pi}\ =\
\bigl{\\{}v\in{\mathbb{F}}^{3}\,\bigm{|}\,\sum_{j=1}^{3}u_{j}v_{j}=0\bigr{\\}}\qquad\mbox{
for all }u\in G_{3,1}{\mathbb{F}}\,,$

which gives $u^{\pi}\in G_{3,2}{\mathbb{F}}$. Using the polarity $\pi$, we can
also introduce homogeneous coordinates for the lines in
$\hbox{PG}({\mathbb{F}}^{3})$ by setting
$[v_{1}:v_{2}:v_{3}]=(v_{1}:v_{2}:v_{3})^{\pi}$.

A nonempty subset $C$ of $G_{3,1}{\mathbb{F}}$ is called a plane
${\mathbb{F}}$-algebraic curve if it is the zero locus of a homogeneous
$3$-variate polynomial over ${\mathbb{F}}$. The defining polynomial of $C$ is
not uniquely determined: if $f$ defines the curve, then so does, for example,
$f^{k}$ for any $k\geq 1$. However, every curve $C$ has a defining polynomial
of minimal degree which is unique up to a constant factor. A curve is said to
be irreducible if it has an irreducible defining polynomial. Since a
polynomial ring over a field is a unique factorisation domain, each algebraic
curve $C$ is the union of finitely many irreducible curves. If
$C_{1},\ldots,C_{k}$ are the irreducible components of $C$ with irreducible
defining polynomials $f_{1},\ldots,f_{k}$, then $f=f_{1}\cdots f_{k}$ is a
defining polynomial of $C$ of minimal degree. We call $f$ a minimal polynomial
of $C$. Note that an irreducible real algebraic curve is not necessarily
connected.

Let $f$ be a minimal polynomial of the algebraic curve

$C=\bigl{\\{}u\in G_{3,1}{\mathbb{F}}\,\bigm{|}\,f(u)=0\bigr{\\}}.$

A point $u\in C$ is called singular or a singularity of $C$ if $({\partial
f}/{\partial u_{j}})(u)=0$ for $j=1,2,3$. Observe that $C$ has at most
finitely many singular points. These are the singular points of the
irreducible components of $C$ together with the points of intersection of any
two of these components. A nonsingular point $u\in C$ is called a simple point
of $C$. The curve $C$ is the topological closure of its simple points. Also,
to every simple point $u\in C$, there exists a neighbourhood of $u$ in which
$C$ admits a smooth parametrization.

Let $C$ be an irreducible plane algebraic curve with minimal polynomial $f$.
At each simple point $u\in C$, we have a unique tangent line to $C$ which is
given by

${\mathcal{T}}_{u}C\ =\ \left[\frac{\partial f}{\partial
u_{1}}(u):\frac{\partial f}{\partial u_{2}}(u):\frac{\partial f}{\partial
u_{3}}(u)\right].$

If $C$ is not a projective line or a point, then it is well-known that the set

$\bigl{\\{}({\mathcal{T}}_{u}C)^{\pi}\,\bigm{|}\,u\in
C\mbox{~{}simple}\bigr{\\}}$

is contained in a unique irreducible algebraic curve $C^{*}$, the so-called
dual curve of $C$. In fact, since an algebraic curve has at most finitely many
singularities, the dual curve is the topological closure of the set
$\bigl{\\{}({\mathcal{T}}_{u}C)^{\pi}\,\bigm{|}\,u\in C\mbox{
simple}\bigr{\\}}$. We have $C^{**}=C$. If $C$ is a projective line, then
$\bigl{\\{}({\mathcal{T}}_{u}C)^{\pi}\,\bigm{|}\,u\in C\bigr{\\}}$ consists of
a single point $u$ in $\hbox{PG}({\mathbb{F}}^{3})$. In this case, we set
$C^{*}=\\{u\\}$ and define $C^{**}$ to be the image under $\pi$ of the set of
all lines in $\hbox{PG}({\mathbb{F}}^{3})$ which pass through $u$. This again
yields $C^{**}=C$. The dual curve of a general plane algebraic curve $C$ is
the union of the dual curves of its irreducible components. In particular, $C$
and $C^{*}$ have the same number of irreducible components.

In general, it is difficult to derive an explicit equation for the dual curve
$C^{*}$ from the given equation of a curve $C$. However, from the above we
obtain the following criterion for a point in $\hbox{PG}({\mathbb{F}}^{3})$ to
belong to $C^{*}$.

######  Lemma 2.1.

Let $(x_{1}:x_{2}:1)\in G_{3,1}{\mathbb{F}}$. If there exists a smooth local
parametrization $\zeta\longmapsto(c(\zeta):s(\zeta):\mu(\zeta))$ of $C$, for
$\zeta$ in an open set $U\subseteq{\mathbb{F}}$, and a point $z\in U$ such
that $x_{1}c(z)+x_{2}s(z)+\mu(z)=0$ and
$x_{1}c^{\prime}(z)+x_{2}s^{\prime}(z)+\mu^{\prime}(z)=0$, then the point
$(x_{1}:x_{2}:1)$ belongs to $C^{*}$.

###### Proof.

The two points $(c(z):s(z):\mu(z))$ and
$(c^{\prime}(z):s^{\prime}(z):\mu^{\prime}(z))$ span the tangent line
${\mathcal{T}}_{(c(z):s(z):\mu(z))}C$ to $C$ at $(c(z):s(z):\mu(z))$. The
equations $x_{1}c(z)+x_{2}s(z)+\mu(z)=0$ and
$x_{1}c^{\prime}(z)+x_{2}s^{\prime}(z)+\mu^{\prime}(z)=0$ imply that
$(x_{1}:x_{2}:1)=({\mathcal{T}}_{(c(z):s(z):\mu(z))}C)^{\pi}$. Hence
$(x_{1}:x_{2}:1)$ belongs to the dual curve $C^{*}$ of $C$. ∎

The details and further information on complex algebraic curves can be found,
for example, in [32]. The literature for the real case is somewhat less easy
to access. As a general reference to the theory of real algebraic geometry,
see [8].

Let $A=A_{1}+iA_{2}\in{\mathcal{L}}({\mathbb{C}}^{N})$. Following R.
Kippenhahn [26], we define the complex algebraic curve $C_{\mathbb{C}}(A)$ in
the complex projective plane $\hbox{PG}({\mathbb{C}}^{3})$ by setting its dual
curve to be

$D(A)\ =\ \bigl{\\{}(c:d:\mu)\in
G_{3,1}{\mathbb{C}}\,\bigm{|}\,\det(cA_{1}+dA_{2}+\mu I)=0\bigr{\\}}.$

In [26], Kippenhahn showed that the real part $C_{\mathbb{R}}(A)$ of the curve
$C_{\mathbb{C}}(A)=D(A)^{*}$ is contained in the affine subplane
$F=\bigl{\\{}(\alpha_{1}:\alpha_{2}:1)\,\bigm{|}\,(\alpha_{1},\alpha_{2})\in{\mathbb{R}}^{2}\bigr{\\}}$
of $\hbox{PG}({\mathbb{R}}^{3})$ and, identifying $F$ with ${\mathbb{R}}^{2}$
in the canonical way, that the convex hull ${\rm co}(C_{\mathbb{R}}(A))$ of
$C_{\mathbb{R}}(A)$ is precisely the numerical range of ${\boldsymbol{A}}$.

The curve $C_{\mathbb{R}}(A)$ considered as a real algebraic curve in
$\hbox{PG}({\mathbb{R}}^{3})$ is the dual curve of the real part of $D(A)$
given by

$D_{\mathbb{R}}(A)\ =\ \bigl{\\{}(c:d:\mu)\in
G_{3,1}{\mathbb{R}}\,\bigm{|}\,\det(cA_{1}+dA_{2}+\mu I)=0\bigr{\\}}.$

Every point $u\in D_{\mathbb{R}}(A)$ has a representation
$(\cos\theta:\sin\theta:\mu)$ for some $\theta\in[0,\pi)$ and
$\mu\in{\mathbb{R}}$. As $u$ is a zero of $\det(cA_{1}+dA_{2}+\mu I)$, it
follows that $-\mu$ is an eigenvalue of the operator
${\mathcal{A}}(\theta)=\cos\theta\,A_{1}+\sin\theta\,A_{2}$.

Note that the points in $D_{\mathbb{R}}(A)$ are in one-to-one correspondence
with the lines $L_{y,t}$ in ${\mathbb{R}}^{2}$ satisfying $\langle
x,t\rangle\in\sigma(\langle{{\boldsymbol{A}}},t\rangle)$ for all $x\in
L_{y,t}$. For $u=(\cos\theta:\sin\theta:\mu)\in D_{\mathbb{R}}(A)$, take
$t=(\cos\theta,\sin\theta)\in{\mathbb{T}}$ and
${\boldsymbol{y}}\in{\mathbb{R}}^{2}$ such that $\langle y,t\rangle=-\mu$.
Then $u^{\pi}$ is the two dimensional subspace

$\bigcup\bigl{\\{}(x_{1}:x_{2}:1)\,\bigm{|}\,(x_{1},x_{2})\in
L_{y,t}\bigr{\\}}$

of ${\mathbb{R}}^{3}$, that is, $L_{y,t}\times\\{1\\}$ is the line in which
the plane $u^{\pi}$ normal to $u$ in ${\mathbb{R}}^{3}$ cuts the plane
$\\{x_{3}=1\\}$.

##  3\. Clifford Analysis

The basic idea of forming a Clifford algebra ${\mathcal{A}}$ with $n$
generators is to take the smallest real or complex algebra ${\mathcal{A}}$
with an identity element $e_{0}$ such that
${\mathbb{R}}\oplus{\mathbb{R}}^{n}$ is embedded in ${\mathcal{A}}$ via the
identification of
$(x_{0},{\boldsymbol{x}})\in{\mathbb{R}}\oplus{\mathbb{R}}^{n}$ with
$x_{0}e_{0}+{\boldsymbol{x}}\in{\mathcal{A}}$ and the identity

${\boldsymbol{x}}^{2}=-|{\boldsymbol{x}}|^{2}e_{0}=-(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})e_{0}$

holds for all ${\boldsymbol{x}}\in{\mathbb{R}}^{n}$. Then we arrive at the
following definition.

Let ${\mathbb{F}}$ be either the field ${\mathbb{R}}$ of real numbers or the
field ${\mathbb{C}}$ of complex numbers. The Clifford algebra
${\mathbb{F}}_{(n)}$ over ${\mathbb{F}}$ is a $2^{n}$-dimensional algebra with
unit defined as follows. Given the standard basis vectors
$e_{0},e_{1},\dots,e_{n}$ of the vector space ${\mathbb{F}}^{n+1}$, the basis
vectors $e_{S}$ of ${\mathbb{F}}_{(n)}$ are indexed by all finite subsets $S$
of $\\{1,2,\dots,n\\}$. The basis vectors are determined by the following
rules for multiplication on ${\mathbb{F}}_{(n)}$:

$\displaystyle e_{0}=1,\qquad$ $\displaystyle e_{j}^{2}=-1,\qquad$
$\displaystyle{\rm for}\quad 1\leq j\leq n$ $\displaystyle
e_{j}e_{k}=-e_{k}e_{j}=e_{\\{j,k\\}},\qquad$ $\displaystyle{\rm for}\quad
1\leq j<k\leq n$ $\displaystyle e_{j_{1}}e_{j_{2}}\cdots
e_{j_{s}}=e_{S},\qquad$ $\displaystyle{\rm if}\quad 1\leq
j_{1}<j_{2}<\dots<j_{s}\leq n$ $\displaystyle\qquad\qquad{\rm and}\quad
S=\\{j_{1},\dots,j_{s}\\}.$

Here the identifications $e_{0}=e_{\emptyset}$ and $e_{j}=e_{\\{j\\}}$ for
$1\leq j\leq n$ have been made.

Suppose that $m\leq n$ are positive integers. The vector space
${\mathbb{R}}^{m}$ is identified with a subspace of ${\mathbb{F}}_{(n)}$ by
virtue of the embedding
$(x_{1},\dots,x_{m})\longmapsto\sum_{j=1}^{m}x_{j}e_{j}$. On writing the
coordinates of $x\in{\mathbb{R}}^{n+1}$ as $x=(x_{0},x_{1},\dots,x_{n})$, the
space ${\mathbb{R}}^{n+1}$ is identified with a subspace of
${\mathbb{F}}_{(n)}$ with the embedding
$(x_{0},x_{1},\dots,x_{n})\longmapsto\sum_{j=0}^{n}x_{j}e_{j}$.

The product of two elements $u=\sum_{S}u_{S}e_{S}$ and
$v=\sum_{S}v_{S}e_{S},v_{S}\in{\mathbb{F}}$ with coefficients
$u_{S}\in{\mathbb{F}}$ and $v_{S}\in{\mathbb{F}}$ is
$uv=\sum_{S,R}u_{S}v_{R}e_{S}e_{R}$. According to the rules for
multiplication, $e_{S}e_{R}$ is $\pm 1$ times a basis vector of
${\mathbb{F}}_{(n)}$. The scalar part of
$u=\sum_{S}u_{S}e_{S},u_{S}\in{\mathbb{F}}$ is the term $u_{\emptyset}$, also
denoted as $u_{0}$.

The Clifford algebras ${\mathbb{R}}_{(0)},{\mathbb{R}}_{(1)}$ and
${\mathbb{R}}_{(2)}$ are the real, complex numbers and the quaternions,
respectively. In the case of ${\mathbb{R}}_{(1)}$, the vector $e_{1}$ is
identified with $i$ and for ${\mathbb{R}}_{(2)}$, the basis vectors
$e_{1},e_{2},e_{1}e_{2}$ are identified with $i,j,k$ respectively.

The conjugate $\overline{{e_{S}}}$ of a basis element $e_{S}$ is defined so
that $e_{S}\overline{e_{S}}=\overline{e_{S}}e_{S}=1$. Denote the complex
conjugate of a number $c\in{\mathbb{F}}$ by $\overline{c}$. Then the operation
of conjugation $u\longmapsto\overline{u}$ defined by
$\overline{u}=\sum_{S}\overline{u_{S}}\,\overline{e_{S}}$ for every
$u=\sum_{S}u_{S}e_{S},u_{S}\in{\mathbb{F}}$ is an involution of the Clifford
algebra ${\mathbb{F}}_{(n)}$ and $\overline{vu}=\overline{u}\,\overline{v}$
for all elements $u$ and $v$ of ${\mathbb{F}}_{(n)}$. Because $e_{j}^{2}=-1$,
the conjugate $\overline{e_{j}}$ of $e_{j}$ is $-e_{j}$. An inner product is
defined on ${\mathbb{F}}_{(n)}$ by the formula $(u,v)=[u\overline{v}]_{0}=\sum
u_{S}\overline{v_{S}}$ for every $u=\sum_{S}u_{S}e_{S}$ and
$v=\sum_{S}v_{S}e_{S}$ belonging to ${\mathbb{F}}_{(n)}$. The corresponding
norm is written as $|\cdot|$.

For a Banach space $X$, the tensor product
$X_{(n)}:=X\otimes{\mathbb{C}}_{(n)}$ denotes the module of all sums
$u=\sum_{S}u_{S}e_{S}$ with coefficients $u_{S}\in X$ endowed with the norm

$\|u\|_{X_{(n)}}=\left(\sum_{S}\|u_{S}\|_{X}^{2}\right)^{\frac{1}{2}}.$

The left product $\lambda u$ and right product $u\lambda$ are defined in the
obvious way for all $\lambda\in{\mathbb{C}}_{(n)}$. The space
${\mathcal{L}}_{(n)}(X_{(n)})$ of right-module homomorphisms is identified
with ${\mathcal{L}}(X)_{(n)}$ by writing

$\left(\sum_{S}T_{S}e_{S}\right)\left(\sum_{S^{\prime}}u_{S^{\prime}}e_{S^{\prime}}\right)=\sum_{S}(T_{S}u_{S^{\prime}})e_{S}e_{S^{\prime}},$

so that $T(u\lambda)=(Tu)\lambda$ for all $u\in X_{(n)}$,
$\lambda\in{\mathbb{C}}_{(n)}$ and $T={\mathcal{L}}(X)_{(n)}$. The linear
subspace $\\{Te_{0}:T\in{\mathcal{L}}(X)\\}$ of
${\mathcal{L}}(X)_{(n)}\equiv{\mathcal{L}}_{(n)}(X_{(n)})$ is identified with
${\mathcal{L}}(X)$. The norm induced on
${\mathcal{L}}_{(n)}(X_{(n)})\equiv{\mathcal{L}}(X)_{(n)}$ by the norm of
$X_{(n)}$ is given by

$\big{\|}\sum_{S}T_{S}e_{S}\big{\|}_{{\mathcal{L}}_{(n)}(X_{(n)})}=\left(\sum_{S}\|T_{S}\|_{{\mathcal{L}}(X)}^{2}\right)^{\frac{1}{2}}.$

For an $n$-tuple ${{\boldsymbol{A}}}$ of bounded linear operators on $X$, it
is convenient to write $\|{{\boldsymbol{A}}}\|$ for the norm
$\|A_{1}e_{1}+\dots+A_{n}e_{n}\|_{{\mathcal{L}}_{(n)}(X_{(n)})}=\big{(}\sum_{j}\|A_{j}\|_{{\mathcal{L}}(X)}^{2}\big{)}^{\frac{1}{2}}$.

Because $x=(x_{0},x_{1},\dots,x_{n})\in{\mathbb{R}}^{n+1}$ is identified with
the element $\sum_{j=0}^{n}x_{j}e_{j}$ of ${\mathbb{R}}_{(n)}$, the conjugate
$\overline{x}$ of $x$ in ${\mathbb{R}}_{(n)}$ is
$x_{0}e_{0}-x_{1}e_{1}-\cdots-x_{n}e_{n}$. A useful feature of Clifford
algebras is that a nonzero vector $x\in{\mathbb{R}}^{n+1}$ has an inverse
$x^{-1}$ in the algebra ${\mathbb{R}}_{(n)}$ (the Kelvin inverse) given by

$x^{-1}=\frac{\overline{x}}{|x|^{2}}=\frac{x_{0}e_{0}-x_{1}e_{1}-\cdots-
x_{n}e_{n}}{x_{0}^{2}+x_{1}^{2}+\cdots+x_{n}^{2}}.$

The vector $x=(x_{0},x_{1},\dots,x_{n})\in{\mathbb{R}}^{n+1}$ will often be
written as $x=x_{0}e_{0}+{\boldsymbol{x}}$ with
${\boldsymbol{x}}=(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}$.

Because ${\mathbb{C}}_{(n)}$ is an algebra over the complex numbers, the
spectrum ${\sigma}(i{\boldsymbol{x}})$ of the vector $i{\boldsymbol{x}}$ with
${\boldsymbol{x}}\in{\mathbb{R}}^{n}$ is the set of all
$\lambda\in{\mathbb{C}}$ for which $\lambda e_{0}-i{\boldsymbol{x}}$ is not
invertible in ${\mathbb{C}}_{(n)}$. The formula

$(\lambda e_{0}-i{\boldsymbol{x}})^{-1}=\frac{\lambda
e_{0}+i{\boldsymbol{x}}}{\lambda^{2}-|{\boldsymbol{x}}|^{2}},\quad\lambda\neq\pm|{\boldsymbol{x}}|,$

ensures that ${\sigma}(i{\boldsymbol{x}})=\\{\pm|{\boldsymbol{x}}|\\}$ if
${\boldsymbol{x}}\neq 0$ and ${\sigma}(0)=\\{0\\}$. For ${\boldsymbol{x}}\neq
0$, the spectral representaion

$i{\boldsymbol{x}}=|{\boldsymbol{x}}|\smash{\raise
2.15277pt\hbox{$\chi$}}_{+}({\boldsymbol{x}})+(-|{\boldsymbol{x}}|)\smash{\raise
2.15277pt\hbox{$\chi$}}_{-}({\boldsymbol{x}})$

holds with respect to the spectral idempotents

$\smash{\raise
2.15277pt\hbox{$\chi$}}_{\pm}({\boldsymbol{x}})=\frac{1}{2}\left(e_{0}\pm
i\frac{{\boldsymbol{x}}}{|{\boldsymbol{x}}|}\right).$

The vector $i{\boldsymbol{x}}$ is actually selfadjoint with respect to the
inner product of ${\mathbb{C}}_{(n)}$ defined above. For every function
$f:\\{\pm|{\boldsymbol{x}}|\\}\to{\mathbb{C}}$ there is an element

(12) $f(i{\boldsymbol{x}})=f(|{\boldsymbol{x}}|)\smash{\raise
2.15277pt\hbox{$\chi$}}_{+}({\boldsymbol{x}})+f(-|{\boldsymbol{x}}|)\smash{\raise
2.15277pt\hbox{$\chi$}}_{-}({\boldsymbol{x}})$

of ${\mathbb{C}}_{(n)}$ associated with $f$ by the functional calculus for
selfadjoint operators. For a polynomial $p(z)=a_{0}+a_{1}z+\dots+a_{k}z^{k}$,
the expression

$p(i{\boldsymbol{x}})=a_{0}e_{0}+a_{1}(i{\boldsymbol{x}})+\dots+a_{k}(i{\boldsymbol{x}})^{k}$

expected in ${\mathbb{C}}_{(n)}$ is obtained. The identities $\smash{\raise
2.15277pt\hbox{$\chi$}}_{{\mathbb{R}}_{\pm}}(i{\boldsymbol{x}})=\smash{\raise
2.15277pt\hbox{$\chi$}}_{\pm}({\boldsymbol{x}})$ hold for the characteristic
functions $\smash{\raise 2.15277pt\hbox{$\chi$}}_{{\mathbb{R}}_{\pm}}$ of the
half lines ${\mathbb{R}}_{+}=\\{t>0\\}$ and ${\mathbb{R}}_{-}=\\{t<0\\}$.

Given an $n$-tuple of operators ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ acting
on a Hilbert space ${\mathcal{H}}$, the expression
${\mathcal{A}}=\sum_{j=1}^{n}e_{j}A_{j}$ acts on
${{\mathcal{H}}}_{(n)}:={\mathcal{H}}\otimes{\mathbb{C}}_{(n)}$ via the
formula

${\mathcal{A}}u=\sum_{j=1}^{n}\sum_{S}(e_{j}e_{S})(A_{j}u_{S}),\quad
u=\sum_{S}u_{S}e_{S}.$

The coefficients $u_{S}$ are elements of the Hilbert space ${\mathcal{H}}$ and
$u_{S}\otimes e_{S}$ is written simply as $u_{S}e_{S}$ for all
$S\subseteq\\{1,\dots,n\\}$ .

Using Fourier theory, the functional calculus for the selfadjoint differential
operator

${\mathcal{D}}=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}$

acting in the Hilbert space
$L^{2}_{(n)}({\mathbb{R}}^{n}):=L^{2}({\mathbb{R}}^{n})\otimes{\mathbb{C}}_{(n)}$
can be calculated explicitly.

If $\hat{u}(\xi)=\int_{{\mathbb{R}}^{n}}e^{-i\langle x,\xi\rangle}u(x)\,dx$
denotes the Fourier transform of $u\in L^{1}({\mathbb{R}}^{n})$, then
according to the Fourier-Plancherel Theorem, the linear map
$u\longmapsto(2\pi)^{-n/2}\hat{u}$, $u\in L^{1}\cap L^{2}({\mathbb{R}}^{n})$,
extends to an isometry of $L^{2}({\mathbb{R}}^{n})$. For each $j=1,\dots,n$,
the selfadjoint operator $\frac{1}{i}\frac{\partial}{\partial x_{j}}$ defined
in $L^{2}({\mathbb{R}}^{n})$ satisfies

$\left(\frac{1}{i}\frac{\partial u}{\partial
x_{j}}\right)\widehat{\phantom{\big{|}}}(\xi)=\xi_{j}\hat{u}(\xi)$

almost everywhere for each $u\in L^{2}({\mathbb{R}}^{n})$ in its domain.
Furthermore, for any bounded measurable function $\varphi$ defined on
${\mathbb{R}}^{n}$, the operator
$\varphi\left(\frac{1}{i}\frac{\partial}{\partial
x_{1}},\dots,\frac{1}{i}\frac{\partial}{\partial x_{n}}\right)$ satisfies

$\left(\varphi\left(\frac{1}{i}\frac{\partial}{\partial
x_{1}},\dots,\frac{1}{i}\frac{\partial}{\partial
x_{n}}\right)u\right)\widehat{\phantom{\big{|}}}(\xi)=\varphi(\xi)\hat{u}(\xi)$

almost everywhere for $u\in L^{2}({\mathbb{R}}^{n})$. Similarly,

$\big{(}f({\mathcal{D}})u)\big{)}\widehat{\phantom{|}}(\xi)=f(i\xi)\hat{u}(\xi),\quad
u\in L^{2}_{(n)}({\mathbb{R}}^{n}),\ \xi\in{\mathbb{R}}^{n},$

is valid for any bounded measurable function $f:{\mathbb{R}}\to{\mathbb{C}}$
with the understanding that for each $\xi\in{\mathbb{R}}^{n}$, the vector
$f(i\xi)\in{\mathbb{R}}^{n}$ is given by the functional calculus (12) of the
selfadjoint element $i\xi$ of the Clifford algebra ${\mathbb{C}}_{(n)}$.

What is usually called Clifford analysis is the study of functions of finitely
many real variables, which take values in a Clifford algebra, and which
satisfy higher dimensional analogues of the Cauchy-Riemann equations.

It is worthwhile to spell out the direction this analogy takes. The Cauchy-
Riemann equations for a complex valued function $f$ defined in an open subset
of the complex plane may be represented as $\overline{\partial}f=0$ for the
operator

$\overline{\partial}=\frac{\partial}{\partial x}+i\frac{\partial}{\partial
y},\quad z=x+iy\in{\mathbb{C}}.$

The fundamental solution $E$ of the operator $\overline{\partial}$ is the
solution in the sense of Schwartz distributions of the equation
$\overline{\partial}E=\delta_{0}$ for the unit point mass $\delta_{0}$ at
zero. Then

$E(z)=\frac{1}{2\pi}\,\frac{1}{z}=\frac{1}{2\pi}\,\frac{\overline{z}}{|z|^{2}},\quad\hbox{for
}z=x+iy\in{\mathbb{C}}\setminus\\{0\\}.$

A function $f$ satisfying $\overline{\partial}f=0$ in a neighbourhood of a
simple closed contour $C$ together with its interior can be represented as

$f(z)=\frac{1}{i}\int_{C}E(\zeta-z)f({\zeta})\,d{\zeta}=\int_{C}E(\zeta-z){\boldsymbol{n}}({\zeta})f({\zeta})\,d|{\zeta}|$

at all points $z$ inside $C$. Here ${\boldsymbol{n}}({\zeta})$ is the outward
unit normal at ${\zeta}\in{\mathbb{C}}$, $d|{\zeta}|$ is arclength measure so
that $i{\boldsymbol{n}}({\zeta})d|{\zeta}|=d{\zeta}$. The higher dimensional
analogue for functions taking values in a Clifford algebra is as follows.

A function $f:U\to{\mathbb{F}}_{(n)}$ defined in an open subset $U$ of
${\mathbb{R}}^{n+1}$ has a unique representation $f=\sum_{S}f_{S}e_{S}$ in
terms of ${\mathbb{F}}$-valued functions $f_{S}$, $S\subseteq\\{1,\dots,n\\}$
in the sense that $f(x)=\sum_{S}f_{S}(x)e_{S}$ for all $x\in U$. Then $f$ is
continuous, differentiable and so on, in the normed space
${\mathbb{F}}_{(n)}$, if and only if for all finite subsets $S$ of
$\\{1,\dots,n\\}$, its scalar component functions $f_{S}$ have the
corresponding property. Let $\partial_{j}$ be the operator of differentiation
of a scalar function in the $j$’th coordinate in ${\mathbb{R}}^{n+1}$ – the
coordinates of $x\in{\mathbb{R}}^{n+1}$ are written as
$x=(x_{0},x_{1},\dots,x_{n})$. For a continuously differentiable function
$f:U\to{\mathbb{F}}_{(n)}$ with $f=\sum_{S}f_{S}e_{S}$ defined in an open
subset $U$ of ${\mathbb{R}}^{n+1}$, the functions $Df$ and $fD$ are defined by

$\displaystyle Df$
$\displaystyle=\sum_{S}\left((\partial_{0}f_{S})e_{S}+\sum_{j=1}^{n}(\partial_{j}f_{S})e_{j}e_{S}\right)$
$\displaystyle fD$
$\displaystyle=\sum_{S}\left((\partial_{0}f_{S})e_{S}+\sum_{j=1}^{n}(\partial_{j}f_{S})e_{S}e_{j}\right).$

Similarly, the conjugate operator is defined by

$\overline{D}=e_{0}\partial_{0}-\sum_{j=1}^{n}e_{j}\partial_{j}$

so that the Laplace operator $\Delta=\partial_{0}^{2}+\dots+\partial_{n}^{2}$
in ${\mathbb{R}}^{n+1}$ has factorizations

$\Delta=D\overline{D}=\overline{D}D.$

Now suppose that $f$ is an ${\mathbb{F}}_{(n)}$-valued, continuously
differentiable function defined in an open subset $U$ of ${\mathbb{R}}^{n+1}$.
Then $f$ is said to be left monogenic in $U$ if $Df(x)=0$ for all $x\in U$ and
right monogenic in $U$ if $fD(x)=0$ for all $x\in U$.

The following result connects ${\mathbb{R}}^{n+1}$-valued monogenic functions
with systems of conjugate harmonic functions.

######  Proposition 3.1.

Let $F=u_{0}e_{0}-\sum_{j=1}^{n}u_{j}e_{j}$ be an ${\mathbb{R}}^{n+1}$-valued
function defined on an open subset $\Omega$ of ${\mathbb{R}}^{n+1}$.
Conditions (1)-(4) below are equivalent in $\Omega$.

  * (1)

The $(n+1)$-tuple $U=(u_{j})_{j=0}^{n}$ is a system of conjugate harmonic
functions in $\Omega$, that is, $U$ satisfies the generalised Cauchy-Riemann
equations $\hbox{\rm div}\,U=0$ and $\hbox{\rm curl}\,U=0$.

  * (2)

$F$ is left monogenic.

  * (3)

$F$ is right monogenic.

  * (4)

The 1-form $\omega:=u_{0}dx_{0}-u_{1}dx_{1}-\dots-u_{n}dx_{n}$ satisfies
$d{\omega}=0$ and $d^{*}{\omega}=0$, where $d$ and $d^{*}$ are the exterior
differential operator and its formal transpose, respectively.

  * (5)

In the case that $\Omega$ is simply connected, then the above conditions are
equivalent to the existence of a real valued harmonic function $v$ in $\Omega$
such that $U=\hbox{\rm grad}\,v$, so that $F=\overline{D}v$.

For each $x\in{\mathbb{R}}^{n+1}$, the function $G({\,\cdot\,},x)$ defined by

(13)
$G(\omega,x)=G_{\omega}(x)=\frac{1}{\Sigma_{n}}\,\frac{\overline{\omega-x}}{|\omega-x|^{n+1}}$

for every $\omega\neq x$ is both left and right monogenic as a function of
$\omega$. Here the volume
${2\pi^{\frac{n+1}{2}}/\Gamma\left(\frac{n+1}{2}\right)}$ of the unit
$n$-sphere in ${\mathbb{R}}^{n+1}$ has been denoted by $\Sigma_{n}$ and we
have used the identification of ${\mathbb{R}}^{n+1}$ with a subspace of
${\mathbb{R}}_{(n)}$ mentioned earlier.

The function $G({\,\cdot\,},x)$, $x\in{\mathbb{R}}^{n+1}$ plays the role in
Clifford analysis of a Cauchy kernel. Rewriting $G(\omega,x)$ as $E(\omega-x)$
for all $\omega\neq x$ in ${\mathbb{R}}^{n}$, it follows that the
${\mathbb{R}}^{n+1}$-valued function

$E(x)=\frac{1}{\Sigma_{n}}\,\frac{\overline{x}}{|x|^{n+1}}$

defined for all $x\neq 0$ belonging to ${\mathbb{R}}^{n+1}$ is the fundamental
solution of the operator $D$, that is, $DE=\delta_{0}e_{0}$ in the sense of
Schwartz distributions, because
$E=\overline{D}{\Gamma}_{n+1}={\Gamma}_{n+1}\overline{D}$ for the fundamental
solution

${\Gamma}_{n+1}(x)=\left\\{\begin{array}[]{rl}\displaystyle-\frac{1}{(n-1)\Sigma_{n}}\frac{1}{|x|^{n-1}},&x\neq
0,\ n\geq 2\\\ &\\\ \displaystyle\frac{1}{2\pi}\log|x|,&x\neq 0,\
n=1,\end{array}\right.$

of the Laplace operator $\Delta$ in ${\mathbb{R}}^{n+1}$. Then a function
satisfying $Df=0$ in an open set can be retrieved from a surface integral
involving $E$.

Suppose that $\Omega\subset{\mathbb{R}}^{n+1}$ is a bounded open set with
smooth boundary $\partial\Omega$ and exterior unit normal
${\boldsymbol{n}}(\omega)$ defined for all $\omega\in\partial\Omega$. For any
left monogenic function $f$ defined in a neighbourhood $U$ of
$\overline{\Omega}$, the Cauchy integral formula

(14)
$\int_{\partial\Omega}G(\omega,x){\boldsymbol{n}}(\omega)f(\omega)\,d\mu(\omega)=\left\\{\begin{array}[]{rl}f(x),&{\rm
if}\quad x\in\Omega;\cr 0,&{\rm if}\quad x\in
U\setminus\overline{\Omega}.\end{array}\right.$

is valid. Here $\mu$ is the surface measure of $\partial\Omega$. The result is
proved in [9, Corollary 9.6] by appealing to by Stoke’s theorem. If $g$ is
right monogenic in $U$ then
$\int_{\partial\Omega}g(\omega){\boldsymbol{n}}(\omega)f(\omega)\,d\mu(\omega)=0$
[9, Corollary 9.3]. The manifold $\partial\Omega$ integral (14) can then be
deformed across any set not containing $x$.

In terms of differential forms, the ${\mathbb{F}}^{n+1}$-valued $n$-form

$\eta=\sum_{j=0}^{n}(-1)^{j}e_{j}\,dx_{0}\wedge\dots\wedge\widehat{dx_{j}}\wedge\dots\wedge
dx_{n}\quad(\,\widehat{\phantom{a}}\equiv\hbox{omitted}),$

is defined on ${\mathbb{R}}^{n+1}$ and its pullback to the orientable
$n$-dimensional manifold $\partial\Omega$ is denoted by the same symbol. Then
for any continuous ${\mathbb{F}}_{(n)}$-valued functions $u,v$ on
$\partial\Omega$, the equality

$\int_{\partial\Omega}u\eta
v=\int_{\partial\Omega}u(\omega){\boldsymbol{n}}(\omega)v(\omega)\,d\mu(\omega)$

holds. Now suppose that $f,g$ are any continuously differentiable
${\mathbb{F}}_{(n)}$-valued functions defined on $U$. The differential of
$g\eta f$ is equal to $\big{(}(gD)f+g(Df)\big{)}\,dx_{0}\wedge\dots\wedge
dx_{n}$, so Stokes’ Theorem gives

$\int_{\Omega}\big{(}(gD)f+g(Df)\big{)}\,d\lambda=\int_{\partial\Omega}g\eta
f$

with respect to Lebesgue measure $\lambda$ on ${\mathbb{R}}^{n+1}$. The Cauchy
integral formula (14) follows by shrinking $\partial\Omega$ to a sphere about
$x\in\Omega$, see [9, Corollary 9.6].

######  Example 3.2.

For the case $n=1$, the Clifford algebra ${\mathbb{R}}_{(1)}$ is identified
with ${\mathbb{C}}$. A continuously differentiable function
$f:U\to{\mathbb{R}}_{(1)}$ defined in an open subset $U$ of ${\mathbb{R}}^{2}$
satisfies $Df=0$ in $U$ if and only if it satisfies the Cauchy-Riemann
equations $\overline{\partial}f=0$ in $U$. For each
$x,\omega\in{\mathbb{R}}^{2}$, $x\neq\omega$, the formula

$G(\omega,x)=\frac{1}{2\pi}\ \frac{1}{\omega-x}$

holds. The inverse is taken in ${\mathbb{C}}$. As indicated above, the tangent
at the point ${\zeta}(t)$ of the portion $\\{{\zeta}(s):a<s<b\\}$ of a
positively oriented rectifiable curve $C$ is $i$ times the normal
${\boldsymbol{n}}({\zeta}(t))$ at ${\zeta}(t)$, so the equality
$d{\zeta}=i.{\boldsymbol{n}}({\zeta})\,d|{\zeta}|$ shows that (14) is the
Cauchy integral formula for a simple closed contour $C$ bounding a region
$\Omega$.

##  4\. The Cauchy kernel

Armed with the Cauchy integral formula (14) for monogenic functions, formula
(25) is established for the $n$-tuple ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$
of bounded linear operators on a Banach space $X$ by substituting the
$n$-tuple ${\boldsymbol{A}}$ for the vector $y\in{\mathbb{R}}^{n}$. If $n$ is
odd and ${\boldsymbol{A}}$ is a commutative $n$-tuple, that is,
$A_{j}A_{k}=A_{k}A_{j}$ for $j,k=1,\dots,n$, and each operator $A_{j}$ has
real spectrum ${\sigma}(A_{j})\subset{\mathbb{R}}$ for $j=1,\dots,n$, then for
suitable $\omega\in{\mathbb{R}}^{n+1}$, the expression

(15) $G_{\omega}({\boldsymbol{A}})=\frac{1}{\Sigma_{n}}|\omega
I-{\boldsymbol{A}}|^{-n-1}(\overline{\omega I-{\boldsymbol{A}}})$

makes sense as an element of ${\mathcal{L}}_{(n)}(X_{(n)})$. For an even
integer $m$

$|\omega
I-{\boldsymbol{A}}|^{-m}=\bigg{(}\big{(}\omega_{0}^{2}I+\sum_{j=1}^{n}(\omega_{j}I-A_{j})^{2}\big{)}^{-1}\bigg{)}^{m/2}$

and $\overline{\omega
I-{\boldsymbol{A}}}=\omega_{0}I-\sum_{j=1}^{n}(\omega_{j}I-A_{j})e_{j}$ for
${\omega}={\omega}_{0}e_{0}+{\boldsymbol{{\omega}}}\in{\mathbb{R}}^{n+1}$,
${\boldsymbol{{\omega}}}=\sum_{j=1}^{n}{\omega}_{j}e_{j}$.

An appeal to the Spectral Mapping Theorem shows that the operator

$\omega_{0}^{2}I+\sum_{j=1}^{n}(\omega_{j}I-A_{j})^{2}$

is invertible in ${\mathcal{L}}(X)$ for each $\omega_{0}\neq 0$, so the
${\mathcal{L}}_{(n)}(X_{(n)})$-valued function ${\omega}\longmapsto
G_{\omega}({\boldsymbol{A}})$ is defined on the set
${\mathbb{R}}^{n+1}\setminus(\\{0\\}\times{\gamma}({\boldsymbol{A}}))$ with

${\gamma}({\boldsymbol{A}})=\left\\{({\omega}_{1},\dots,{\omega}_{n}):\sum_{j=1}^{n}(\omega_{j}I-A_{j})^{2}\hbox{
is not invertible in }{\mathcal{L}}(X)\,\right\\}.$

Off ${\gamma}({\boldsymbol{A}})$, the function ${\omega}\longmapsto
G_{\omega}({\boldsymbol{A}})$ is left and right monogenic and formula (25)
defines a functional calculus $f\longmapsto f({\boldsymbol{A}})$ which
coincides with Taylor’s functional calculus
$\tilde{f}\longmapsto\tilde{f}({\boldsymbol{A}})$. Any left monogenic function
$f$ defined in a neighbourhood $U$ of ${\gamma}({\boldsymbol{A}})$ in
${\mathbb{R}}^{n+1}$ has a holomorphic counterpart $\tilde{f}$ defined in a
neighborhood $\tilde{U}$ of ${\gamma}({\boldsymbol{A}})$ in ${\mathbb{C}}^{n}$
by taking the power series expansion about points of
$U\cap(\\{0\\}\times{\mathbb{R}}^{n})$ [9]. The left monogenic function $f$ is
referred to as the Cauchy-Kowaleski extension of
$\tilde{f}\restriction(\tilde{U}\cap{\mathbb{R}}^{n})$ to ${\mathbb{R}}^{n+1}$
[9].

In the case of even $n=2,4,\dots$, the operator $|\omega
I-{\boldsymbol{A}}|^{-n-1}$ needs to be defined suitably. The direct
formulation employs Taylor’s functional calculus, but by using the plane wave
decomposition of the Cauchy kernel (see Section 4.2), the case of even $n$ and
noncommuting operators can be treated simultaneously.

For an $n$-tuple ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ of commuting bounded
linear operators on a Banach space $X$ with real spectra, the nonempty compact
subset ${\gamma}({\boldsymbol{A}})$ of ${\mathbb{R}}^{n}$ coincides with
Taylor’s joint spectrum defined in terms of the Koszul complex [37, Definition
III.6.4].

In general, the symbol ${\gamma}({\boldsymbol{A}})$ is used to denote the set
of points of ${\mathbb{R}}^{n+1}$ in the complement of the domain where the
Cauchy kernel ${\omega}\longmapsto G_{\omega}({\boldsymbol{A}})$ is defined
and monogenic in the Banach module ${\mathcal{L}}_{(n)}(X_{(n)})$. For a
single operator $A$, its spectrum ${\sigma}(A)$ is precisely the set of
singularities of the Cauchy kernel or resolvent $\lambda\longmapsto(\lambda
I-A)^{-1}$ for $\lambda\in{\mathbb{C}}$, that is, the set of
$\lambda\in{\mathbb{C}}$ for which $\lambda I-A$ is not invertible in
${\mathcal{L}}(X)$.

A commuting $n$-tuple ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ of bounded
selfadjoint operators on a Hilbert space $H$ has a ready-made functional
calculus given by formula (25). The support of the joint spectral measure
$P_{{\boldsymbol{A}}}$ is naturally interpreted as the joint spectrum
${\sigma}({\boldsymbol{A}})$ of ${\boldsymbol{A}}$. The observation that
${\sigma}({\boldsymbol{A}})$ is actually the Gelfand spectrum of the
commutative $C^{*}$-algebra generated by ${\boldsymbol{A}}$ lends credence to
the interpretation of ${\sigma}({\boldsymbol{A}})$ as the joint spectrum. By
setting
$G_{\omega}({\boldsymbol{A}})=\int_{{\sigma}({\boldsymbol{A}})}G_{\omega}(\lambda)\,dP_{{\boldsymbol{A}}}(\lambda)$
for all
${\omega}\in{\mathbb{R}}^{n+1}\setminus(\\{0\\}\times{\sigma}({\boldsymbol{A}}))$,
it is easy to check by the vector valued version of Fubini’s Theorem, that
with the assumptions of formula (25) below for the left monogenic function
$f:U\to{\mathbb{C}}_{(n)}$ and the open set $\Omega$, the equalities

$\displaystyle\int_{\partial\Omega}G_{\omega}({\boldsymbol{A}}){\boldsymbol{n}}({\omega})f({\omega})\,d\mu({\omega})$
$\displaystyle=$
$\displaystyle\int_{\partial\Omega}\left(\int_{{\sigma}({\boldsymbol{A}})}G_{\omega}(\lambda)\,dP_{{\boldsymbol{A}}}(\lambda)\right){\boldsymbol{n}}({\omega})f({\omega})\,d\mu({\omega})$
$\displaystyle=$
$\displaystyle\int_{{\sigma}({\boldsymbol{A}})}\left(\int_{\partial\Omega}G_{\omega}(\lambda){\boldsymbol{n}}({\omega})f({\omega})\,d\mu({\omega})\right)\,dP_{{\boldsymbol{A}}}(\lambda)$
$\displaystyle=$
$\displaystyle\int_{{\sigma}({\boldsymbol{A}})}f(\lambda)\,dP_{{\boldsymbol{A}}}(\lambda)$
$\displaystyle=$ $\displaystyle f({\boldsymbol{A}})$

hold. Furthermore ${\gamma}({\boldsymbol{A}})={\sigma}({\boldsymbol{A}})$.

###  4.1. The Weyl calculus

There is an operator valued distribution ${\mathcal{W}}_{{\boldsymbol{A}}}$ on
${\mathbb{R}}^{n}$ associated with any $n$-tuple
${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ of selfadjoint operators on a Hilbert
space $H$. Now the operators $A_{1},\dots,A_{n}$ need not commute with each
other. The distribution ${\mathcal{W}}_{{\boldsymbol{A}}}$ is a substitute for
the joint spectral measure $P_{{\boldsymbol{A}}}$. If ${\boldsymbol{A}}$ is a
commuting $n$-tuple of bounded selfadjoint operators, then
${\mathcal{W}}_{{\boldsymbol{A}}}:f\longmapsto\int_{{\sigma}({\boldsymbol{A}})}f\,dP_{{\boldsymbol{A}}}$
for all smooth functions defined in a neighbourhood of
${\sigma}({\boldsymbol{A}})$ in ${\mathbb{R}}^{n}$.

Suppose that $T:C^{\infty}({\mathbb{R}}^{n})\to{\mathcal{L}}(X)$ is an
operator valued distribution with compact support supp$(T)$ acting on a Banach
space $X$. For any smooth ${\mathbb{C}}_{(n)}$-valued function function
$f=\sum_{S}f_{S}e_{S}$ defined in a neighborhood of supp$(T)$ in
${\mathbb{R}}^{n}$, the element $Tf$ of ${\mathcal{L}}_{(n)}(X_{(n)})$ is
defined by $Tf=\sum_{S}e_{S}T(f_{S})$

The Cauchy integral formula (14) may be viewed as an equality

$f=\int_{\partial\Omega}G_{\omega}{\boldsymbol{n}}(\omega)f(\omega)\,d\mu(\omega)$

between smooth ${\mathbb{C}}_{(n)}$-valued functions defined in a
neighbourhood $U\cap{\mathbb{R}}^{n}$ of the support of $T$ when $f$ is left
monogenic on $U$, supp$(T)\subset\Omega$ and $\overline{\Omega}\subset U$, so
that

$\displaystyle Tf$ $\displaystyle=$ $\displaystyle
T\int_{\partial\Omega}G_{\omega}{\boldsymbol{n}}(\omega)f(\omega)\,d\mu(\omega)$
$\displaystyle=$
$\displaystyle\int_{\partial\Omega}T(G_{\omega}){\boldsymbol{n}}(\omega)f(\omega)\,d\mu(\omega).$

The last inequality is a property of the Bochner integral for functions with
values in the Fréchet space $C^{\infty}_{(n)}({\mathbb{R}}^{n})$. The
${\mathcal{L}}_{(n)}(X_{(n)})$-valued function ${\omega}\longmapsto
T(G_{\omega})$ is left and right monogenic in ${\mathbb{R}}^{n+1}$ away from
supp$(T)$. If $\varphi$ is a smooth function with compact support in a
neighbourhood of supp$(T)$, $x\in X$ and $\xi\in X^{\prime}$, then according
to [9, Theorem 27.7],

$\langle(T\varphi)x,\xi\rangle=\lim_{t\to
0+}\int_{{\mathbb{R}}^{n}}\langle\left(T(G_{u+te_{0}})-T(G_{u-te_{0}})\right)x,\xi\rangle\varphi(u)\,du,$

which may be compared with equation (8).

For any $n$-tuple ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ of bounded
selfadjoint operators on a Hilbert space $H$, the Weyl functional calculus is
the ${\mathcal{L}}(H)$-valued distribution

(16)
${\mathcal{W}}_{{\boldsymbol{A}}}=\frac{1}{(2\pi)^{n}}\left(e^{i\langle\xi,{\boldsymbol{A}}\rangle}\right)\hat{}\phantom{|}.$

The operator
$\langle\xi,{\boldsymbol{A}}\rangle=\langle{\boldsymbol{A}},\xi\rangle=\sum_{j=1}^{n}\xi_{j}A_{j}$
is selfadjoint for each $\xi\in{\mathbb{R}}^{n}$ and the Fourier transform is
taken with respect $\xi$ in the sense of distributions. If ${\boldsymbol{A}}$
is a commuting system, then the distribution
${\mathcal{W}}_{{\boldsymbol{A}}}$ is integration with respect to the joint
spectral measure $P_{\boldsymbol{A}}$.

Setting
$G_{\omega}({\boldsymbol{A}})={\mathcal{W}}_{{\boldsymbol{A}}}(G_{\omega}\restriction{\mathbb{R}}^{n})$
for
${\omega}\in{\mathbb{R}}^{n+1}\setminus(\\{0\\}\times\hbox{supp}({\mathcal{W}}_{{\boldsymbol{A}}}))$,
the equality
$f({\boldsymbol{A}})={\mathcal{W}}_{{\boldsymbol{A}}}(f\restriction{\mathbb{R}}^{n})$
holds for the element $f({\boldsymbol{A}})$ of ${\mathcal{L}}_{(n)}(H_{(n)})$
defined by formula (25) below and
${\gamma}({\boldsymbol{A}})=\hbox{supp}({\mathcal{W}}_{{\boldsymbol{A}}})$. E.
Nelson [28] has identified ${\gamma}({\boldsymbol{A}})$ with the Gelfand
spectrum of a certain commutative Banach algebra.

######  Example 4.1.

Let $n=3$ and consider the simplest noncommuting example of the Pauli
matrices,

(17) $\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\qquad\sigma_{2}=\left(\begin{array}[]{cc}0&{-i}\\\
{i}&0\end{array}\right),\qquad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&{-1}\end{array}\right),$

viewed as linear transformations acting on $H={\mathbb{C}}^{2}$. Set
${\boldsymbol{\sigma}}=(\sigma_{1},\sigma_{2},\sigma_{3})$. Then
$\langle\xi,{\boldsymbol{{\sigma}}}\rangle^{2}=|\xi|^{2}I$, so the exponential
series gives

$e^{it\langle\xi,{\boldsymbol{{\sigma}}}\rangle}=\cos(t|\xi|)I+i\langle\xi,{\boldsymbol{{\sigma}}}\rangle\frac{\sin(t|\xi|)}{|\xi|},\quad
t\in{\mathbb{R}}.$

Because $(2\pi)^{-3}({\sin(t|\xi|)}/|\xi|)\,\hat{}=t\mu_{t}$, with $\mu_{t}$
the unit surface measure on the sphere $tS^{2}$ of radius $t>0$ centred at
zero in ${\mathbb{R}}^{3}$ and

$\cos(t|\xi|)=\frac{d}{ds}\,\frac{\sin(s|\xi|)}{|\xi|}\bigg{|}_{s=t},$

for each $f\in C^{\infty}({\mathbb{R}}^{3})$, the matrix
${\mathcal{W}}_{t{\boldsymbol{{\sigma}}}}(f)$ is given by

(18)
${\mathcal{W}}_{t{\boldsymbol{{\sigma}}}}(f)=I\int_{tS^{2}}\left(f+{\boldsymbol{n}}\cdot\nabla
f\right)\,d\mu_{t}+t\int_{tS^{2}}{{\boldsymbol{\sigma}}}\cdot\nabla
f\,d\mu_{t},\quad t>0.$

Here ${\boldsymbol{n}}(x)$ is the outward unit normal at $x\in S^{2}$. Thus,
$\hbox{\rm supp}\left({\mathcal{W}}_{{\boldsymbol{{\sigma}}}}\right)=S^{2}$.

For all $\omega={\omega}_{0}+{\boldsymbol{{\omega}}}\in{\mathbb{R}}^{4}$ such
that ${\boldsymbol{{\omega}}}\notin S^{2}\subset{\mathbb{R}}^{3}$, the Cauchy
kernel
$G_{\omega}({\boldsymbol{{\sigma}}})\in{\mathcal{L}}({\mathbb{C}}^{2})_{(3)}$
is given by

$G_{\omega}({\boldsymbol{{\sigma}}})={\mathcal{W}}_{{\boldsymbol{{\sigma}}}}(G_{\omega})=I\int_{S^{2}}\left(G_{\omega}+{\boldsymbol{n}}\cdot\nabla
G_{\omega}\right)\,d\mu+\int_{S^{2}}{{\boldsymbol{\sigma}}}\cdot\nabla
G_{\omega}d\mu.$

By the operator valued version of the Paley-Wiener Theorem (see [31, Theorem
7.23] for the scalar version), the distribution (16) exists for $n$-tuple
${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ of bounded linear operators on a Banach
space $X$ provided that the exponential growth estimate

(19)
$\left\|e^{i\langle{\zeta},{\boldsymbol{A}}\rangle}\right\|_{{\mathcal{L}}(X)}\leq
C(1+|\xi|)^{s}e^{r|\eta|},\quad{\zeta}=\xi+i\eta,\
\xi,\eta\in{\mathbb{R}}^{n},$

for $\langle{\zeta},{\boldsymbol{A}}\rangle=\sum_{j=1}^{n}{\zeta}_{j}A_{j}$
holds, for some positive numbers $C,s,r$ independent of
${\zeta}\in{\mathbb{C}}^{n}$. Then supp$({\mathcal{W}}_{{\boldsymbol{A}}})$ is
contained in the ball of radius $r>0$ centred at zero in ${\mathbb{R}}^{n}$.
The estimate (19) holds if ${\boldsymbol{A}}$ is finite system of
simultaneously triangularisable matrices with real eigenvalues [21, Theorem
5.10].

The Weyl functional calculus ${\mathcal{W}}_{{\boldsymbol{A}}}$ has the
property that operator products are symmetrically ordered. For an $n$-tuple
${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ of bounded selfadjoint operators on a
Hilbert space $H$, other choices of operator ordering define an operator
valued distribution ${\mathcal{F}}_{{\boldsymbol{A}},{\boldsymbol{\mu}}}$. The
weighting for operator products is determined by an $n$-tuple
${\boldsymbol{\mu}}=(\mu_{1},\dots,\mu_{n})$ of continuous Borel probability
measures on $[0,1]$, see [21, Chapter 7].

The collection of single operators $A$ satisfying the bound (19) is called the
class of generalized scalar operators and these have been extensively studied
[13].

###  4.2. Plane wave decomposition of the Cauchy kernel

The exponential growth estimate (19) leads to a $C^{\infty}$-functional
calculus ${\mathcal{W}}_{\boldsymbol{A}}$, so it is desirable to have a
condition weaker than (19) for which the Cauchy kernel can be defined in a way
that agrees with the preceding definition.

It turns out that it suffices for ${{\boldsymbol{A}}}$ to be hyperbolic to
make sense of the Cauchy kernel $G_{\omega}({\boldsymbol{A}})$ in the Cauchy
integral formula (25). For matrices, this is equivalent to the bound (19),
because as noted in the Introduction it says that

$I\frac{\partial}{\partial t}+\sum_{j=1}^{n}A_{j}\frac{\partial}{\partial
x_{j}}$

is a hyperbolic differential operator on ${\mathbb{R}}^{n+1}$.

The key to constructing the Cauchy kernel $G_{\omega}({\boldsymbol{A}})$ for a
hyperbolic $n$-tuple ${\boldsymbol{A}}$ of bounded linear operators is the
plane wave decomposition of the fundamental solution

$E:x\longmapsto\frac{1}{\Sigma_{n}}\frac{\overline{x}}{|x|^{n+1}},\quad
x\in{\mathbb{R}}^{n+1}\setminus\\{0\\},$

of the generalised Cauchy-Riemann operator
$D=\sum_{j=0}^{n}e_{j}\partial_{j}$. The plane wave decomposition of $E$ was
first given by F. Sommen [34] and is most simply realised with the proof of
Li, McIntosh, Qian [27] using Fourier analysis. The unit hypersphere $S^{n-1}$
in ${\mathbb{R}}^{n}$ is the set $\\{s\in{\mathbb{R}}^{n}:|s|=1\,\\}$.

The Fourier transform of $u\in L^{1}\cap L^{2}({\mathbb{R}}^{n})$ is
$\hat{u}(\xi)=\int_{{\mathbb{R}}^{n}}e^{-i\langle x,\xi\rangle}u(x)\,dx$ and
the inverse map $u(x)=(2\pi)^{-n}\int_{{\mathbb{R}}^{n}}e^{i\langle
x,\xi\rangle}\hat{u}(\xi)\,d\xi$ recovers $u$ from $\hat{u}$ when $\hat{u}\in
L^{1}\cap L^{2}({\mathbb{R}}^{n})$. For any $\xi\in{\mathbb{R}}^{n}$, the
linear function ${\boldsymbol{x}}\longmapsto
i\langle{\boldsymbol{x}},\xi\rangle$, ${\boldsymbol{x}}\in{\mathbb{R}}^{n}$,
defined in ${\mathbb{R}}^{n}$ extends monogenically to ${\mathbb{R}}^{n+1}$ to
the function

$x\longmapsto i\langle{\boldsymbol{x}},\xi\rangle
e_{0}-i{\boldsymbol{x}}x_{0},\quad
x=x_{0}e+{\boldsymbol{x}},x_{0}\in{\mathbb{R}},\
{\boldsymbol{x}}\in{\mathbb{R}}^{n}.$

According to the functional calculus for the selfadjoint element
$i{\boldsymbol{x}}$ of ${\mathbb{C}}_{(n)}$, the unique monogenic extension of
the function ${\boldsymbol{x}}\longmapsto
e^{i\langle{\boldsymbol{x}},\xi\rangle}$,
${\boldsymbol{x}}\in{\mathbb{R}}^{n}$, is given by

$\displaystyle\exp(i\langle{\boldsymbol{x}},\xi\rangle
e_{0}-i{\boldsymbol{x}}x_{0})$ $\displaystyle=$ $\displaystyle
e^{i\langle{\boldsymbol{x}},\xi\rangle-|{\boldsymbol{x}}|x_{0}}\smash{\raise
2.15277pt\hbox{$\chi$}}_{+}({\boldsymbol{x}})+e^{i\langle{\boldsymbol{x}},\xi\rangle+|{\boldsymbol{x}}|x_{0}}\smash{\raise
2.15277pt\hbox{$\chi$}}_{-}({\boldsymbol{x}})$ $\displaystyle=$ $\displaystyle
e_{+}(x,\xi)+e_{-}(x,\xi)$

for $x=x_{0}e+{\boldsymbol{x}}$ with $x_{0}\in{\mathbb{R}}$ and
${\boldsymbol{x}}\in{\mathbb{R}}^{n}$.

######  Theorem 4.2.

Let $x=x_{0}e_{0}+{\boldsymbol{x}}$ be an element of ${\mathbb{R}}^{n+1}$ with
${\boldsymbol{x}}\in{\mathbb{R}}^{n}$. If $x_{0}>0$, then

(20)
$E(x)=\frac{(n-1)!}{2}\left(\frac{i}{2\pi}\right)^{n}\int_{S^{n-1}}(e_{0}+is)\left(\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s\right)^{-n}\/ds.$

If $x_{0}<0$, then

(21)
$E(x)=(-1)^{n+1}\frac{(n-1)!}{2}\left(\frac{i}{2\pi}\right)^{n}\int_{S^{n-1}}(e_{0}+is)\left(\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s\right)^{-n}\/ds.$

Now suppose that ${\boldsymbol{A}}$ is a hyperbolic $n$-tuple of bounded
linear operators on a Banach space $X$. Then
$\langle{{\boldsymbol{A}}},s\rangle-x_{0}sI$ is an element of the space
${\mathcal{L}}_{(n)}(X_{(n)})$ of module homomorphisms for each $s\in
S^{n-1}$. When $x_{0}\neq 0$ and $a\in{\mathbb{R}}$, the inverse of
$(aI-\langle{{\boldsymbol{A}}},s\rangle)e_{0}-x_{0}sI$ in the Clifford module
${\mathcal{L}}_{(n)}(X_{(n)})$ is given by

$((aI-\langle{{\boldsymbol{A}}},s\rangle)e_{0}-x_{0}sI)^{-1}=\left((aI-\langle{{\boldsymbol{A}}},s\rangle)+x_{0}sI\right)((aI-\langle{{\boldsymbol{A}}},s\rangle)^{2}+x_{0}^{2}I)^{-1}.$

Because ${\sigma}(\langle\xi,{\boldsymbol{A}}\rangle)\subset{\mathbb{R}}$, the
Spectral Mapping Theorem ensures that the bounded linear operator
$(aI-\langle{{\boldsymbol{A}}},s\rangle)^{2}+x_{0}^{2}I$ is invertible and

${\sigma}((aI-\langle{{\boldsymbol{A}}},s\rangle)^{2}+x_{0}^{2}I)=\varphi_{a}({\sigma}(\langle
s,{{\boldsymbol{A}}}\rangle))$

for the function $\varphi_{a}:t\longmapsto(a-t)^{2}+x_{0}^{2}$,
$t\in{\mathbb{R}}$. Moreover,

$((aI-\langle{{\boldsymbol{A}}},s\rangle)e_{0}-x_{0}sI)^{-n}=\left((aI-\langle{{\boldsymbol{A}}},s\rangle)e_{0}+x_{0}sI\right)^{n}((aI-\langle{{\boldsymbol{A}}},s\rangle)^{2}+x_{0}^{2}I)^{-n}$

in ${\mathcal{L}}_{(n)}(X_{(n)})$. To define $G_{x}({\boldsymbol{A}})$, the
promised substitution ${\boldsymbol{y}}\longrightarrow{\boldsymbol{A}}$ in the
Cauchy kernel $G_{x}({\boldsymbol{y}})=E(x-{\boldsymbol{y}})$ is now made by
setting

(22)
$G_{x}({\boldsymbol{A}})=\frac{(n-1)!}{2}\left(\frac{i}{2\pi}\right)^{n}\int_{S^{n-1}}(e_{0}+is)\left((\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{-n}\/ds$

for $x=x_{0}e_{0}+{\boldsymbol{x}}$ with ${\boldsymbol{x}}\in{\mathbb{R}}^{n}$
and $x_{0}>0$, and

(23)
$G_{x}({\boldsymbol{A}})=(-1)^{n+1}\frac{(n-1)!}{2}\left(\frac{i}{2\pi}\right)^{n}\int_{S^{n-1}}(e_{0}+is)\left((\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{-n}\/ds$

for $x_{0}<0$. The expression $\left((\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{-n}$ is left and right
monogenic in ${\mathcal{L}}_{(n)}(X_{(n)})$ for the variable
$x=x_{0}e_{0}+{\boldsymbol{x}}$ with $x_{0}\neq 0$, so differentiating under
the integral sign shows that $x\longmapsto G_{x}({\boldsymbol{A}})$ is itself
two sided monogenic for $x_{0}\neq 0$.

Note that for $n$ even, symmetry in the integral gives

$\displaystyle\int_{S^{n-1}}(e_{0}+is)\left((\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{-n}\/ds=e_{0}\int_{S^{n-1}}\left((\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{-n}\/ds$
$\displaystyle\qquad=e_{0}\int_{S^{n-1}}\left((\langle{{\boldsymbol{x}}},s\rangle+x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{n}\left((\langle{{\boldsymbol{x}}},s\rangle
I-\langle{{\boldsymbol{A}}},s\rangle)^{2}+x_{0}^{2}I\right)^{-n}\/ds$

and for $n$ odd

$\displaystyle\int_{S^{n-1}}(e_{0}+is)\left((\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{-n}\/ds=i\int_{S^{n-1}}s\left((\langle{{\boldsymbol{x}}},s\rangle-
x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{-n}\/ds$
$\displaystyle\qquad=i\int_{S^{n-1}}\left((\langle{{\boldsymbol{x}}},s\rangle+x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{n}\left((\langle{{\boldsymbol{x}}},s\rangle
I-\langle{{\boldsymbol{A}}},s\rangle)^{2}+x_{0}^{2}I\right)^{-n}\/s\,ds.$

###  4.3. The McIntosh functional calculus

For a general matrix or operator $A$, the Riesz-Dunford formula

(24) $f(A)=\frac{1}{2\pi i}\int_{C}(\lambda I-A)^{-1}f(\lambda)\,d\lambda$

is valid for all functions $f$ holomorphic in a neighbourhood of the spectrum
${\sigma}(A)$ in the complex plane. The simple closed contour $C$ surrounds
${\sigma}(A)$ and is contained in the domain of $f$. The higher-dimensional
analogue of the Riesz-Dunford formula (24) is then

(25)
$f({\boldsymbol{A}})=\int_{\partial\Omega}G_{x}({\boldsymbol{A}}){\boldsymbol{n}}(x)f(x)\,d\mu(x)$

for the hyperbolic $n$-tuple ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ of bounded
linear operators on a Banach space $X$, simply by substituting the $n$-tuple
${\boldsymbol{A}}$ for the vector $x\in\\{0\\}\times{\mathbb{R}}^{n}$ in the
Cauchy integral formula (14). The price paid is that Clifford regular
functions have values in the Clifford algebra ${\mathbb{C}}_{(n)}$, which is
noncommutative for $n=2,3,\dots$ and the correspondence between Clifford
regular functions and their holomorphic counterparts needs to be investigated.
The algebra ${\mathbb{R}}_{(1)}$ is isomorphic to ${\mathbb{C}}$ and formulas
(25) and (24) coincide in the case $n=1$.

Let ${\boldsymbol{A}}$ is a hyperbolic $n$-tuple of bounded linear operators
on a Banach space $X$ so that $G_{\cdot}({\boldsymbol{A}})$ is defined by the
plane wave decomposition. If $x\longmapsto G_{x}({\boldsymbol{A}})$ has a
continuous extension to a neighborhood in ${\mathbb{R}}^{n+1}$ of a point
${\boldsymbol{a}}\in{\mathbb{R}}^{n}$, then $G_{\cdot}({\boldsymbol{A}})$ is
actually monogenic in a neighborhood of ${\boldsymbol{a}}$ in
${\mathbb{R}}^{n+1}$ by the monogenic analogue of Painlevé’s Theorem [9,
Theorem 10.6 p. 64].

The joint monogenic spectrum ${\gamma}({\boldsymbol{A}})$ of the $n$-tuple
${\boldsymbol{A}}$ is the subset of ${\mathbb{R}}^{n}$ for which
$\\{0\\}\times{\gamma}({\boldsymbol{A}})$ is the set of singularities of the
Cauchy kernel $G_{\cdot}({\boldsymbol{A}})$. In the case that the $n$-tuple
${\boldsymbol{A}}$ satisfies the exponential growth estimates (19), the joint
spectrum ${\gamma}({\boldsymbol{A}})$ coincides with
supp$({\mathcal{W}}_{{\boldsymbol{A}}})$ [21, Theorem 4.8]. In the case that
the $n$-tuple ${\boldsymbol{A}}$ consists of bounded selfadjoint operators,
${\gamma}({\boldsymbol{A}})$ equals the Gelfand spectrum of a certain
commutative Banach algebra (operants) associated with ${\boldsymbol{A}}$ [28].
The monogenic analogue of Liouville’s Theorem ensures that the joint spectrum
${\gamma}({\boldsymbol{A}})$ is nonempty and compact [21, Theorem 4.16].

The joint spectral radius
$r({\boldsymbol{A}})=\sup\\{|x|:x\in{\gamma}({{\boldsymbol{A}}})\,\\}$ of the
hyperbolic $n$-tuple ${{\boldsymbol{A}}}$ is the radius of the smallest ball
centred at zero containing ${\gamma}({{\boldsymbol{A}}})$.

The joint spectrum ${\gamma}({\boldsymbol{A}})$ is the analogue of the
spectrum ${\sigma}(A)$ of a single operator $A$ in the sense that it is the
set of singularities of the Cauchy kernel $\lambda\longmapsto(\lambda
I-A)^{-1}$ in the Riesz-Dunford formula (24), that is, the set of
$\lambda\in{\mathbb{C}}$ for which $\lambda I-A$ is not invertible — the first
is the analytic viewpoint and the second is the algebraic viewpoint.

Given a left monogenic function $f$ defined in a neighbourood $U$ of the joint
spectrum ${\gamma}({\boldsymbol{A}})$, the element $f({\boldsymbol{A}})$ of
${\mathcal{L}}_{(n)}(X_{(n)})$ is defined by formula $(\ref{eqn:fc})$
independently of the oriented $n$-manifold $\partial\Omega$ such that
${\gamma}({\boldsymbol{A}})\subset\Omega$ and $\overline{\Omega}\subset U$.

A real analytic function $f:V\to{\mathbb{C}}$ defined in a neighbourhood $V$
of ${\gamma}({\boldsymbol{A}})$ in ${\mathbb{R}}^{n}$ has a unique two-side
monogenic extension $\tilde{f}$ (the Cauchy-Kowaleski extension) to a
neighborhood $U$ of $\\{0\\}\times{\gamma}({\boldsymbol{A}})$ in
${\mathbb{R}}^{n+1}$. The extension is provided by an expansion in a series of
monogenic polynomials [9]. Then the definition
$f({\boldsymbol{A}}):=\tilde{f}({\boldsymbol{A}})$ makes sense and does not
depend on the domain $U$ of $\tilde{f}$ containing
$\\{0\\}\times{\gamma}({\boldsymbol{A}})$.

It is important to know that $f({\boldsymbol{A}})\in{\mathcal{L}}(X)$ (where
${\mathcal{L}}(X)\equiv{\mathcal{L}}(X)e_{0}$) and what the bounded linear
operator $p({\boldsymbol{A}})\in{\mathcal{L}}(X)$ is in the case that $p$ is a
polynomial in $n$ real variables. The following results are taken from [21,
§4.3].

######  Theorem 4.3.

Let ${\boldsymbol{A}}$ be a hyperbolic $n$-tuple of bounded operators acting
on a Banach space $X$.

  1. (i)

Let $p:{\mathbb{C}}\to{\mathbb{C}}$ be a polynomial and
$\zeta\in{\mathbb{C}}^{n}$. Set $f(z)=p(\langle z,\zeta\rangle)$, for all
$z\in{\mathbb{C}}^{n}$. Then
$f({\boldsymbol{A}})=p(\langle{\boldsymbol{A}},\zeta\rangle).$

  2. (ii)

Suppose that $k_{1},\dots,k_{n}=0,1,2,\dots,k=k_{1}+\cdots+k_{n}$ and
$f(x)=x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}$ for all
$x=(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}$. Then

$f({\boldsymbol{A}})=\frac{k_{1}!\cdots k_{n}!}{k!}\sum_{\pi}A_{\pi(1)}\cdots
A_{\pi(k)},$

where the sum is taken over every map $\pi$ of the set $\\{1,\dots,k\\}$ into
$\\{1,\dots,n\\}$ which assumes the value $j$ exactly $k_{j}$ times, for each
$j=1,\dots,n$.

  3. (iii)

Let $\Omega$ be an open set in ${\mathbb{R}}^{n+1}$ containing
${\gamma}({\boldsymbol{A}})$ with a smooth boundary $\partial\Omega$. Then for
all $\omega\notin\overline{\Omega}$,
$G_{\omega}({\boldsymbol{A}})=\int_{\partial\Omega}G_{\zeta}({\boldsymbol{A}}){\boldsymbol{n}}(\zeta)G_{\omega}(\zeta)\,d\mu(\zeta).$

  4. (iv)

Suppose that $U$ is an open neighbourhood of ${\gamma}({\boldsymbol{A}})$ in
${\mathbb{R}}^{n}$ and $f:U\to{\mathbb{C}}$ is an analytic function. Then
$f({\boldsymbol{A}})\in{\mathcal{L}}(X)$.

For a commuting $n$-tuple ${\boldsymbol{A}}$ of bounded operators with real
spectra, the McIntosh functional calculus $f\longmapsto f({\boldsymbol{A}})$
given by formula $(\ref{eqn:fc})$ coincides with Taylor’s functional calculus
$\tilde{f}\longmapsto\tilde{f}({\boldsymbol{A}})$ for the holomorphic
counterpart $\tilde{f}:\tilde{U}\to{\mathbb{C}}$ of the monogenic function
$f:U\to{\mathbb{R}}^{n+1}$, that is, $\tilde{U}$ is an open subset of
${\mathbb{C}}^{n}$ containing ${\gamma}({\boldsymbol{A}})$ and $U$ is an open
subset of ${\mathbb{R}}^{n+1}$ containing
$\\{0\\}\times{\gamma}({\boldsymbol{A}})$ such that
$\tilde{f}({\boldsymbol{x}})=f({\boldsymbol{x}})$ for every element
${\boldsymbol{x}}$ of the open subset
$(\tilde{U}\cap{\mathbb{R}}^{n})\cap\pi_{{\mathbb{R}}^{n}}(U\cap(\\{0\\}\times{\mathbb{R}}^{n})$
of ${\mathbb{R}}^{n}$. Here $\pi_{{\mathbb{R}}^{n}}$ is the projection
$\pi_{{\mathbb{R}}^{n}}(x_{0},x_{1},\dots,x_{n})=(x_{1},\dots,x_{n})$ for
$x_{1},\dots,x_{n}\in{\mathbb{R}}$.

######  Theorem 4.4.

Let ${\boldsymbol{A}}$ be a commuting $n$-tuple of bounded operators acting on
a Banach space $X$ such that ${\sigma}(A_{j})\subset{\mathbb{R}}$ for all
$j=1,\dots,n$.

Then ${\gamma}({\boldsymbol{A}})$ is the complement in ${\mathbb{R}}^{n}$ of
the set of all $\lambda\in{\mathbb{R}}^{n}$ for which the operator
$\sum_{j=1}^{n}(\lambda_{j}I-A_{j})^{2}$ is invertible in ${\mathcal{L}}(X)$.

Moreover, ${\gamma}({\boldsymbol{A}})$ is the Taylor spectrum of
${\boldsymbol{A}}$. If the complex valued function $f$ is real analytic in a
neighbourhood of ${\gamma}({\boldsymbol{A}})$ in ${\mathbb{R}}^{n}$, then the
operator $f({\boldsymbol{A}})\in{\mathcal{L}}(X)$ coincides with the operator
obtained from Taylor’s functional calculus [37].

In the noncommuting case, there is no homomorphism properties for the McIntosh
functional calculus, but it does enjoy symmetry properties similar to those of
the Weyl calculus ${\mathcal{W}}_{{\boldsymbol{A}}}$ when it exists, that is,
when the exponential estimate (19) obtains.

The following general properties of the Weyl functional calculus [1, Theorem
2.9], suitably interpreted, are also enjoyed by the McIntosh functional
calculus.

######  Theorem 4.5.

Let ${\boldsymbol{A}}$ be an hyperbolic $n$-tuple of bounded operators acting
on a Banach space $X$.

  1. (i)

Affine covariance: if $L:{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ is an affine
map, then ${\gamma}(L{\boldsymbol{A}})\subseteq L{\gamma}({\boldsymbol{A}})$
and for any function $f$ analytic in a neighbourhood in ${\mathbb{R}}^{m}$ of
$L{\gamma}({\boldsymbol{A}})$, the equality $f(L{\boldsymbol{A}})=(f\circ
L)({\boldsymbol{A}})$ holds.

  2. (ii)

Consistency with the one-dimensional calculus: if
$g:{\mathbb{R}}\to{\mathbb{C}}$ is analytic in a neighbourhood of the
projection $\pi_{1}{\gamma}({\boldsymbol{A}})$ of ${\gamma}({\boldsymbol{A}})$
onto the first ordinate, and $f=g\circ\pi_{1}$, then
$f({\boldsymbol{A}})=g(A_{1})$. We also have consistency with the
$k$-dimensional calculus, $1<k<n$.

  3. (iii)

Continuity: The mapping $(T,f)\longmapsto f(T)$ is continuous for
$T=\sum_{j=1}^{n}T_{j}e_{j}$ from ${\mathcal{L}}_{(n)}(X_{(n)})\times
M({\mathbb{R}}^{n+1},{\mathbb{C}}_{(n+1)})$ to ${\mathcal{L}}_{(n)}(X_{(n)})$
and from ${\mathcal{L}}(X)\times H_{M}({\mathbb{R}}^{n})$ to
${\mathcal{L}}(X)$.

  4. (iv)

Covariance of the Range: If $T$ is an invertible continuous linear map on $X$
and $TAT^{-1}$ denotes the $n$-tuple with entries $TA_{j}T^{-1}$ for
$j=1,\dots n$, then ${\gamma}(TAT^{-1})={\gamma}({\boldsymbol{A}})$ and
$f(TAT^{-1})=Tf({\boldsymbol{A}})T^{-1}$ for all functions $f$ analytic in a
neighbourhood of ${\gamma}({\boldsymbol{A}})$ in ${\mathbb{R}}^{n}$.

It is time to make the first sentence of the Introduction precise by
specifying the term hyperbolic for a system ${{\boldsymbol{A}}}$ of unbounded
operators in a Banach space $X$. The object is to ensure that the Cauchy
kernel $G_{x_{0}e_{0}+{\boldsymbol{x}}}({\boldsymbol{A}})$ is sensibly defined
for all $x_{0}\neq 0$ and ${\boldsymbol{x}}\in{\mathbb{R}}^{n}$.

The set of $s\in S^{n-1}$ with nonzero coordinates $s_{j}$ for every
$j=1,\dots,n$ is denoted by $S_{0}^{n-1}.$ Then $S_{0}^{n-1}$ is a dense open
subset of $S^{n-1}$ with full surface measure.

The notation ${\mathcal{D}}(f)$ is used for the domain of a function $f$.

######  Definition 4.6.

An $n$-tuple ${{\boldsymbol{A}}}=(A_{1},\dots,A_{n})$ of densely defined
operators in a Banach space $X$ is called hyperbolic provided that

  1. (a)

$\cap_{j=1}^{n}{\mathcal{D}}(A_{j})$ is dense in $X$,

  2. (b)

${\langle{{\boldsymbol{A}}},s\rangle}$ is closable on
$\cap_{j=1}^{n}{\mathcal{D}}(A_{j})$ for every $s\in S_{0}^{n-1}$ and

  3. (c)

${\sigma}\big{(}\,{\overline{{\langle{{\boldsymbol{A}}},s\rangle}}}\,\big{)}\subset{\mathbb{R}}$
for every $s\in S_{0}^{n-1}$.

The operator $\left((\langle{{\boldsymbol{x}}},s\rangle
I-\langle{{\boldsymbol{A}}},s\rangle)^{2}+x_{0}^{2}I\right)^{-n}$ makes sense
for $s\in S_{0}^{n-1}$ because we can interpret it as the bounded linear
operator

$\left((\langle{{\boldsymbol{x}}},s\rangle+i|x_{0}|)I-{\overline{\langle{{\boldsymbol{A}}},s\rangle}}\right)^{-n}\left((\langle{{\boldsymbol{x}}},s\rangle-i|x_{0}|)I-{\overline{\langle{{\boldsymbol{A}}},s\rangle}}\right)^{-n}$

and also

$\left((\langle{{\boldsymbol{x}}},s\rangle+x_{0}s)I-\langle{{\boldsymbol{A}}},s\rangle\right)^{n}\left((\langle{{\boldsymbol{x}}},s\rangle+i|x_{0}|)I-{\overline{\langle{{\boldsymbol{A}}},s\rangle}}\right)^{-n}$

is a bounded linear operator for every $x_{0}\neq 0$ and
${\boldsymbol{x}}\in{\mathbb{R}}^{n}$.

Because $S_{0}^{n-1}$ is a set of full measure $x\mapsto
G_{x}({{\boldsymbol{A}}})$ is defined off $\\{0\\}\times{\mathbb{R}}^{n}$ by
formulas (22) and (23). Differentiation under the integral sign ensures that
$x\mapsto G_{x}({{\boldsymbol{A}}})$ is two-sided monogenic in
${\mathcal{L}}_{(n)}(X_{(n)})$ so that the joint spectrum
${\gamma}({{\boldsymbol{A}}})$ is a closed and nonempty subset of
${\mathbb{R}}^{n}$. The same argument works if $S_{0}^{n-1}$ is replaced by a
set of full Hausdorff measure. If $X$ is a Hilbert space and elements of
${{\boldsymbol{A}}}$ are selfadjoint, then (b) implies (c) in the Definition
4.6.

The next step is to verify that (25) produces the right result for elementary
functions. For bounded operators the case of polynomials is treated in Theorem
4.3. For unbounded operators we should check functions like $x\mapsto
p({\langle x,\xi\rangle})(\lambda-{\langle x,\xi\rangle})^{-k}$,
$x\in{\mathbb{R}}^{n}$, for $\lambda\in{\mathbb{C}}$, $\Im\lambda\neq 0$ with
$p$ a polynomial of degree less than or equal to $k=1,2,\dots$ . As we are
concerned here only with matrices we won’t go any further into the case of
unbounded operators.

##  5\. Systems of Matrices

Let ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ be a hyperbolic $n$-tuple of
$N\times N$ matrices. If $A_{1},\dots,A_{n}$ are hermitian then the
exponential bound (19) follows from the Lie-Kato-Trotter product formula with
$C=1$, $s=0$ and $r=\|{\boldsymbol{A}}\|$ [36, Theorem 1].

In general, the exponential bound (19) follows from the properties of
hyperbolic polynomials considered below. It is instructive to see this
directly. The characteristic polynomial of a square matrix $B$ is
$p_{B}(z)=\det(B-zI)$, $z\in{\mathbb{C}}$.

######  Proposition 5.1.

Let ${{\boldsymbol{A}}}$ be a hyperbolic $n$-tuple of $N\times N$ matrices.
Then for each $r>\|{\boldsymbol{A}}\|$, there exists $C>0$ such that the
exponential bound (19) holds with $s=N-1$.

###### Proof.

Following [23, V §2], to see directly that ${{\boldsymbol{A}}}$ satisfies the
bound (19) we can take the Fourier transform of (2), estimate the factor
associated with a differential operator of order $s=N-1$ and look at the
function

$Z(\xi,t)=\frac{1}{2\pi}\int_{C(\xi)}\frac{e^{izt}}{p_{{\langle{{\boldsymbol{A}}},\xi\rangle}}(-z)}\,dz,$

for a suitable contour $C(\xi)$ with $\xi\in{\mathbb{R}}^{n}$. Let $D_{r}$ be
the closed disk of radius $r\geq 0$ centred at zero in ${\mathbb{C}}$
($D_{0}=\\{0\\}$). If we know that
${\sigma}({\langle{{\boldsymbol{A}}},\xi\rangle})\subset D_{r(\xi)}$ with
$r(\xi)\geq 0$ for $\xi\in{\mathbb{R}}^{n}$ and

$r=\sup_{|\xi|=1}r(\xi)<\infty,$

then we can take $C(\xi)$ to be that part of the circle of radius $|\xi|(r+1)$
centred at zero in the half-plane $\\{z:\Im z>-1\\}$ joined with a segment of
the line $\\{z:\Im z=-1\\}$, oriented clockwise. Then
$\text{dist}(C(\xi),{\sigma}({\langle{{\boldsymbol{A}}},\xi\rangle}))\geq 1$
for each $\xi\in{\mathbb{R}}^{n}$, so

$|Z(\xi,t)|\leq\frac{e^{t}}{2\pi}\int_{C{(\xi)}}\frac{|dz|}{|p_{{\langle{{\boldsymbol{A}}},\xi\rangle}}(-z)|}\leq
e^{t}|\xi|(r+1),\qquad\xi\in{\mathbb{R}}^{n}.$

With $t=1$, an appeal to the Payley-Weiner Theorem shows that
${\boldsymbol{A}}$ is of type $(s,r)$, the Weyl fuctional calculus
${\mathcal{W}}_{{{\boldsymbol{A}}}}$ exists and
${\gamma}({{\boldsymbol{A}}})=\hbox{\rm
supp}({\mathcal{W}}_{{{\boldsymbol{A}}}})$. It suffices to take

$r(\xi)=\|{\langle{{\boldsymbol{A}}},\xi\rangle}\|\leq\|{\boldsymbol{A}}\|\,|\xi|.$

The number $r>0$ in the bound (19) may be taken to be any number greater than
the spectral radius
$r({{\boldsymbol{A}}})=\sup\\{|x|:x\in{\gamma}({{\boldsymbol{A}}})\\}$ and
$s=N-1$. ∎

The speed of propagation is equal to the joint spectral radius
$r({\boldsymbol{A}})$ [23, p. 131]. The inclusion

${\sigma}({\langle{{\boldsymbol{A}}},{\zeta}\rangle})\subset
r({\zeta})D_{|{\zeta}|},\quad{\zeta}\in{\mathbb{C}}^{n}$

with the $r({\zeta})$ given by equation

$r({\zeta})=\sup\\{|x|:x\in{\gamma}((\langle{\boldsymbol{A}},\xi\rangle,\langle{\boldsymbol{A}},\eta\rangle)\\},\quad{\zeta}=\xi+i\eta,\
\xi,\eta\in{\mathbb{R}}^{n},$

is a general fact about bounded hyperbolic operators on a Banach space [21,
Theorem 5.7]. The relevant result for hyperbolic polynomials is [20, Theorem
12.5.1].

###  5.1. The Fundamental Solution

Let

$\displaystyle
P^{{\boldsymbol{A}}}({\zeta}_{0},{\zeta}_{1},\cdots,{\zeta}_{n})$
$\displaystyle=$
$\displaystyle\det({\zeta}_{0}I+{\zeta}_{1}A_{1}+\cdots+{\zeta}_{n}A_{n})$
$\displaystyle=$ $\displaystyle
p_{\langle{\boldsymbol{A}},{\boldsymbol{{\zeta}}}\rangle}(-{\zeta}_{0}),$

for all ${\zeta}\in{\mathbb{C}}^{n+1}$ with the representation
${\zeta}={\zeta}_{0}e_{0}+{\boldsymbol{{\zeta}}}$,
${\boldsymbol{{\zeta}}}\in{\mathbb{C}}^{n}$.

Let ${\mathbb{RP}}^{n}$ be real $n$-dimensional projective space. Then

(26) $\Xi({\boldsymbol{A}})=\
\bigl{\\{}(\xi_{0}:\xi_{1}:\cdots:\xi_{n})\in{\mathbb{RP}}^{n}\,\bigm{|}\,P^{{\boldsymbol{A}}}(\xi_{0},\xi_{1},\cdots,\xi_{n})=0\bigr{\\}}$

is an algebraic hypersurface. Identifying elements of ${\mathbb{RP}}^{n}$ with
lines $[b]$ in ${\mathbb{R}}^{n+1}$, $b\in{\mathbb{R}}^{n+1}\setminus\\{0\\}$,
let ${\Gamma}(P^{{\boldsymbol{A}}})$ denote the open connected component of
${\mathbb{R}}^{n+1}\setminus\Xi({\boldsymbol{A}})$ containing $e_{0}$. The
convex cone ${\Gamma}(P^{{\boldsymbol{A}}})$ in ${\mathbb{R}}^{n+1}$ is called
the hyperbolicity cone of ${\boldsymbol{A}}$. The trace of the dual cone
$K(P^{{\boldsymbol{A}}})$ of ${\Gamma}({\boldsymbol{A}})$ on the set
$x_{0}=1$, referred to as the propagation set of ${\boldsymbol{A}}$, is the
given by

(27) $K({\boldsymbol{A}})=\
\bigl{\\{}{\boldsymbol{x}}\in{\mathbb{R}}^{n}\,\bigm{|}\,\langle
e_{0}+{\boldsymbol{x}},\xi\rangle\geq 0,\
\forall\xi\in{\Gamma}({\boldsymbol{A}})\ \bigr{\\}}.$

In the case that $n=2$ and ${\boldsymbol{A}}=(A_{1},A_{2})$ is a pair of
hermitian matrices, then the result of Kippenhahn mentioned in Section 2
ensures that the set $K({\boldsymbol{A}})$ can be identified with the
numerical range of the matrix $A=A_{1}+iA_{2}$. Writing
${\mathcal{F}}\phi=\hat{\phi}$, $\phi\in{\mathcal{S}}({\mathbb{R}}^{n+1})$ for
the Fourier transform, its inverse map is

$\big{(}{\mathcal{F}}^{-1}\psi\big{)}(x)=\frac{1}{(2\pi)^{n+1}}\int_{{\mathbb{R}}^{n+1}}e^{i{\langle
x,\xi\rangle}}\psi(\xi)\,d\xi,\quad\psi\in{\mathcal{S}}({\mathbb{R}}^{n+1}),\
x\in{\mathbb{R}}^{n+1}$

and by a common abuse of notation the dual linear maps on
${\mathcal{S}}^{\prime}({\mathbb{R}}^{n+1})$ are denoted by the same symbols.

The distribution

${\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}\right)(te_{0}+{\boldsymbol{x}})$

is the fundamental solution of $P^{{\boldsymbol{A}}}(\tau
e_{0}+{\boldsymbol{D}})={\delta}_{0}$ in the sense that

${\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i{\epsilon}e_{0})}\right)(te_{0}+{\boldsymbol{x}})$

is defined as a distribution for all ${\epsilon}>0$ and is independent of
${\epsilon}$ [20, (12.5.3)]. By the Paley-Wiener-Schwartz theorem

(28) $\operatorname{co}\left(\hbox{\rm
supp}\left({\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}\right)\right)\right)=K(P^{{\boldsymbol{A}}})$

[20, Theorem 12.5.1], so that

${\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i{\epsilon}e_{0})}\right)(te_{0}+{\boldsymbol{x}})=0,\quad
t<0.$

Furthermore

$\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}=\lim_{{\epsilon}\to
0+}\frac{1}{P^{{\boldsymbol{A}}}(\xi-i{\epsilon}e_{0})}$

as the limit of distributions in ${\mathcal{S}}^{\prime}({\mathbb{R}}^{n})$
[3, Theorem 4.1]. Similarly,
${\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi+i0)}\right)$ is
supported in $-K(P^{{\boldsymbol{A}}})$. The equality (28) also establishes
Proposition 5.1.

###  5.2. The Numerical Range Distribution

Let $n_{{\boldsymbol{A}}}$ be the joint numerical range map (6) for the
$n$-tuple ${{\boldsymbol{A}}}$ of $(N\times N)$ hermitian matrices and $\mu$
the unitarily invariant probability on $S({\mathbb{C}}^{N})$. Then the Borel
probability measure $\nu_{{\boldsymbol{A}}}=\mu\circ
n_{{\boldsymbol{A}}}^{-1}$ is supported by the joint numerical range
$N_{\boldsymbol{A}}=n_{{\boldsymbol{A}}}(S({\mathbb{C}}^{N}))$ of
${{\boldsymbol{A}}}$. The probability measure $\nu_{{\boldsymbol{A}}}$ on
$({\mathbb{R}}^{n},{\mathcal{B}}({\mathbb{R}}^{n}))$ is called the numerical
range distribution of the $n$-tuple ${{\boldsymbol{A}}}$. For general
hyperbolic matrices ${{\boldsymbol{A}}}$, the measure $\nu_{{\boldsymbol{A}}}$
lives on $({\mathbb{C}}^{n},{\mathcal{B}}({\mathbb{C}}^{n}))$.

Let
$P^{{\boldsymbol{A}}}(\xi)=\det(\xi_{0}I+{\langle{{\boldsymbol{{{\boldsymbol{A}}}}}},{\boldsymbol{\xi}}\rangle})$,
$\xi\in{\mathbb{R}}^{n+1}$. Nelson’s representation (5.2) suggests that the
numerical range distribution $\nu_{{\boldsymbol{A}}}$ and the fundamental
solution of $F_{P^{{\boldsymbol{A}}}}$ of
$P^{{\boldsymbol{A}}}(\tau,{\boldsymbol{D}})={\delta}_{0}$ must be related.
The relationship is curiously the basis of the generalisation of the Cauchy
integral formula considered in the paper [22] which intentionally does not
mention the fundamental solution $F_{P^{{\boldsymbol{A}}}}$.

The adjugate of an invertible square matrix $B$ is the matrix adj$B=(\det
B)B^{-1}$, that is, the transpose of the matrix of signed minor determinants
of $B$. The same rules apply if $B$ is a matrix of differential operators
acting on ${\mathcal{S}}^{\prime}({\mathbb{R}}^{m})$, $m=1,2,\dots$ .

######  Theorem 5.2.

Let ${{\boldsymbol{A}}}$ be an $n$-tuple of $(N\times N)$ hermitian matrices.
The equality

$t^{N-1}\nu_{t{\boldsymbol{A}}}=(-i)^{N}(N-1)!{\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}\right)(te_{0}+{\,\cdot\,}),\quad
t>0.$

holds in the sense of distributions on $(0,\infty)\times{\mathbb{R}}^{n}$ and

$\nu_{{\boldsymbol{A}}}=(-i)^{N}(N-1)!{\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}\right)(e_{0}+{\,\cdot\,})$

as distributions on ${\mathbb{R}}^{n}$. Furthermore

$\displaystyle{\mathcal{W}}_{t{{\boldsymbol{A}}}}$ $\displaystyle=$
$\displaystyle i\text{\rm
adj}(\partial_{t}I+{\langle{{\boldsymbol{A}}},\nabla\rangle}){\mathcal{F}}^{-1}\left(\frac{1}{P^{{{\boldsymbol{A}}}}(\xi-i0)}\right)(te_{0}+{\boldsymbol{x}})$
$\displaystyle=$ $\displaystyle i^{N}\text{\rm adj}(\tau
I+{\langle{{\boldsymbol{A}}},{\boldsymbol{D}}\rangle}){\mathcal{F}}^{-1}\left(\frac{1}{P^{{{\boldsymbol{A}}}}(\xi-i0)}\right)(te_{0}+{\boldsymbol{x}}),\quad
t>0.$

###### Proof.

According to [21, equation (5.7)] we have

$\frac{1}{2\pi
i}\int_{C(\xi)}\frac{e^{iz}}{p_{\langle{\boldsymbol{A}},\xi\rangle}(z)}\,dz=-\frac{(-i)^{N-1}(2\pi)^{n}}{(N-1)!}\check{\nu}_{{\boldsymbol{A}}}(\xi)$

for $\xi\in{\mathbb{R}}^{n}$ and a simple closed contour $C(\xi)$ about the
real eigenvalues ${\sigma}({\langle{{\boldsymbol{A}}},\xi\rangle})$.

This equation arises in a proof of Nelson’s representation for the Weyl
calculus [21, Proposition 5.4] (a forward slash is obviously missing in
equations (5.4) and (5.5) there) but was not mentioned by Nelson himself. For
$N\geq 2$, the characteristic polymomial has only real roots so that the left
hand side is

$\frac{1}{2\pi
i}\int_{({\mathbb{R}}-i\eta)+(-{\mathbb{R}}+i\eta)}\frac{e^{iz}}{p_{\langle{\boldsymbol{A}},\xi\rangle}(z)}\,dz$

for any $\eta>0$, with the understanding that ${\mathbb{R}}$ is oriented from
$-\infty$ to $\infty$.

Taking a Fourier transform and substituting $z=-{\zeta}$ yields

$\frac{1}{2\pi
i}\int_{({\mathbb{R}}-i\eta)+(-{\mathbb{R}}+i\eta)}\int_{{\mathbb{R}}^{n}}\frac{e^{-i{\zeta}}\hat{\phi}(\xi)}{p_{\langle{\boldsymbol{A}},\xi\rangle}(-{\zeta})}\,d\xi\,d{\zeta}=\frac{(-i)^{N-1}(2\pi)^{n}}{(N-1)!}\int_{{\mathbb{R}}^{n}}\hat{\phi}(\xi)\check{\nu}_{{\boldsymbol{A}}}(\xi)\,d\xi$
$\frac{1}{(2\pi)^{n+1}}\int_{({\mathbb{R}}-i\eta)+(-{\mathbb{R}}+i\eta)}\int_{{\mathbb{R}}^{n}}\frac{e^{-itz}\hat{\phi}(\xi)}{P^{{\boldsymbol{A}}}(ze_{0}+\xi)}\,d\xi\,dz=-\frac{(-i)^{N}t^{N-1}}{(N-1)!}\int_{{\mathbb{R}}^{n}}\hat{\phi}(\xi)\check{\nu}_{t{\boldsymbol{A}}}(\xi)\,d\xi,\quad
t>0,$

and all $\phi\in{\mathcal{S}}({\mathbb{R}}^{n})$.

$\displaystyle t^{N-1}\nu_{t{\boldsymbol{A}}}$ $\displaystyle=$
$\displaystyle-i^{N}(N-1)!{\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}-\frac{1}{P^{{\boldsymbol{A}}}(\xi+i0)}\right)(-te_{0}-{\boldsymbol{x}})$
$\displaystyle=$
$\displaystyle{i^{N}(N-1)!}{\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi+i0)}\right)(-te_{0}-{\boldsymbol{x}})$
$\displaystyle=$
$\displaystyle{(-i)^{N}(N-1)!}{\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}\right)(te_{0}+{\boldsymbol{x}}).$

as distributions. The distribution
${\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}\right)$ is
supported in $K(P^{{\boldsymbol{A}}})$ while
$-te_{0}-{\boldsymbol{x}}\in-K(P^{{\boldsymbol{A}}})$ and
${\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi+i0)}\right)$ is
supported in $-K(P^{{\boldsymbol{A}}})$ as mentioned above. Convolution with
${\delta}_{1}\otimes{\delta}_{0}$ gives the second equality. Using similar
reasoning

$\displaystyle e^{it{\langle{{\boldsymbol{A}}},\xi\rangle}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\int_{C(\xi)}e^{it\lambda}(\lambda-{\langle{{\boldsymbol{A}}},\xi\rangle})^{-1}\,d\lambda$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi
i}\int_{({\mathbb{R}}-i\eta)+(-{\mathbb{R}}+i\eta)}e^{it\lambda}\frac{\text{adj}(-\lambda
I+{\langle{{\boldsymbol{A}}},\xi\rangle})}{P^{{\boldsymbol{A}}}(-\lambda
e_{0}+\xi)}\,d\lambda$

from which formula (5.2) follows by taking the Fourier transform. ∎

######  Example 5.3.

If ${\boldsymbol{{\sigma}}}$ are the Pauli matrices, then
$P^{{\boldsymbol{{\sigma}}}}(\xi)=\xi_{0}^{2}-|{\boldsymbol{\xi}}|^{2}$ for
$\xi\in{\mathbb{R}}^{n+1}$. According to [17, p. 204], the solution of the
wave equation in ${\mathbb{R}}^{4}$ is

$\delta(r-t)/(4\pi
t)=-{\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{{\sigma}}}}(\xi-i0)}\right)(te_{0}+{\boldsymbol{x}}).$

The negative sign in the formula is because the fundamental solution of the
wave equation satisfies $p(\tau,{\boldsymbol{D}})E=\delta_{0}$ for the
hyperbolic polynomial $p(\xi)=-\xi_{0}^{2}+|{\boldsymbol{\xi}}|^{2}.$ The
convention is required when considering nonhomogeneous hyperbolic polynomials
$p$—these are not needed in the present context of hyperbolic systems
${{\boldsymbol{A}}}$.

Computing $\nu_{t{\boldsymbol{{\sigma}}}}$ requires the evaluation of an
integral in polar coordinates for $S({\mathbb{R}}^{4})$ but we can simply read
it off formula (18) and Nelson’s formula [21, Theorem 5.1] given by

$\displaystyle{\mathcal{W}}_{t{\boldsymbol{{\sigma}}}}$
$\displaystyle=\sum_{k=0}^{1}\sum_{j=0}^{1-k}\sum_{m=0}^{j}(-1)^{k+m}\left(\begin{array}[]{c}j\\\
m\end{array}\right)\frac{1}{(1-j+m)!}\times$ $\displaystyle\hskip
28.45274pt\langle
t{\boldsymbol{{\sigma}}},\nabla\rangle^{k}\phi_{1-j-k}(\langle
t{\boldsymbol{{\sigma}}},\nabla\rangle)(\nabla\cdot
id)^{m}\nu_{t{\boldsymbol{{\sigma}}}}$
$\displaystyle=\sum_{j=0}^{1}\sum_{m=0}^{j}(-1)^{m}\left(\begin{array}[]{c}j\\\
m\end{array}\right)\frac{1}{(1-j+m)!}\times$ $\displaystyle\hskip
28.45274pt\phi_{1-j}(\langle
t{\boldsymbol{{\sigma}}},\nabla\rangle)(\nabla\cdot
id)^{m}\nu_{t{\boldsymbol{{\sigma}}}}-\langle
t{\boldsymbol{{\sigma}}},\nabla\rangle\nu_{t{\boldsymbol{{\sigma}}}}$
$\displaystyle=\phi_{1}(\langle
t{\boldsymbol{{\sigma}}},\nabla\rangle)\nu_{t{\boldsymbol{{\sigma}}}}+\nu_{t{\boldsymbol{{\sigma}}}}-(\nabla\cdot
id)\nu_{t{\boldsymbol{{\sigma}}}}-\langle
t{\boldsymbol{{\sigma}}},\nabla\rangle\nu_{t{\boldsymbol{{\sigma}}}}$
$\displaystyle=\nu_{t{\boldsymbol{{\sigma}}}}-(\nabla\cdot
id)\nu_{t{\boldsymbol{{\sigma}}}}-\langle
t{\boldsymbol{{\sigma}}},\nabla\rangle\nu_{t{\boldsymbol{{\sigma}}}}.$

to verify

$t\nu_{t{\boldsymbol{{\sigma}}}}=t\delta(r-t)/(4\pi
t^{2})=-{\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{{\sigma}}}}(\xi-i0)}\right)(te_{0}+{\boldsymbol{x}}).$

Further identities follow from the Nelson formula

$\displaystyle{\mathcal{W}}_{t{\boldsymbol{A}}}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N-1}\sum_{j=0}^{N-k-1}\sum_{m=0}^{j}(-1)^{k+m}\left(\begin{array}[]{c}j\\\
m\end{array}\right)\frac{1}{(N-1-j+m)!}\times$ $\displaystyle\hskip
85.35826pt\langle t{\boldsymbol{A}},\nabla\rangle^{k}\phi_{N-j-k-1}(\langle
t{\boldsymbol{A}},\nabla\rangle)(\nabla\cdot id)^{m}\nu_{t{\boldsymbol{A}}}$

by equating coefficients in powers of $t$ with $i\text{\rm
adj}(\partial_{t}I+{\langle{{\boldsymbol{A}}},\nabla\rangle}){\mathcal{F}}^{-1}\left(\frac{1}{P^{{{\boldsymbol{A}}}}(\xi-i0)}\right)(te_{0}+{\boldsymbol{x}})$.

##  6\. Lacunas

The Herglotz-Petrovsky-Leray formula for the fundamental solution $E$ of the
hyperbolic differential operator $P^{{\boldsymbol{A}}}(\tau,{\boldsymbol{D}})$
for
$P^{{\boldsymbol{A}}}({\zeta})=\det({\zeta}_{0}I+\langle{\boldsymbol{A}},{\boldsymbol{{\zeta}}}\rangle)$,
${\zeta}\in{\mathbb{R}}^{n+1}$ is derived from the distribution

${\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}\right)$

by the explicit integration of the radial variable. In particular for the
hermitian case of $N\geq n+1$, it turns out that
${d\nu_{{{\boldsymbol{A}}}}}/{d\lambda}$ is a polynomial of degree $N-n-1$ in
regions where the Petrovsky cycle vanishes. Such regions are called lacunas
because they make up the connected components of
$\operatorname{co}(N_{{\boldsymbol{A}}})^{\circ}\setminus\hbox{\rm
supp}({\mathcal{W}}_{{\boldsymbol{A}}})$.

In the hermitian cases $n=2$, $N\geq 2$ and $n=3$, $N\geq 3$, the joint
numerical range $N_{{\boldsymbol{A}}}$ is convex and the set
$(N_{{\boldsymbol{A}}})^{\circ}\setminus\hbox{\rm
supp}({\mathcal{W}}_{{\boldsymbol{A}}})$ consists of the gaps between the
joint numerical range $N_{{\boldsymbol{A}}}$ and the support $\hbox{\rm
supp}({\mathcal{W}}_{{\boldsymbol{A}}})$ of the Weyl functional calculus. The
trivial gaps are eliminated by restricting the elements of
${{\boldsymbol{A}}}$ to the subspaces $X$ of ${\mathbb{C}}^{N}$ on which
${{\boldsymbol{A}}}$ has no further nontrivial joint invariant subspaces of
$X$. By spectral theory and induction ${\mathbb{C}}^{N}$ can be written as the
orthogonal sum of subspaces on which $\hbox{\rm
supp}({\mathcal{W}}_{{\boldsymbol{A}}})$ has no further decomposition.

The linearised equations of magnetohydrodynamics correspond to the case $n=2$
and $N=7$ for hermitian matrices. A beautiful instrument drawing before the
era of consumer computer graphics appears in Figures 1a, 1b of the fundamental
paper of J. Bazer and D. Yen [5]. The corresponding symmetric matrices have no
joint eigenvalues or nontrivial joint invariant subspaces.

Let $n$ be an even integer and ${\boldsymbol{A}}=(A_{1},\dots,A_{n})$ a
hyperbolic $n$-tuple of $(N\times N)$ matrices. Modifications required for the
case of odd $n$ are indicated later. The purpose of this section is to outline
a general method using Clifford analysis for establishing that a point
${\boldsymbol{x}}\in{\mathbb{R}}^{n}$ belongs to the joint spectrum
${\gamma}({\boldsymbol{A}})$ or not and to determine if $\hbox{\rm
supp}({\mathcal{W}}_{{\boldsymbol{A}}})=K({{\boldsymbol{A}}})$ when
${{\boldsymbol{A}}}$ has no nontrivial joint invariant subspaces. Because the
propagation set $K({{\boldsymbol{A}}})$ is convex by construction and
$\hbox{\rm supp}({\mathcal{W}}_{{\boldsymbol{A}}})\subseteq
K({{\boldsymbol{A}}})$ [3, Theorem 4.1], the set
$K({{\boldsymbol{A}}})\setminus\hbox{\rm
supp}({\mathcal{W}}_{{\boldsymbol{A}}})$ consists of genuine lacunas for the
Weyl functional calculus.

Roughly speaking, the approach of Atiyah, Bott and Gärding [3] is interpreted
in the present matrix setting,[21, Section 5.3] and we see that the detailed
explanation given in [21] for the fundamental case $n=2$ may be generalised by
using the appropriate tools from algebraic topology. The presentation of this
section is based on the summary of the Herglotz-Petrovsky-Leray formulas [3]
given by Y. Berest in [7]. Another brief account is given in [20, Section
12.6].

A general element $x=(x_{0},x_{1},\dots,x_{n})$ of ${\mathbb{R}}^{n+1}$ will
be written as $x={\boldsymbol{x}}+x_{0}e_{0}$ with
${\boldsymbol{x}}=\sum_{j=1}^{n}x_{j}e_{j}$. Because $n$ is assumed to be an
even integer,

$\int\limits_{S^{n-1}}s\left(\langle{\boldsymbol{x}}I-{\boldsymbol{A}},s\rangle-
x_{0}s\right)^{-n}\/ds=0$

and the plane wave decomposition (22), (23) for the Cauchy kernel is

(35) $\displaystyle G_{x}({\boldsymbol{A}})$ $\displaystyle=$
$\displaystyle{\mathcal{W}}_{{\boldsymbol{A}}}(G_{x})$ $\displaystyle=$
$\displaystyle\frac{(n-1)!}{2}\left(\frac{i}{2\pi}\right)^{n}\hbox{sgn}(x_{0})\int\limits_{S^{n-1}}\left(\langle{\boldsymbol{x}}I-{\boldsymbol{A}},s\rangle-
x_{0}s\right)^{-n}\/ds,$

for $x\in{\mathbb{R}}^{n+1}$ with $x_{0}\neq 0$. For ease of notation, an
element $xI$ of ${\mathcal{L}}_{(n)}({\mathbb{C}}^{N})$ for
$x\in{\mathbb{C}}_{(n)}$ will often be written as $x$. Because $x\longmapsto
G_{x}({\boldsymbol{A}})$ is actually the Cauchy transform of the Weyl calculus
${\mathcal{W}}_{{\boldsymbol{A}}}$ off ${\mathbb{R}}^{n}$, we have

(36) ${\mathcal{W}}_{{\boldsymbol{A}}}=\lim_{{\epsilon}\to
0+}G_{{\boldsymbol{x}}+{\epsilon}e_{0}}({\boldsymbol{A}})-G_{{\boldsymbol{x}}-{\epsilon}e_{0}}({\boldsymbol{A}})$

in the sense of distributions. Consequently, if the limit on the right hand
side of equation (36) exists uniformly for all ${\boldsymbol{x}}$ in a
neighbourhood of a point ${\boldsymbol{a}}\in{\mathbb{R}}^{n}$ and is zero
there, then ${\boldsymbol{a}}$ lies outside the support of the matrix valued
distribution ${\mathcal{W}}_{{\boldsymbol{A}}}$, that is,
${\boldsymbol{a}}\in{\gamma}({\boldsymbol{A}})^{c}$. We shall seek conditions
which guarantee that the limit

(37) $\lim_{{\epsilon}\to
0+}\int\limits_{S^{n-1}}\left(\langle{\boldsymbol{x}}I-{\boldsymbol{A}},s\rangle-{\epsilon}s\right)^{-n}+\left(\langle{\boldsymbol{x}}I-{\boldsymbol{A}},s\rangle+{\epsilon}s\right)^{-n}\/ds$

exists uniformly for all elements ${\boldsymbol{x}}$ of an open subset of
${\mathbb{R}}^{n}$. In any case ${\mathcal{W}}_{{\boldsymbol{A}}}$ is
$\frac{(n-1)!}{2}\left(\frac{i}{2\pi}\right)^{n}$ times the limit (37) in the
distributional sense.

For the case $n=2$ considered in [21, Section 5.3], the integral (37) was
calculated in an elementary manner by converting it into a contour integral
and actually computing the residues associated with the spectral
representation of the hermitian matrix $\langle{\boldsymbol{A}},s\rangle$
following the analysis of Bazer and Yen [5].

###  6.1. Hyperbolic polynomials

In this subsection properties and concepts of hyperbolic polynomials are
stated in the context of the determinantal polynomial

$P^{{\boldsymbol{A}}}({\zeta})=\det({\zeta}_{0}I+{\langle{{\boldsymbol{A}}},{\boldsymbol{{\zeta}}}\rangle}),\quad{\zeta}={\zeta}_{0}+{\boldsymbol{{\zeta}}}\in{\mathbb{R}}^{n+1}$

associated with a hyperbolic $n$-tuple ${{\boldsymbol{A}}}$ of matrices. Most
phenomena are already exhibited in the classes of simultaneously
triangularisable matrices with real spectra, hermitian matrices and their
direct sums.

A localisation $P^{{\boldsymbol{A}}}_{\xi}$ of $P^{{\boldsymbol{A}}}$ at
$\xi\in{\mathbb{R}}^{n+1}$, is the lowest nonzero term of the polynomial

$t\mapsto
P^{{\boldsymbol{A}}}(\xi+t\zeta)=t^{\mu_{\xi}}P^{{\boldsymbol{A}}}_{\xi}(\zeta)+{\mathcal{O}}(t^{\mu_{\xi}+1})\
,\quad\mu_{\xi}=\deg\,P^{{\boldsymbol{A}}}_{\xi}\ .$

Let ${\boldsymbol{A}}$ and $\xi\in{\mathbb{R}}^{n+1}$ be fixed. Consider the
localisation $P^{{\boldsymbol{A}}}_{\xi}$ of $P^{{\boldsymbol{A}}}$ at $\xi$.
The local hyperbolicity cone and the local propagation set of
$P^{{\boldsymbol{A}}}$ at $\xi$ are defined by setting, respectively,

$\Gamma_{\xi}({{\boldsymbol{A}}}):=\Gamma(P^{{\boldsymbol{A}}}_{\xi}),\qquad
K_{\xi}({{\boldsymbol{A}}}):=\\{{\boldsymbol{x}}\in{\mathbb{R}}^{n}:[e_{0}+{\boldsymbol{x}}]\in
K(P^{{\boldsymbol{A}}}_{\xi})\ \\}\ .$

Here the polynomial $P^{{\boldsymbol{A}}}$ has been replaced by
$P^{{\boldsymbol{A}}}_{\xi}$ in the definitions (26) and (27). A similar
notation is used for the real lineality $\Lambda(P^{{\boldsymbol{A}}}_{\xi})$
of the polynomial $P^{{\boldsymbol{A}}}_{\xi}$.

Clearly, $\Gamma_{\xi}({{\boldsymbol{A}}})\supseteq\Gamma({\boldsymbol{A}})$
and, hence, $K_{\xi}({{\boldsymbol{A}}})\subseteq K({\boldsymbol{A}})$ for all
$\xi\in{\mathbb{R}}^{n+1}$. More precisely, the mapping
$(\xi,{{\boldsymbol{A}}})\mapsto\Gamma_{\xi}({{\boldsymbol{A}}})$ (and
$(\xi,{{\boldsymbol{A}}})\mapsto K_{\xi}({{\boldsymbol{A}}})$) is inner
(resp., outer) continuous in the sense that
$\Gamma_{\xi}({{\boldsymbol{A}}})\,\cap\,\Gamma_{\tilde{\xi}}(\tilde{{\boldsymbol{A}}})$
(resp.,
$K_{\xi}({{\boldsymbol{A}}})\,\cup\,K_{\tilde{\xi}}(\tilde{{\boldsymbol{A}}})$)
is close to $\Gamma_{\xi}({{\boldsymbol{A}}})$ (resp.,
$K_{\xi}({{\boldsymbol{A}}})$) when $(\tilde{\xi},\tilde{{\boldsymbol{A}}})$
is close to $(\xi,{{\boldsymbol{A}}})$ with
$\xi,\tilde{\xi}\in{\mathbb{R}}^{n+1}$ and
${{\boldsymbol{A}}},\tilde{{\boldsymbol{A}}}$ hyperbolic.

######  Example 6.1.

Suppose that ${{\boldsymbol{A}}}$ consists of two hermitian matrices. Then
$\xi\in D_{\mathbb{R}}({\boldsymbol{A}})$ is a simple point if and only if
$P^{{\boldsymbol{A}}}_{\xi}({\zeta})=\langle b,{\zeta}\rangle$ where
$b_{0}\neq 0$ and $b$ is the tangent vector at $\xi$, that is

$[b]=\left[\frac{\partial
P^{{\boldsymbol{A}}}}{\partial\xi_{0}}(\xi):\frac{\partial
P^{{\boldsymbol{A}}}}{\partial\xi_{1}}(\xi):\frac{\partial
P^{{\boldsymbol{A}}}}{\partial\xi_{2}}(\xi)\right].$

Then
$K_{\xi}({{\boldsymbol{A}}})=\\{{\boldsymbol{x}}:[e_{0}+{\boldsymbol{x}}]=[b]\\}$
and

$C({\boldsymbol{A}})=\overline{\bigcup_{\mu_{\xi}({\boldsymbol{A}})=1}K_{\xi}({{\boldsymbol{A}}})}.$

[4, Theorem 14.20], shows that
ss$({\mathcal{W}}_{{\boldsymbol{A}}})=C({\boldsymbol{A}})$.

If $\xi\in D_{\mathbb{R}}({\boldsymbol{A}})$ and $\deg
P^{{\boldsymbol{A}}}_{\xi}=2$ (the multiplicity of tangent vectors to
$D_{\mathbb{R}}({\boldsymbol{A}})$ at $\xi$) and

$P^{{\boldsymbol{A}}}_{\xi}({\zeta})=a({\zeta}_{0}+\langle
b_{1},{\boldsymbol{{\zeta}}}\rangle)({\zeta}_{0}+\langle
b_{2},{\boldsymbol{{\zeta}}}\rangle)$

for some $a\in{\mathbb{R}}$, $b_{1},b_{2}\in{\mathbb{R}}^{2}$ , then
$b_{1},b_{2}\in C({{\boldsymbol{A}}})$ and

${\Gamma}(P^{{\boldsymbol{A}}}_{\xi})=\\{{\zeta}_{0}+\langle
b_{1},{\boldsymbol{{\zeta}}}\rangle>0,\ {\zeta}_{0}+\langle
b_{2},{\boldsymbol{{\zeta}}}\rangle>0\ \\},\
K_{\xi}({{\boldsymbol{A}}})=\operatorname{co}\\{b_{1},b_{2}\\}.$

Hence $K_{\xi}({{\boldsymbol{A}}})$ is the line segment joining $b_{1}$ and
$b_{2}$ lying on the “double tangent” corresponding to
$\xi\in{\mathbb{R}}^{3}$.

At this stage, we need to take into account that the homogeneous polynomial
$P^{{\boldsymbol{A}}}$ may not depend on all variables in
${\mathbb{C}}^{n+1}$. For example, one of the matrices $A_{j}$ could be the
zero matrix.

The real lineality $\Lambda({\boldsymbol{A}})$ of ${{\boldsymbol{A}}}$, is the
maximal linear subspace of ${\mathbb{R}}^{n+1}$ such that the restriction of
$P^{{\boldsymbol{A}}}$ the quotient
${\mathbb{R}}^{n+1}/\Lambda({\boldsymbol{A}})$ is again a polynomial. Then
$\Lambda({\boldsymbol{A}})$ coincides with the edge of the hyperbolicity cone
$\Gamma({\boldsymbol{A}})$, so that $\Gamma+\Lambda=\Gamma$, and
$K({\boldsymbol{A}})$ spans the intersection of its orthogonal complement
$\Lambda^{\perp}({\boldsymbol{A}})$ in ${\mathbb{R}}^{n+1}$ with the plane
$x_{0}=1$.

The system ${\boldsymbol{A}}$ is called complete if ${{\boldsymbol{A}}}$ has a
trivial lineality. In this case, $P^{{\boldsymbol{A}}}_{\xi}(\zeta)\equiv
P^{{\boldsymbol{A}}}(\zeta)$ implies $\xi=0$, the cone
$\Gamma({\boldsymbol{A}})$ is proper (peaked) in the sense that
$\overline{\Gamma({\boldsymbol{A}})}$ does not contain any straight lines, and
then $K({\boldsymbol{A}})$ has a non-empty interior
$K^{\circ}({\boldsymbol{A}})$ in ${\mathbb{R}}^{n}$.

The wave front surface $W({{\boldsymbol{A}}})$ of the system
${{\boldsymbol{A}}}$ of matrices is generated by the union of local
propagation cones:

(38)
$W({{\boldsymbol{A}}}):=\bigcup\limits_{0\not=\xi\in{\mathbb{R}}^{n+1}}\,K_{\xi}({{\boldsymbol{A}}})\
.$

######  Theorem 6.2 (Joswig and Straub [25]).

Let ${{\boldsymbol{A}}}$ be two $(N\times N)$ hermitian matrices. The set of
critical points of the numerical range map
$n_{{{\boldsymbol{A}}}}:S({\mathbb{C}}^{N})\to{\mathbb{R}}^{2}$ is
$n_{{\boldsymbol{A}}}^{-1}(W({{{\boldsymbol{A}}}}))$.

By the implicit function theorem, the numerical range distribution
$\nu_{{\boldsymbol{A}}}$ has a real analytic density with respect to Lebesgue
measure outside the wave front set $W({{{\boldsymbol{A}}}})$. Although the
distribution ${\mathcal{W}}_{{\boldsymbol{A}}}$ consists of a differential
operator of order $(N-1)$ acting on the numerical range distribution
$\nu_{{\boldsymbol{A}}}$ as may be seen from Nelson’s representation (5.2),
the equality ss$({\mathcal{W}}_{{\boldsymbol{A}}})=C({\boldsymbol{A}})$ proved
in [4, Theorem 14.20], shows that ${\mathcal{W}}_{{\boldsymbol{A}}}$ does not
see the double tangents mentioned in Example 6.1 in the hermitian case—a fact
already apparent from the simplest of examples.

For an ordered set $a=(a_{1},\dots,a_{N})$ of real numbers,
$\operatorname{diag}a$ denotes the $(N\times N)$ diagonal matrix with entries
$a_{1},\dots,a_{N}$ down the diagonal.

######  Example 6.3.

  1. (a)

Let $A_{1}=\operatorname{diag}(1,0)$, $A_{2}=\operatorname{diag}(0,1)$ and
${{\boldsymbol{A}}}=(A_{1},A_{2})$. Then

${\mathcal{W}}_{{{\boldsymbol{A}}}}={\delta}_{(1,0)}P_{1}+{\delta}_{(0,1)}P_{2},\quad
P_{1}:x\mapsto\left(\begin{matrix}x_{1}\\\ 0\end{matrix}\right),\
P_{2}:x\mapsto\left(\begin{matrix}0\\\ x_{2}\end{matrix}\right),\
x\in{\mathbb{R}}^{2},$

but
$W({{\boldsymbol{A}}})=K_{(1,1)}({{\boldsymbol{A}}})=\operatorname{co}\left\\{\left(\begin{matrix}1\\\
0\end{matrix}\right),\left(\begin{matrix}0\\\ 1\end{matrix}\right)\right\\}$.

  2. (b)

Let $A_{1}=\left(\begin{matrix}0&1\\\ 0&0\end{matrix}\right)$ and
$A_{2}=\left(\begin{matrix}0&1\\\ 0&1\end{matrix}\right)$. Then as in [21,
Example 5.31], we have

${\mathcal{W}}_{{{\boldsymbol{A}}}}=\left(\begin{matrix}0&\delta_{0}^{\prime}\otimes\chi_{[0,1]}\\\
0&\delta_{0}\otimes\delta_{1}\end{matrix}\right)+\left(\begin{matrix}0&\delta_{0}\otimes(\delta_{0}-\delta_{1})\\\
0&\delta_{0}\otimes\delta_{1}\end{matrix}\right)$

so $\hbox{\rm
supp}({\mathcal{W}}_{{\boldsymbol{A}}})=\\{0\\}\times[0,1]=K({{\boldsymbol{A}}})=W({{\boldsymbol{A}}})$.
The polynomial $P^{{\boldsymbol{A}}}$ is not complete [21, Example 5.36]. The
spectral projections associated with
${\langle{{\boldsymbol{A}}},{\zeta}\rangle}$ for ${\zeta}\in{\mathbb{C}}^{2}$
have poles on the unit circle centred at zero—a phenomenon forbidden by
Rellich’s lemma in the hermitian case [25].

  3. (c)

Not so obvious is that the operator valued distribution
${\mathcal{W}}_{{\boldsymbol{A}}}$ vanishes across the double tangent of the
Kippenhahn curve $C({{\boldsymbol{A}}})$ associated with

${{\boldsymbol{A}}}=\left(\left(\begin{matrix}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{matrix}\right),\left(\begin{matrix}0&0&1\\\ 0&0&1\\\
1&1&0\end{matrix}\right)\right).$

The curve $C({{\boldsymbol{A}}})$ is the third type in Kippenhahn’s
classification [26, Theorem 26] of numerical range of $(3\times 3)$ matrices,
see [25, Figure X]. The argument goes as follows. The couple
${{\boldsymbol{A}}}=A_{1}+iA_{2}$ has no joint invariant subspaces otherwise
$\hbox{\rm supp}({\mathcal{W}}_{{\boldsymbol{A}}})$ would be an elliptical
region or an elliptical region together with an outside point—the joint
eigenvalue, or just finitely many points corresponding to the case when
$A_{1}+iA_{2}$ is a normal matrix, which it is not.

The elliptical region for two $(3\times 3)$ matrices ${\boldsymbol{B}}$
derives from a two dimensional joint invariant subspace $X$, should one exist,
on which ${\boldsymbol{B}}$ is represented by $(2\times 2)$ hermitian
matrices. Denote the restriction of ${\boldsymbol{B}}$ to the two dimensional
joint invariant subspace $X$ by ${\boldsymbol{B}}_{X}$. The Pauli matrices
${\boldsymbol{{\sigma}}}$ and the identity $I_{2}$ form a linear basis of the
$(2\times 2)$ hermitian matrices, so there exists an affine transformation
$L:x\mapsto Tx+a$, $x\in{\mathbb{R}}^{3}$ with rank two linear $T$ and
$a\in{\mathbb{R}}^{2}$ such that
${\boldsymbol{B}}_{X}=T{\boldsymbol{{\sigma}}}+aI_{2}$. By Theorem 4.5 (i),
${\mathcal{W}}_{{{\boldsymbol{B}}}_{X}}$ is an affine transformation of
${\mathcal{W}}_{{\boldsymbol{{\sigma}}}}$ and
${\gamma}({\boldsymbol{B}}_{X})=LS({\mathbb{R}}^{3})$ is an elliptical region
in ${\mathbb{R}}^{2}$. However, $C({{\boldsymbol{A}}})$ is a cardioid so the
set $(N_{{\boldsymbol{A}}})^{\circ}\setminus\hbox{\rm
supp}({\mathcal{W}}_{{\boldsymbol{A}}})$ is a nontrivial lacuna of
${\mathcal{W}}_{{\boldsymbol{A}}}$.

To check the double tangent, observe that

${\zeta}_{0}I+\langle{\boldsymbol{A}},{\boldsymbol{{\zeta}}}\rangle=\begin{pmatrix}{\zeta}_{0}+{\zeta}_{1}&0&{\zeta}_{2}\\\
0&{\zeta}_{0}-{\zeta}_{1}&{\zeta}_{2}\\\
{\zeta}_{2}&{\zeta}_{2}&{\zeta}_{0}-{\zeta}_{1}\end{pmatrix}$

and

$\displaystyle P^{{\boldsymbol{A}}}({\zeta})$ $\displaystyle=$
$\displaystyle({\zeta}_{0}+{\zeta}_{1})(({\zeta}_{0}-{\zeta}_{1})^{2}-{\zeta}_{2}^{2})-({\zeta}_{0}-{\zeta}_{1}){\zeta}_{2}^{2}$
$\displaystyle=$
$\displaystyle({\zeta}_{0}^{2}-{\zeta}_{1}^{2})({\zeta}_{0}-{\zeta}_{1})-2{\zeta}_{0}{\zeta}_{2}^{2}.$

Then $P^{{\boldsymbol{A}}}((1,1,0))=0$ and
$P_{(1,1,0)}^{{\boldsymbol{A}}}({\zeta})=2(({\zeta}_{0}-{\zeta}_{1})^{2}-{\zeta}_{2}^{2})$,
so a line parallel to $\\{x_{1}=0\\}$ is tangential to $C({{\boldsymbol{A}}})$
at two points satisfying $x_{2}=\pm(x_{1}-1)$ in ${\mathbb{R}}^{2}$.

We finish Part I of this paper with an explicit calculation of
$E_{{\boldsymbol{A}}}={\mathcal{F}}^{-1}\left(\frac{1}{P^{{\boldsymbol{A}}}(\xi-i0)}\right)$
for two $(3\times 3)$ hermitian matrices ${{\boldsymbol{A}}}$ with a joint
eigenvalue. Here $E_{{\boldsymbol{A}}}$ has a Petrovsky lacuna on which
${\mathcal{W}}_{{\boldsymbol{A}}}$ vanishes. In this example, it is easy to
write down the Weyl functional calculus:

${\mathcal{W}}_{{\boldsymbol{A}}}=\delta_{a}P_{a}\oplus{\mathcal{W}}_{({\sigma}_{1},{\sigma}_{2})}P$

for selfadjoint projections $P_{a}$, $P$ with $P_{a}+P=I$ on
${\mathbb{C}}^{3}$ and as mentioned above
${\mathcal{W}}_{({\sigma}_{1},{\sigma}_{2})}$ is the affine image of
${\mathcal{W}}_{{\boldsymbol{{\sigma}}}}$ so $\hbox{\rm
supp}({\mathcal{W}}_{{{\boldsymbol{A}}}})$ is the union of $\\{a\\}$ and the
closed unit disk centred at zero. The numerical range $N_{{\boldsymbol{A}}}$
is the convex hull of $\hbox{\rm supp}({\mathcal{W}}_{{{\boldsymbol{A}}}})$.
On the other hand calculating the fundamental solution $E_{{\boldsymbol{A}}}$
takes more effort.

######  Example 6.4.

Let ${{\boldsymbol{A}}}=(a_{1}\oplus{\sigma}_{1},a_{2}\oplus{\sigma}_{2})$ for
the Pauli matrices (17) and $a\in{\mathbb{R}}^{2}$. Then

$P^{{\boldsymbol{A}}}(\xi)=\det(\xi_{0}I+\langle{\boldsymbol{A}},\xi\rangle)=(\xi_{0}+\langle
a,{\boldsymbol{\xi}}\rangle)(\xi_{0}^{2}-|{\boldsymbol{\xi}}|^{2}),$

The Kippenhahn curve $C({\boldsymbol{A}})$ is the union of the singleton
$\\{a\\}$ and the unit circle centred at $0$ and $N_{{\boldsymbol{A}}}$ is the
convex hull of $C({\boldsymbol{A}})$.

Then by [7, equation (4.1)] we have

$\displaystyle{\mathcal{F}}^{-1}\left(\frac{1}{P(\xi-i0)}\right)$
$\displaystyle=\frac{i}{2\pi^{\frac{1}{2}}{\Gamma}(\frac{1}{2})}H(x_{0}+\langle
a,{\boldsymbol{x}}\rangle)\delta({\boldsymbol{x}}-x_{0}a)*H(x_{0})(x_{0}^{2}-|{\boldsymbol{x}}|^{2})_{+}^{-\frac{1}{2}}$
$\displaystyle=\frac{i}{2\pi^{\frac{1}{2}}{\Gamma}(\frac{1}{2})}\int_{0}^{\infty}H(x_{0}-y_{0})(y_{0}^{2}-|{\boldsymbol{x}}-(x_{0}-y_{0})a|^{2})_{+}^{-\frac{1}{2}}\,dy_{0}$
$\displaystyle=\frac{i}{2\pi^{\frac{1}{2}}G(\frac{1}{2})}\int_{0}^{x_{0}}(y_{0}^{2}-|{\boldsymbol{x}}-(x_{0}-y_{0})a|^{2})_{+}^{-\frac{1}{2}}\,dy_{0},$

where $H$ is the Heaviside function and $\delta$ is the unit point mass at
zero, both interpreted as distributions. The notation means that we are
integrating the function

$y_{0}\longmapsto(y_{0}^{2}-|{\boldsymbol{x}}-(x_{0}-y_{0})a|^{2})_{+}^{-\frac{1}{2}}$

with values in the space of distributions over the interval $[0,x_{0}]$.

Suppose $x_{0}=1$, $1<|x|<|a|$. Then $\langle a,{\boldsymbol{x}}-a\rangle<0$
and $|\langle a,{\boldsymbol{x}}-a\rangle|>\sqrt{|a|^{2}-1}|x-a|$ because the
angle $\theta$ between $a$ and $x-a$ satisfies
$\cos\theta>\sqrt{|a|^{2}-1}/|a|$.

$y_{0}^{2}-|{\boldsymbol{x}}-(1-y_{0})a|^{2}=-(|{\boldsymbol{x}}-a|^{2}+2\langle
a,{\boldsymbol{x}}-a\rangle y_{0}+(|a|^{2}-1)y_{0}^{2}):=p(y_{0}).$
$-(|a|^{2}-1)\big{(}(y_{0}+\langle
a,{\boldsymbol{x}}-a\rangle/(|a|^{2}-1))^{2}-(\langle
a,{\boldsymbol{x}}-a\rangle^{2}/(|a|^{2}-1)^{2}-|{\boldsymbol{x}}-a|^{2}/(|a|^{2}-1))\big{)}$

If $|\langle a,{\boldsymbol{x}}-a\rangle|<\sqrt{|a|^{2}-1}|x-a|$, then the
expression is negative and the integral is zero. Suppose that $|\langle
a,{\boldsymbol{x}}-a\rangle|>\sqrt{|a|^{2}-1}|x-a|$.

$y_{0}=\left(-\langle a,{\boldsymbol{x}}-a\rangle\pm\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}\right)\vee
0/(|a|^{2}-1)$ $\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}=(|a|^{2}\cos\theta-(|a|^{2}-1))|{\boldsymbol{x}}-a|^{2}>0.$

If $|x|>1$, then the limits of integration are

$(y_{0}+\langle a,{\boldsymbol{x}}-a\rangle/(|a|^{2}-1))=\pm\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}/(|a|^{2}-1).$

If $|x|<1$, then $-\langle a,{\boldsymbol{x}}-a\rangle+\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}>(|a|^{2}-1)$
and

$0<-\langle a,{\boldsymbol{x}}-a\rangle-\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}<(|a|^{2}-1)$

because the polynomial $p$ is equal to $1-|x|^{2}$ at $y_{0}=1$, so it has a
zero to the right of 1.

If $|x|>1$ and $|\langle a,{\boldsymbol{x}}-a\rangle|>|a|^{2}-1$, that is, on
the other side of the unit circle to $a$, then the polynomial $p$ has zeros on
the positive axis because it has the values $-|x-a|^{2}$ at $y_{0}=0$,
$1-|x|^{2}$ at $y_{0}=1$ and the maximum value is at

$y_{0}=|\langle a,{\boldsymbol{x}}-a\rangle|/(|a|^{2}-1)>1.$

Hence the least zero of $p$ satisfies the inequality

$1<\frac{-\langle a,{\boldsymbol{x}}-a\rangle-\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}}{|a|^{2}-1}<\frac{|\langle
a,{\boldsymbol{x}}-a\rangle|}{|a|^{2}-1}$

and $p(y_{0})=y_{0}^{2}-|{\boldsymbol{x}}-(1-y_{0})a|^{2}<0$ for all $0\leq
y_{0}\leq 1$. The polynomial $p$ is increasing on $[0,1]$ where
$(y_{0}^{2}-|{\boldsymbol{x}}-(x_{0}-y_{0})a|^{2})_{+}$ vanishes.

Let $u(x)=\left(\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}\right)/(|a|^{2}-1)$.

$\int\frac{1}{\sqrt{b^{2}-x^{2}}}\,dx=\sin^{-1}(x/b)$

Then for $|x|>1$, $\cos\theta>\sqrt{|a|^{2}-1}/|a|$ and $\langle
a,{\boldsymbol{x}}-a\rangle<0$, the integral

$\int_{0}^{1}(y_{0}^{2}-|{\boldsymbol{x}}-(x_{0}-y_{0})a|^{2})_{+}^{-\frac{1}{2}}\,dy_{0}$

equals

$\left[\sin^{-1}\left(t(|a|^{2}-1)/\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}\right)\right]_{t=-u(x)}^{t=u(x)}=\pi.$

For $|x|\leq 1$,
$\int_{0}^{1}(y_{0}^{2}-|{\boldsymbol{x}}-(x_{0}-y_{0})a|^{2})_{+}^{-\frac{1}{2}}\,dy_{0}$
equals

$\left[\sin^{-1}\left(t(|a|^{2}-1)/\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}\right)\right]_{t=-u(x)}^{t=1+\langle
a,{\boldsymbol{x}}-a\rangle/(|a|^{2}-1)}$ $=\pi/2+\sin^{-1}\left((\langle
a,{\boldsymbol{x}}\rangle-1)/\sqrt{\langle
a,{\boldsymbol{x}}-a\rangle^{2}-(|a|^{2}-1)|{\boldsymbol{x}}-a|^{2}}\right).$

Also
${\mathcal{W}}_{{\boldsymbol{A}}}=\delta_{a}P_{a}\oplus{\mathcal{W}}_{({\sigma}_{1},{\sigma}_{2})}P_{2}$

To calculate the wave front set, we note that

$(\xi_{0}+t{\zeta}_{0})^{2}-|{\boldsymbol{\xi}}+t{\boldsymbol{{\zeta}}}|^{2}=\xi_{0}^{2}-|{\boldsymbol{\xi}}|^{2}+2t(\xi_{0}{\zeta}_{0}-\langle{\boldsymbol{\xi}},{\boldsymbol{{\zeta}}}\rangle)+t^{2}({\zeta}_{0}^{2}-|{\boldsymbol{{\zeta}}}|^{2}).$

For $\xi_{0}^{2}-|{\boldsymbol{\xi}}|^{2}=0$,
$\eta=\pm|{\boldsymbol{\xi}}|e_{0}-{\boldsymbol{\xi}}$, we have
${\Gamma}_{\eta}=\\{\pm\langle\eta,{\zeta}\rangle>0\\}$, $K_{\eta}=\\{\pm
t\eta|t>0\\}$. $\cup_{\eta\neq 0}K_{\eta}$ is a circular cone in
${\mathbb{R}}^{3}$ and

$\displaystyle P(\xi)$ $\displaystyle=(\xi_{0}+\langle
a,{\boldsymbol{\xi}}\rangle)(\xi_{0}^{2}-|{\boldsymbol{\xi}}|^{2})$
$\displaystyle{\Gamma}(P)$ $\displaystyle=\\{\xi_{0}+\langle
a,{\boldsymbol{\xi}}\rangle>0,\ \xi_{0}-|{\boldsymbol{\xi}}|>0,\
\xi_{0}+|{\boldsymbol{\xi}}|>0\\}$ $\displaystyle K(P)$
$\displaystyle=\text{co}\\{t(e_{0}+a),t(e_{0}+\xi):|\xi|=1,\xi\in{\mathbb{R}}^{n},\
t>0\ \\}$
$P(\xi+t{\zeta})=(\xi_{0}^{2}-|{\boldsymbol{\xi}}|^{2})(\xi_{0}+\langle
a,{\boldsymbol{\xi}}\rangle)+2t(\frac{1}{2}(\xi_{0}^{2}-|{\boldsymbol{\xi}}|^{2})({\zeta}_{0}+\langle
a,{\boldsymbol{{\zeta}}}\rangle)+(\xi_{0}{\zeta}_{0}-\langle{\boldsymbol{\xi}},{\boldsymbol{{\zeta}}}\rangle)(\xi_{0}+\langle
a,{\boldsymbol{\xi}}\rangle))$
$+t^{2}(({\zeta}_{0}^{2}-|{\boldsymbol{{\zeta}}}|^{2})(\xi_{0}+\langle
a,{\boldsymbol{\xi}}\rangle)+2(\xi_{0}{\zeta}_{0}-\langle{\boldsymbol{\xi}},{\boldsymbol{{\zeta}}}\rangle)({\zeta}_{0}+\langle
a,{\boldsymbol{{\zeta}}}\rangle))+t^{3}({\zeta}_{0}^{2}-|{\boldsymbol{{\zeta}}}|^{2})({\zeta}_{0}+\langle
a,{\boldsymbol{{\zeta}}}\rangle)$ $\xi_{0}+\langle
a,{\boldsymbol{\xi}}\rangle=0,\ \eta=e_{0}+a.$ $|a|>1,\
\xi_{0}^{2}=|{\boldsymbol{\xi}}|^{2},\ \langle
a,{\boldsymbol{\xi}}\rangle=-\xi_{0}.$
$P_{\xi}({\zeta})=2(\xi_{0}{\zeta}_{0}-\langle{\boldsymbol{\xi}},{\boldsymbol{{\zeta}}}\rangle)({\zeta}_{0}+\langle
a,{\boldsymbol{{\zeta}}}\rangle)$
$\Xi_{\xi^{\prime}}({\zeta})=\\{{\zeta}:|{\boldsymbol{\xi}}^{\prime}|{\zeta}_{0}-\langle{\boldsymbol{\xi}}^{\prime},{\boldsymbol{{\zeta}}}\rangle=0\\}\cup\\{{\zeta}_{0}+\langle
a,{\boldsymbol{{\zeta}}}\rangle=0\\},\ \xi_{0}=|{\boldsymbol{\xi}}^{\prime}|$

Because $-{\boldsymbol{\xi}}^{\prime}={\boldsymbol{\xi}}$ for one of the two
solutions of $\langle a,{\boldsymbol{\xi}}\rangle=|{\boldsymbol{\xi}}|$, this
set is equal to

$\Xi_{\xi}({\zeta})=\\{{\zeta}:|{\boldsymbol{\xi}}|{\zeta}_{0}+\langle{\boldsymbol{\xi}},{\boldsymbol{{\zeta}}}\rangle=0\\}\cup\\{{\zeta}_{0}+\langle
a,{\boldsymbol{{\zeta}}}\rangle=0\\},\ \xi_{0}=-|{\boldsymbol{\xi}}|\\}$

Now choose $\ |{\boldsymbol{\xi}}|=1$. Then

$\displaystyle{\Gamma}_{\xi}$
$\displaystyle=\\{{\zeta}:{\zeta}_{0}+\langle{\boldsymbol{\xi}},{\boldsymbol{{\zeta}}}\rangle>0,\
{\zeta}_{0}+\langle a,{\boldsymbol{{\zeta}}}\rangle>0\\},$ $\displaystyle
K_{\xi}$ $\displaystyle=\text{co}\\{{\boldsymbol{\xi}},a\\},\ \xi_{0}=-1,$

for each of the two solutions of $\langle a,{\boldsymbol{\xi}}\rangle=1.$ The
wave front set $W({\boldsymbol{A}})$ is the unit circle plus $\\{a\\}$ as well
as segments tangential to the circle meeting at $\\{a\\}$.

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