the Riesz Representation theorem A.1.21 satisfying

$\displaystyle\braket{A^{\dagger}\phi}{\psi}_{1}=\begin{cases}\braket{\phi}{A\psi}_{2}&\text{
if }A\text{ is linear }\\\ \braket{\phi}{A\psi}_{2}^{*}&\text{ if }A\text{ is
antilinear }\end{cases}.$ (A.24)

An operator is called self-adjoint or Hermitian if $A=A^{\dagger}$.

######  Example A.5.

If $A\in\mathcal{B}(\mathbb{C}^{d})$, then its adjoint is the complex
conjugate transpose,

$\displaystyle(A^{\dagger})_{ij}=A_{ji}^{*}.$ (A.25)

$\triangle$

######  Definition A.1.23.

Given a Hilbert space, $\mathcal{H}$, a projector, $P$, is a linear idempotent
satisfying

$\displaystyle PP=P.$ (A.26)

An orthogonal projector is a self-adjoint projector.

######  Theorem A.1.24.

For every closed linear subspace of a Hilbert space, $\mathcal{K}$, there
exists a unique orthogonal projection whose range equals $\mathcal{K}$. I
refer to this projector as $\mathscr{P}_{\mathcal{K}}$.

######  Definition A.1.25.

A linear operator, $D$, is diagonal in the basis $x_{i}$ if and only if the
basis elements are eigenvectors of $D$. A linear operator is diagonalizable in
the basis $x_{i}$ if and only if it is similar to a diagonal operator. An
operator is diagonalizable in a finite-dimensional space if and only if it is
diagonalizable in every linearly independent basis. An operator on a finite-
dimensional space is defective if and only if it is not diagonalizable.

######  Proposition A.1.26 (Spectral Theorem for Compact Normal Operators).

Every compact normal operator is unitarily diagonalizable.

######  Definition A.1.27.

Consider a set of operators, $\mathcal{S}$, on a vector space. A subspace $V$
is an invariant subspace if the image of $V$ under every element
$s\in\mathcal{S}$ is contained in $V$, or more explicitly, $s(V)\subseteq V$.
If $\mathcal{S}$ consists of a single operator, then $V$ is an invariant
subspace of that operator.

######  Example A.6.

If the domain and codomain of an operator are equal, then this domain is an
invariant subspace of the operator. The eigenspaces of an operator are always
invariant subspaces, as well as the set containing the zero vector.

######  Lemma A.1.28.

If $V$ is an invariant subspace of a bounded operator on a Hilbert space,
$A\mathrel{\mathop{\ordinarycolon}}\mathcal{H}\to\mathcal{H}$, then
$V^{\perp}$ is an invariant subspace of this operator’s adjoint,
$A^{\dagger}$.

####  A.1.4 Invertibility

This section discusses the inverses of operators. Firstly, I remind the reader
of what invertibility means for functions.

######  Definition A.1.29.

A function, $f\mathrel{\mathop{\ordinarycolon}}X\to Y$, is

  * •

surjective, or onto, if the range of $f$ is $Y$.

  * •

injective, or one-to-one, if $f(x_{1})=f(x_{2})\,\Rightarrow x_{1}=x_{2}$

  * •

bijective if $f$ is injective and surjective.

A function is bijective if and only if it has an inverse function,
$f^{-1}\mathrel{\mathop{\ordinarycolon}}Y\to X$, such that $f\circ
f^{-1}=\text{id}_{Y}$ and $f^{-1}\circ f=\text{id}_{X}$, where $\text{id}_{S}$
denotes the identity map on a set $S$.

The next goal of this section is to analyse invertibility of linear maps.
Injectivity is discussed in the next lemma.

######  Lemma A.1.30.

A linear map is injective if and only if its kernel consists of only the zero
vector.

More can be said for bounded linear maps between Banach spaces. A first
result, which can be generalized to topological vector spaces, is given below

######  Theorem A.1.31 (Banach bounded inverse theorem).

A bijective bounded linear operator whose domain and codomain are Banach
spaces has a bounded inverse [221, Thm. 14.5.1].

One useful class of operators that helps clarify when an inverse exists is the
class of bounded below operators.

######  Definition A.1.32.

An operator $A$ between normed spaces is bounded below if

$\displaystyle\text{inf}\left\\{\frac{||A\psi||}{||\psi||}\,|\,\psi\in\text{Dom}(A)\setminus\\{0\\}\right\\}>0.$
(A.27)

An equivalent characterization of bounded below operators is as follows.

######  Theorem A.1.33.

A bounded linear operator between Banach spaces is bounded below if and only
if this operator is injective and has a closed range. [1, Thm. 2.5]

Since a surjective map between Banach spaces necessarily has a closed range, a
bounded bijective linear operator is necessarily bounded below. The next
result regards bounded linear operators on Hilbert spaces.

######  Theorem A.1.34.

A bounded linear operator on a Hilbert space, $A\in\mathcal{B}(\mathcal{H})$,
is bijective if and only if $A$ and $A^{\dagger}$ are bounded below.

###### Proof.

This is a straightforward corollary of theorem A.1.33 and the identity

$\displaystyle\mathcal{R}(A)^{\perp}=\ker(A^{\dagger}).$ (A.28)

∎

###  A.2 Associative Algebras

In this section, we will introduce relevant mathematical structures from
functional analysis. More detailed introductions to the theory of
$C^{*}$-algebras can be found in [45, chp. 2], [250, chp. 2.1], and [206, 154,
294, 295].

######  Definition A.2.1.

An associative algebra, $\mathfrak{A}$, over $\mathbb{C}$ is a complex vector
space equipped with an associative bilinear operation,
$\cdot\mathrel{\mathop{\ordinarycolon}}\mathfrak{A}\times\mathfrak{A}\to\mathfrak{A}$,
referred to as the product, which distributes over addition. Given
$x,y\in\mathfrak{A}$, I use the shorthand $xy$ to denote $x\cdot y$. A normed
algebra is an associative algebra and a normed space, where the norm must
satisfy the product inequality,

$\displaystyle\forall x,y\in\mathfrak{A},||xy||\leq||x||\,||y||.$ (A.29)

A Banach algebra is a normed algebra and a Banach space in the same norm.

The reason why the product inequality is imposed is so that the product is
continuous in the product norm topology.

######  Example A.7.

Given a normed space, $X$, $\mathcal{B}(X)$ is a normed algebra, where the
product is function composition, and the norm is the operator norm.
$\triangle$

######  Definition A.2.2.

An involutive complex algebra, also referred to as a ∗-algebra,
$\mathfrak{A}$, is an associative algebra over $\mathbb{C}$ equipped with a
map, $*\mathrel{\mathop{\ordinarycolon}}\mathfrak{A}\rightarrow\mathfrak{A}$,
which satisfies the following properties for every $a,b\in\mathfrak{A}$,
$\alpha,\beta\in\mathbb{C}$:

  * •

$*$ is an involution, so $(a^{*})^{*}=a$.

  * •

$*$ is antilinear, so $(\alpha a+\beta
b)^{*}=\overline{\alpha}a^{*}+\overline{\beta}b^{*}$, where
$\overline{\alpha}$ denotes the complex conjugate of $\alpha$.

  * •

$*$ is an algebra antihomomorphism, which means $(ab)^{*}=b^{*}a^{*}$.

A $C^{*}$-algebra is a ∗-algebra and a Banach algebra for which the $C^{*}$
identity holds, namely

$\displaystyle||a^{*}a||=||a^{*}||\,||a||.$ (A.30)

######  Example A.8.

The prototypical example of a $C^{*}$-algebra is the set of bounded linear
operators, $\mathcal{B}(\mathcal{H})$, on a Hilbert space, $\mathcal{H}$,
where the involution is the adjoint, and the norm is the operator norm.
Understanding this example is crucial to understanding the general case
$C^{*}$-algebra, due to the existence of representations on closed subalgebras
of $\mathcal{B}(\mathcal{H})$ given by the Gelfand-Naimark theoremA.2.6.
$\triangle$

######  Definition A.2.3.

Given a set of subsets of a $C^{*}$-algebra, $S_{i}\in\mathfrak{A}$, the
$C^{*}$-algebra generated by the collection $S_{i}$ is, intuitively, the
smallest $C^{*}$-algebra containing the union of $S_{i}$. Explicitly, the
generated algebra is the intersection of every $C^{*}$-subalgebra of
$\mathfrak{A}$ which contains every $S_{i}$. This algebra is denoted by
$\vee_{i}S_{i}$. The case of a $C^{*}$-algebra generated by two subsets,
$S_{1},S_{2}$ is denoted by $S_{1}\vee S_{2}$.

######  Definition A.2.4.

Given a subset, $S$, of a ring, $R$, the commutant, denoted by $S^{\prime}$,
of $S$ is the set of all elements in $R$ which commute with respect to
multiplication with each element of $S$,

$\displaystyle S^{\prime}\mathrel{\mathop{\ordinarycolon}}=\\{r\in
R\,|\,\forall s\in S,rs=sr\\}.$ (A.31)

####  A.2.1 Homomorphisms

######  Definition A.2.5.

Given two associative algebras, $\mathfrak{A}$ and $\mathfrak{B}$, a linear
map,
$\phi\mathrel{\mathop{\ordinarycolon}}\mathfrak{A}\rightarrow\mathfrak{B}$,
satisfying

$\displaystyle\phi(xy)$ $\displaystyle=\phi(x)\phi(y)$ (A.32)

is an algebra homomorphism. If $\mathfrak{A}$ and $\mathfrak{B}$ are
$*$-algebras, a ∗-homomorphism is an algebra homomorphism which satisfies

$\displaystyle\phi(x^{*})$ $\displaystyle=\phi(x)^{*}.$ (A.33)

A bijective ∗-homomorphism is a ∗-isomorphism, and a ∗-isomorphism whose
codomain is equal to its domain is a ∗-automorphism.

Homomorphisms can be interpreted as structure preserving maps between
algebras. Often, one can utilize a homomorphism to map the solution to a
problem in one algebra to a problem in another algebra.

######  Theorem A.2.6 (Gelfand-Naimark [103]).

Every $C^{*}$-algebra is isometrically $*$-isomorphic to a subalgebra of
bounded linear operators on a Hilbert space.

####  A.2.2 Unital Algebras

The importance of unital algebras stems from their ability to define inverses
and, consequently, spectra of algebra elements.

######  Definition A.2.7.

A unital associative algebra is an associative algebra, $\mathfrak{A}$ with a
two-sided identity with respect to the multiplication, 𝟙, satisfying
$\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}a=a=a\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}$
for all $a\in\mathfrak{A}$.

The identity of a unital associative algebra is unique. The identity of a
∗-algebra satisfies

$\displaystyle\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}=\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(A.34)

In a $C^{*}$-algebra,

$\displaystyle||\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(A.35)

It is often safe to assume that an algebra is unital, since for every algebra
which is not unital, we can define a unital algebra, $\tilde{\mathfrak{A}}$,
such that $\mathfrak{A}$ is isometrically $*$-isomorphic to a subalgebra of
$\tilde{\mathfrak{A}}$.

######  Definition A.2.8.

Let $\mathfrak{A}$ be an algebra which is not unital. Consider the set of
pairs,

$\displaystyle\tilde{\mathfrak{A}}\mathrel{\mathop{\ordinarycolon}}=\\{(\alpha,a)\,|\,\alpha\in\mathbb{C},a\in\mathfrak{A}\\}.$
(A.36)

Given $\alpha,\beta\in\mathbb{C}$, $a,b\in\mathfrak{A}$ the following rules
define addition and multiplication in $\tilde{\mathfrak{A}}$:

$\displaystyle\beta(\alpha,a)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=(\alpha\beta,\beta a)$ (A.37)
$\displaystyle(\alpha,a)(\beta,b)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=(\alpha\beta,\beta a+\alpha
b+ab)$ (A.38) $\displaystyle(\alpha,a)+(\beta,b)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=(\alpha+\beta,a+b)$ (A.39)

The set $\tilde{\mathfrak{A}}$ with addition and multiplication defined as
above is an algebra, referred to as the unitization of $\mathfrak{A}$.
$\tilde{\mathfrak{A}}$ is a unital algebra with the identity
$\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}=(1,0)$.

######  Proposition A.2.9.

Let $\mathfrak{A}$ be an associative algebra without an identity. If
$\mathfrak{A}$ is a ∗-algebra, then $\tilde{\mathfrak{A}}$ with the involution

$\displaystyle(\alpha,a)^{*}\mathrel{\mathop{\ordinarycolon}}=(\alpha^{*},a^{*})$
(A.40)

is a ∗-algebra. If $\mathfrak{A}$ is a normed algebra, then
$\tilde{\mathfrak{A}}$ is a normed algebra with the norm

$\displaystyle||(\alpha,a)||$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\sup\\{||\alpha
b+ab||\,|\,b\in\mathfrak{A}\text{ and }||b||=1\\}.$ (A.41)

If $\mathfrak{A}$ is a $C^{*}$-algebra, then so is $\tilde{\mathfrak{A}}$ [45,
prop. 2.1.5].

######  Definition A.2.10.

A map, $\phi$, whose domain and codomains are unital associative algebras is
called unital if
$\phi(\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}})=\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}$.

######  Example A.9.

Let $\mathfrak{A}$ be a unital normed algebra. Then, the map
$\pi_{l}\mathrel{\mathop{\ordinarycolon}}\mathfrak{A}\to\mathcal{B}(\mathfrak{A})$
defined by

$\displaystyle\pi_{l}(\xi)\eta=\xi\eta$ (A.42)

is a unital algebra homomorphism and an isometry.  $\triangle$

To conclude this section, I discuss the notion of invertibility for unital
associative algebras.

######  Definition A.2.11.

Given a unital associative algebra, $\mathfrak{A}$, an element,
$a\in\mathfrak{A}$, is invertible if there exists an element,
$b\in\mathfrak{A}$, such that $ab=ba=\mathbb{1}$, and $b$ is called an inverse
of $a$. If an element is invertible, its inverse is unique, and is denoted by
$a^{-1}$. The set of all invertible elements in $\mathfrak{A}$ is a group,
referred to as the general linear group of $\mathfrak{A}$, denoted by
$\text{GL}(\mathfrak{A})$. The general linear group of a unital associative
algebra always contains the identity, since $\mathbb{1}^{-1}=\mathbb{1}$.

######  Definition A.2.12.

Two elements in a unital associative algebra, $a,b\in\mathfrak{A}$, are
similar if there exists $s\in\text{GL}(\mathfrak{A})$, referred to as a
similarity transform, such that $sas^{-1}=b$.

######  Lemma A.2.13.

Suppose $\phi$ is a $*$-homomorphism whose domain and co-domain are unital
$C^{*}$-algebras. Then $\phi$ is unital if and only if $\phi(\mathbb{1})$ is
invertible.

###### Proof.

By A.2.11, the identity in the co-domain of $\phi$ is invertible. Thus, we
only need to demonstrate that if $\phi(\mathbb{1})$ is invertible, then
$\phi(\mathbb{1})=\mathbb{1}$. This follows from the following brief
computation,

$\displaystyle\mathbb{1}$
$\displaystyle=\phi(\mathbb{1})^{-1}\phi(\mathbb{1})$ (A.43)
$\displaystyle=\phi(\mathbb{1})^{-1}\phi(\mathbb{1}\mathbb{1})$ (A.44)
$\displaystyle=\phi(\mathbb{1})^{-1}\phi(\mathbb{1})\phi(\mathbb{1})$ (A.45)
$\displaystyle=\phi(\mathbb{1}).$ (A.46)

∎

####  A.2.3 Spectral Theory

######  Definition A.2.14 (Spectrum).

For every element of a unital associative algebra, $a\in\mathfrak{A}$, we
define its spectrum, $\sigma_{\mathfrak{A}}(a)$, as

$\displaystyle\sigma_{\mathfrak{A}}(a)\mathrel{\mathop{\ordinarycolon}}=\\{\lambda\in\mathbb{C}\,|\,\lambda\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}-a\notin\text{GL}(\mathfrak{A})\\}.$
(A.47)

If $\mathfrak{A}$ is not unital, then we define the spectrum of
$a\in\mathfrak{A}$ through unitization,

$\displaystyle\sigma_{\mathfrak{A}}(a)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\sigma_{\tilde{\mathfrak{A}}}(a).$
(A.48)

If $A$ is an operator on a Banach space, $X$, the spectrum of $A$ is the set

$\displaystyle\sigma(A)=\left\\{\lambda\in\mathbb{C}|\lambda\mathbb{1}-A\notin\text{GL}(\mathcal{B}(X))\right\\}.$
(A.49)

The spectral radius, spr, of an element of a Banach algebra is the supremum of
the norm of the elements in the spectrum of this element,

$\displaystyle\text{spr}(a)\mathrel{\mathop{\ordinarycolon}}=\sup\\{|\lambda|\,|\,\lambda\in\sigma(a)\\}.$
(A.50)

The following theorem can be found in [294, prop. 2.8].

######  Theorem A.2.15 (Spectral Mapping).

For every element of a unital Banach algebra, $a\in\mathfrak{A}$, and for
every polynomial, $p$, the following identity holds:222The spectral mapping
theorem can be generalized to apply to functions which are holomorphic on an
open neighbourhood of the spectrum of $a$, although we have no need to do so
in this thesis.

$\displaystyle\sigma_{\mathfrak{A}}(p(a))$
$\displaystyle=p(\sigma_{\mathfrak{A}}(a)).$ (A.51)

If $a$ is invertible,

$\displaystyle\sigma_{\mathfrak{A}}(a^{-1})$
$\displaystyle=(\sigma_{\mathfrak{A}}(a))^{-1}.$ (A.52)

If $a$ is an element of a $C^{*}$-algebra, then

$\displaystyle\sigma(a^{*})$ $\displaystyle=(\sigma(a))^{*}.$ (A.53)

######  Theorem A.2.16 (Spectral Radius).

For every element, $a$, of a Banach algebra,

  1. 1.

${\text{spr}}(a)\leq||a||$,

  2. 2.

${\text{spr}}(a)=\lim_{n\to\infty}||a^{n}||^{1/n}$.

######  Corollary A.2.16.1.

If $a$ is an element of a $C^{*}$-algebra, then
$||a||=\sqrt{{\text{spr}}(a^{*}a)}$.

###### Proof.

One proof of this corollary utilizes the fact that the involution in a
$C^{*}$-algebra is an isometry [72, Thm. 16.1]. ∎

######  Theorem A.2.17.

If $a\in\mathfrak{A}$ and $a\in\mathfrak{B}$, where
$\mathfrak{A},\mathfrak{B}$ are $C^{*}$-algebras, then
$\sigma_{\mathfrak{A}}(a)=\sigma_{\mathfrak{B}}(a)$. Thus, in the case where
$a$ is an element of a $C^{*}$-algebra, $\mathfrak{A}$, the shorthand
$\sigma(a)\mathrel{\mathop{\ordinarycolon}}=\sigma_{\mathfrak{A}}(a)$ can be
safely used.

######  Example A.10.

For every $C^{*}$-algebra, $\mathfrak{A}$, and $n\in\mathbb{N}$, we can
construct a $C^{*}$-algebra denoted by Mn(A). This algebra is the set of
$n\times n$ matrices whose matrix elements are elements of $\mathfrak{A}$,
where the product of $M,M^{\prime}\in\mathfrak{A}$ is given by the matrix
product,

$\displaystyle(MM^{\prime})_{ik}=\sum_{j=1}^{n}M_{ij}M^{\prime}_{jk}.$ (A.54)

Given a matrix with entries $M_{ij}$, the involution in Mn(A) is defined as
$(M^{*})_{ij}=(M_{ji})^{*}$. The norm is defined via a $*$-homomorphism from
Mn(A) to $\oplus_{i=1}^{n}\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is a
Hilbert space given by the Gelfand-Naimark theorem(A.2.6), see [239, p.2-3]
for details. When $\mathfrak{A}=\mathbb{C}$, the norm is the spectral norm of
eq. A.19 and the involution is complex conjugate transposition, as in eq.
A.25.

If $\mathfrak{A}$ is a unital $C^{*}$-algebra, then so is Mn(A). In this case,
we will use the shorthand

$\displaystyle\lx@glossaries@gls@link{symbols}{GLnA}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\text{GL}_{n}(\mathfrak{A})}}}\mathrel{\mathop{\ordinarycolon}}=\text{GL}(\mathfrak{M}_{n}(\mathfrak{A})).$
(A.55)

$\triangle$

######  Theorem A.2.18.

If $\phi$ is a ∗-homomorphism whose domain and codomain are $C^{*}$-algebras,
then $\phi$ is a contraction. If $\phi$ is a unital ∗-homomorphism, then
$\sigma(\phi(a))\subseteq\sigma(a)$ for all $a\in\text{Dom}(\phi)$. If $\phi$
is a unital ∗-isomorphism, then $\phi$ is an isometry and
$\sigma(\phi(a))=\sigma(a)$ [154, Thm. 4.1.8].

######  Definition A.2.19.

Given an operator on a Banach space,
$A\mathrel{\mathop{\ordinarycolon}}\text{Dom}(A)\to X$, with
$\text{Dom}(A)\subseteq X$, $\lambda\in\mathbb{C}$ is referred to as a
eigenvalue of $A$ if $\\{0\\}\subsetneq\ker(\lambda\mathbb{1}-A)$. A vector in
$\ker(\lambda\mathbb{1}-A)$ is referred to as an eigenvector associated to
$\lambda$. The set of all eigenvalues of $A$ is referred to as the point
spectrum of $A$, and is denoted by $\sigma_{p}(A)$. Denoting the range of
$\lambda\mathbb{1}-A$ by $\mathcal{R}(\lambda\mathbb{1}-A)$, we can further
partition the point spectrum into a union of disjoint subsets,

$\displaystyle\sigma_{p,1}(A)$
$\displaystyle=\\{\lambda\in\sigma_{p}(A)\,|\,\text{cl}(\mathcal{R}(\lambda\mathbb{1}-A))\neq
X\\}$ (A.56) $\displaystyle\sigma_{p,2}(A)$
$\displaystyle=\\{\lambda\in\sigma_{p}(A)\,|\,\text{cl}(\mathcal{R}(\lambda\mathbb{1}-A))=X\\}.$
(A.57)

######  Proposition A.2.20.

Given an operator, $A$, on a finite-dimensional Banach space,
$\sigma(A)=\sigma_{p,1}(A)$.

######  Definition A.2.21.

Given an operator on a Banach space,
$A\mathrel{\mathop{\ordinarycolon}}\text{Dom}(A)\to X$, with
$\text{Dom}(A)\subseteq X$, the continuous spectrum, denoted by
$\sigma_{c}(A)$, is the set of all $\lambda\in\sigma(A)$ such that
$\lambda\mathbb{1}-A$ is injective, but not surjective, and has dense range.
The set of all elements of the spectrum of $A$ which are not included in the
point or continuous spectrum is referred to as the residual spectrum, and is
denoted by $\sigma_{r}(A)$.

####  A.2.4 Positivity

I initiate this section’s discussion of positivity in the abstract setting of
a $C^{*}$-algebra. Then, I relate the notions of positivity to the specific
$C^{*}$-algebra of bounded operators on a Hilbert space.

######  Definition A.2.22.

An element, $a$, of a $C^{*}$-algebra, $\mathfrak{A}$, is normal if and only
if $aa^{*}=a^{*}a$, Hermitian if and only if $a=a^{*}$, and positive if and
only if $a=b^{*}b$ for some $b\in\mathfrak{A}$. All Hermitian elements are
normal, and all positive elements are Hermitian. I use the abbreviation $a\geq
0$ to mean that $a$ is positive and the set of positive elements of
$\mathfrak{A}$ is denoted as $\mathfrak{A}^{+}$.

######  Theorem A.2.23.

The following definitions for a positive element, $a\in\mathfrak{A}^{+}$, of a
$C^{*}$-algebra, $\mathfrak{A}$, are equivalent [45, chp. 2.2.2]:

  1. 1.

There exists $b\in\mathfrak{A}$ such that $a=b^{*}b$ (A.2.22).

  2. 2.

There exists a unique positive element, $b\in\mathfrak{A}^{+}$, which is the
square root of $a$, $a=b^{2}$.

  3. 3.

$a$ is normal and $\sigma(a)\subseteq[0,\infty)$.

######  Definition A.2.24.

An element of a unital $C^{*}$-algebra is called strictly-positive if and only
if it is positive and invertible.

######  Lemma A.2.25.

The inverse of a strictly-positive element is also strictly-positive.

######  Definition A.2.26.

An operator, $\eta$, on a Hilbert space, $\mathcal{H}$, with
$\text{Dom}(\eta)\subseteq\mathcal{H}$ is called positive-definite if and only
if for every $\psi\in\text{Dom}(\eta)\setminus\\{0\\}$,

$\displaystyle\braket{\psi}{\eta\psi}>0$ (A.58)

######  Theorem A.2.27.

A bounded operator on a Hilbert space, $a\in\mathcal{B}(\mathcal{H})$ is:

  1. 1.

positive if and only if for every $\xi\in\mathcal{H}$, $\braket{\xi}{a\xi}\geq
0$ [253, §104].

  2. 2.

positive-definite if and only if it is positive and injective.

  3. 3.

strictly-positive if and only if there exists a positive real number, $c>0$,
such that $\braket{\xi}{a\xi}\geq c$ for all $\xi\in\mathcal{H}$.

###### Proof.

While the proof is straightforward, I mention that item 3 follows from the
Banach bounded inverse theorem A.1.31. ∎

####  A.2.5 Complete Positivity

Completely positive maps are used in physics to define quantum channels. One
textbook devoted to complete positivity is [239].

######  Definition A.2.28.

A map,
$Q\mathrel{\mathop{\ordinarycolon}}\mathfrak{A}\rightarrow\mathfrak{B}$, where
$\mathfrak{A},\mathfrak{B}$ are both $C^{*}$-algebras333Completely positive
maps are defined in the same way for maps whose domain is an operator system,
which is a $*$-closed vector subspace of a $C^{*}$-algebra., is called
positive if $a\geq 0$ implies $Q(a)\geq 0$. $Q$ is completely positive if for
all
$n\in\lx@glossaries@gls@link{symbols}{Z+}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{Z}_{+}}}}$,
the map
$Q_{n}\mathrel{\mathop{\ordinarycolon}}\lx@glossaries@gls@link{symbols}{MnA}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathfrak{M}_{n}(\mathfrak{A})}}}\rightarrow\mathfrak{M}_{n}(\mathfrak{B})$
defined by $(Q_{n}(M))_{ij}=(Q(M_{ij}))$ is positive.

######  Example A.11.

∗-homomorphisms are completely positive.  $\triangle$

######  Example A.12.

Given a unital $C^{*}$-algebra, $\mathfrak{A}$, and an element
$b\in\mathfrak{A}$, the map
$Q\mathrel{\mathop{\ordinarycolon}}\mathfrak{A}\to\mathfrak{A}$ defined by
$Q(a)=b^{*}ab$ is completely positive.  $\triangle$

When $b$ from A.12 is a unitary on a Hilbert space, corresponding to time
evolution, a physicist would identify the operation $Q(a)$ as the effect of
evolving $a$ in the Heisenberg picture.

######  Theorem A.2.29 (Choi’s theorem [59]).

$\Phi\mathrel{\mathop{\ordinarycolon}}\mathfrak{M}_{n}(\mathbb{C})\to\mathfrak{M}_{m}(\mathbb{C})$
is completely positive if and only if

$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n}E_{ij}\otimes\Phi(E_{ij})\geq 0,$
(A.59)

where $E_{ij}\in\mathfrak{M}_{n}(\mathbb{C})$ is the matrix with entries
$(E_{ij})_{kl}\mathrel{\mathop{\ordinarycolon}}=\delta_{ik}\delta_{jl}$.

######  Definition A.2.30.

A normalized linear functional on a $C^{*}$-algebra,
$\varphi\mathrel{\mathop{\ordinarycolon}}\mathfrak{A}\rightarrow\mathbb{C}$,
satisfies $||\varphi||=1$. A positive normalized linear functional is referred
to as a state.

######  Lemma A.2.31.

A positive linear functional on a unital $C^{*}$-algebra is a state if and
only if the functional is unital.

The following theorem characterizes states on the algebra of finite-
dimensional matrices as density matrices, which are positive trace-class
operators with trace equal to one.

######  Theorem A.2.32.

$\varphi$ is a state on $\mathfrak{M}_{n}(\mathbb{C})$ if and only if there
exists $\rho\in\mathfrak{M}_{n}(\mathbb{C})^{+}$ satisfying
$\text{Tr}(\rho)=1$ such that

$\displaystyle\varphi(a)={\text{Tr}{}{}}(\rho a).$ (A.60)

###### Proof.

One proof is given in [34]. I present an alternative here.

Given a linear functional of the form eq. A.60, the assumption
$\text{Tr}(\rho)=1$ implies $\varphi$ is normalized. By the spectral theorem
for positive matrices, there exists a decomposition
$\rho=\sum_{k}p_{k}\ket{k}\bra{k}$, where $p_{k}\geq 0$ and
$\ket{k}\in\mathbb{C}^{n}$ is an orthonormal basis, so the proof that
$\varphi$ is positive follows from $\text{Tr}(\rho
M)=\sum_{k}p_{k}\braket{k}{M}{k}$.

For the other direction of the proof, note $\mathfrak{M}_{n}(\mathbb{C})$ is a
Hilbert space with the Frobenius inner product, which is defined in eq. A.12.
Thus, by the Riesz representation theorem A.1.21, every linear functional
$\varphi\mathrel{\mathop{\ordinarycolon}}\mathfrak{M}_{n}(\mathbb{C})\to\mathbb{C}$
can be expressed as

$\displaystyle\varphi(M)=\braket{\rho}{M}_{HS}$ (A.61)

for some $\rho\in\mathfrak{M}_{n}(\mathbb{C})$. For $\varphi$ to be
normalized, we must have $\text{Tr}(\rho)=1$. Assuming $\varphi$ is positive
implies $\rho\geq 0$, as proven by contradiction: If $\rho$ is not positive,
then there exists $\ket{\xi}\in\mathbb{C}^{n}$ such that

$\displaystyle\varphi(\ket{\xi}\bra{\xi})$
$\displaystyle=\text{Tr}(\rho\ket{\xi}\bra{\xi})$ (A.62)
$\displaystyle=\braket{\xi}{\rho}{\xi}<0,$ (A.63)

which is a contradiction since $\ket{\xi}\bra{\xi}$ is positive. ∎

###  A.3 Measure Theory

######  Definition A.3.1.

A $\sigma$-algebra over the set $X$ is a collection of subsets,
$\Sigma\subseteq\mathbb{P}(X)$, satisfying

  1. 1.

$X\in\Sigma$

  2. 2.

$E\in\Sigma\,\Rightarrow\,X\setminus E\in\Sigma$

  3. 3.

If
$\\{E_{i}\\}_{i\in\lx@glossaries@gls@link{symbols}{N}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{N}}}}}\subseteq\Sigma$,
then
$\cup_{i\in\lx@glossaries@gls@link{symbols}{N}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{N}}}}}E_{i}\in\Sigma$.

######  Definition A.3.2.

The $\sigma$-algebra generated by a collection of subsets,
$S\in\mathbb{P}(X)$, is the intersection of every $\sigma$-algebra over $X$
which contains every element of $S$. Given a topological space, $X$, the Borel
$\sigma$-algebra, denoted by $\mathfrak{B}_{X}$, is the $\sigma$-algebra
generated by the topology on $X$.

######  Example A.13.

The Borel $\sigma$-algebra generated by a discrete topological space is the
discrete topology.  $\triangle$

######  Definition A.3.3.

Given a $\sigma$-algebra, $\Sigma$, a positive measure is a map,
$\mu\mathrel{\mathop{\ordinarycolon}}\Sigma\to[0,+\infty]$ which satisfies
$\mu(\emptyset)=0$ and

$\displaystyle\mu(\sum_{i\in\lx@glossaries@gls@link{symbols}{N}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{N}}}}}E_{i})=\sum_{i\in\lx@glossaries@gls@link{symbols}{N}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{N}}}}}\mu(E_{i})$
$\displaystyle\text{if
}\\{E_{i}\\}_{i\in\lx@glossaries@gls@link{symbols}{N}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{N}}}}}\subseteq\Sigma\text{
and }E_{i}\cap E_{j}=\emptyset\text{ when }i\neq j.$ (A.64)

######  Example A.14.

The Lebesgue measure is a measure over the $\sigma$-algebra of subsets of
$\mathbb{R}^{n}$ satisfying the Carathéodory criterion, which contains the
Borel $\sigma$-algebra generated by the Euclidean topology on
$\mathbb{R}^{n}$. The Lebesgue measure was introduced in [176] and
Carathéodory’s criterion was introduced in [51, 52].  $\triangle$

######  Definition A.3.4.

Let $\Sigma$ be a $\sigma$-algebra and let $\mathcal{H}$ be a Hilbert space.
An operator valued measure, $E$, is a map,
$E\mathrel{\mathop{\ordinarycolon}}\Sigma\to\mathcal{B}(\mathcal{H})$ that is
weakly countably additive. More precisely, if $B_{i}\in\Sigma$ is a countable
collection of disjoint sets with union $\cup_{i}B_{i}=B$, then

$\displaystyle\braket{E(B)x,y}=\sum_{i}\braket{E(B_{i})x,y}.$ (A.65)

for all $x,y\in\mathcal{H}$. Furthermore,

  * •

$E$ is self-adjoint if $E(B)$ is self-adjoint for all $B$.

  * •

$E$ is spectral if $E(B_{1}\cap B_{2})=E(B_{1})E(B_{2})$ for all
$B_{1},B_{2}\in\Sigma$.

  * •

$E$ is projection-valued if it is self-adjoint, spectral, and
$E(X)=\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}$.

######  Example A.15.

Let $\mathfrak{X}$ be an orthonormal basis of a Hilbert space. The induced
norm topology on $\mathfrak{X}$ is the discrete topology on $\mathfrak{X}$.
The map
$E\mathrel{\mathop{\ordinarycolon}}\mathbb{P}(B)\to\mathcal{B}(\mathcal{H})$
defined by
$E(S)\mathrel{\mathop{\ordinarycolon}}=\mathscr{P}_{\text{cl}(\text{span}(S)}$
is a projective valued measure.  $\triangle$

######  Example A.16.

Given two Hilbert spaces, $\mathcal{H}$ and $\mathcal{K}$, a unital
$*$-homomorphism,
$\pi\mathrel{\mathop{\ordinarycolon}}\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$,
and a projective valued measure
$E\mathrel{\mathop{\ordinarycolon}}\mathfrak{B}_{X}\to\mathcal{B}(H)$, the map
$\pi\circ E$ is a projective valued measure.  $\triangle$

Integration is a linear map on (measurable) functions over measure spaces.
Integration can be performed with respect to measures either or with respect
to projective-valued measures. No integrals with respect to operator-valued
measures are explicitly performed in this thesis; thus, instead of explicitly
defining integration, I refer the reader to a reference [206, §8.3.2].

######  Theorem A.3.5 (Spectral Theorem).

For every self-adjoint linear operator on a Hilbert space, $A$, there exists a
unique projection-valued measure,
$P_{A}\mathrel{\mathop{\ordinarycolon}}\mathfrak{B}_{\sigma(A)}\to\mathcal{B}(\mathcal{H})$
such that

$\displaystyle A=\int_{\sigma(A)}\lambda\,dP_{A}(\lambda).$ (A.66)

Furthermore, the spectrum of $A$ is a subset of the reals,

$\displaystyle\sigma(A)\subseteq\mathbb{R}.$ (A.67)

[119, Thm. 10.4]

###  A.4 Chebyshev Polynomials

This section tabulates the formulas satisfied by the Chebyshev polynomials
that are applied in this thesis. Many more properties of Chebyshev polynomials
can be located in resources such as [199, 232, 194]. The Chebyshev polynomials
are sequences of orthogonal polynomials. All four polynomials, $P_{n}(x)$, are
solutions to the linear recurrence relation

$\displaystyle P_{n}(x)=2xP_{n-1}(x)-P_{n-2}(x),$ (A.68)

where $n\in\mathbb{Z}$ and $x\in\mathbb{C}$. These polynomials merely differ
in prescriptions for the initial conditions of the recurrence relation. The
four kinds of Chebyshev polynomials and their root sets,
$P_{n}^{-1}(\\{0\\})$, are displayed in table A.1.

Kind, $P_{n}$ | First, $T_{n}$ | Second, $U_{n}$ | Third, $V_{n}$ | Fourth, $W_{n}$
---|---|---|---|---
$P_{n}(\cos\theta)$ | $\cos(n\theta)$ | $\dfrac{\sin((n+1)\theta)}{\sin\theta}$ | $\dfrac{\cos((n+1/2)\theta)}{\cos(\theta/2)}$ | $\dfrac{\sin((n+1/2)\theta)}{\sin(\theta/2)}$
$P_{n}\left(\dfrac{z+z^{-1}}{2}\right)$ | $\dfrac{z^{n}+z^{-n}}{2}$ | $\dfrac{z^{n+1}-z^{-n-1}}{z-z^{-1}}$ | $\dfrac{z^{n+1/2}+z^{-n-1/2}}{z^{1/2}+z^{-1/2}}$ | $\dfrac{z^{n+1/2}-z^{-n-1/2}}{z^{1/2}-z^{-1/2}}$
$\arccos(P_{n}^{-1}(\\{0\\}))$ $k\in\\{1,,n\\}$ | $\dfrac{(2k-1)\pi}{n}$ | $\dfrac{k\pi}{n+1}$ | $\dfrac{(2k-1)\pi}{2n+1}$ | $\dfrac{2k\pi}{2n+1}$
$P_{n}(1)$ | $1$ | $n+1$ | $1$ | $2n+1$
$P_{n}(-1)$ | $(-1)^{n}$ | $(-1)^{n}(n+1)$ | $(-1)^{n}(2n+1)$ | $(-1)^{n}$
Table A.1: Definitions and roots of the Chebyshev polynomials, see [194, chp.
2] for more details. A root for a Chebyshev polynomial is derived from an
entry in the second row by applying the cosine.

Further linear relations between the Chebyshev polynomials include

$\displaystyle 2T_{n}(x)$ $\displaystyle=U_{n}(x)-U_{n-2}(x)$ [194, eq. (1.7)]
(A.69) $\displaystyle V_{n}(x)$ $\displaystyle=U_{n}(x)-U_{n-1}(x)$ [194, eq.
(1.17)] (A.70) $\displaystyle W_{n}(x)$ $\displaystyle=U_{n}(x)+U_{n-1}(x)$
[194, eq. (1.18)] (A.71) $\displaystyle U_{n\pm 1}(x)$
$\displaystyle=xU_{n}(x)\pm T_{n+1}(x)$ (A.72)

Consequently, properties of the any kind of Chebyshev polynomial can often be
determined from corresponding properties of Chebyshev polynomials of the
second kind.

The derivative is

$\displaystyle U_{n}^{\prime}(x)$
$\displaystyle=\begin{cases}\dfrac{-nxU_{n}(x)+(n+1)U_{n+1}(x)}{1-x^{2}}&\text{if
}x\neq\pm 1\\\ (\pm 1)^{n+1}\left(\dfrac{n(n+1)(n+2)}{3}\right)&\text{if
}x=\pm 1\end{cases}$ (A.73)

An interesting composition identity of the Chebyshev polynomials is444This
formula appeared as a textbook problem before the publication of both
references mentioned here, see [194, §1.5]. [328, Lemma 3][44, Thm. 5]

$\displaystyle U_{mk-1}(x)=U_{k-1}\left(T_{m}(x)\right)U_{m-1}\left(x\right),$
(A.74)

The resultant between two Chebyshev polynomials of the second kind is [142,
189]

$\displaystyle\text{Res}_{x}\left(U_{n}(x),U_{m}(x)\right)$
$\displaystyle=\begin{cases}0&\text{if }\text{gcd}(m+1,n+1)>1\\\
(-1)^{mn/2}*2^{mn}&\text{if }\text{gcd}(m,n)=1\end{cases}.$ (A.75)

Some nifty product formulae include

$\displaystyle U_{n-m-1}(x)$
$\displaystyle=U_{n-1}(x)U_{m}(x)-U_{m-1}(x)U_{n}(x)$ (A.76) $\displaystyle
U_{2m}(x)$ $\displaystyle=U_{m}^{2}(x)-U_{m-1}^{2}(x).$ (A.77)

A series expansion is

$\displaystyle U_{n}(x)$ $\displaystyle=\sum^{\lfloor
n/2\rfloor}_{i=0}(-1)^{m}\frac{(n-m)!}{m!(n-2m)!}(2x)^{n-2m}.$ (A.78)

###  A.5 Proof of Theorem2.3.6

As a reminder, one of the claims proven in theorem 2.4.5 is that a matrix is
similar to a matrix with real elements if and only if it has an involutive
antilinear symmetry.

This section references the Jordan normal form of an operator and the notion
of unbroken antilinear symmetry. In the interest of remaining self-contained,
I review these concepts before their usage.

######  Theorem A.5.1 (Jordan Normal Form).

Given a matrix $A\in\mathfrak{M}_{n}(\mathbb{C})$, there exists a set of
projectors, $P_{\lambda}$, and a set of nilpotents555An operator is nilpotent
if there exists a finite power of this operator which equals zero.,
$D_{\lambda}$, such that [157, chp. I §5]

$\displaystyle A=\sum_{\lambda\in\sigma(A)}\lambda P_{\lambda}+D_{\lambda}.$
(A.79)

The projectors and nilpotents satisfy the identities

$\displaystyle P_{\lambda}D_{\lambda^{\prime}}$
$\displaystyle=D_{\lambda}P_{\lambda^{\prime}}=\delta^{\lambda}_{\lambda^{\prime}}P_{\lambda}$
(A.80) $\displaystyle D_{\lambda}D_{\lambda^{\prime}}$
$\displaystyle=\delta^{\lambda}_{\lambda^{\prime}}D^{2}_{\lambda}$ (A.81)
$\displaystyle P_{\lambda}P_{\lambda^{\prime}}$
$\displaystyle=\delta^{\lambda}_{\lambda^{\prime}}P_{\lambda}$ (A.82)
$\displaystyle\sum_{\lambda\in\sigma(A)}P_{\lambda}$
$\displaystyle=\mathbb{1}.$ (A.83)

######  Definition A.5.2.

Given an antilinear operator on a Hilbert space, $\Theta$, a linear operator,
$A\mathrel{\mathop{\ordinarycolon}}\text{Dom}(A)\to\mathcal{H}$, is called
$\Theta$-unbroken if and only if $\Theta$ is an antilinear symmetry of $A$ and
the eigenspaces of $A$ are invariant subspaces of $\Theta$,

$\displaystyle\Theta\ker(\lambda\mathbb{1}-A)\subseteq\ker(\lambda\mathbb{1}-A)$
$\displaystyle\forall\lambda\in\sigma_{p}(A).$ (A.84)

######  Theorem A.5.3.

The spectrum of a matrix, $A\in\mathfrak{M}_{n}(\mathbb{C})$, is a subset of
the reals, $\sigma(A)\subset\mathbb{R}$, if and only if there exists a
involutive antilinear symmetry, $\Theta$, so that
$\Theta^{2}=\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}$,
such that $A$ is $\Theta$-unbroken. If $A$ is $\Theta$-unbroken for one
invertible antilinear symmetry, then $A$ is $\Theta$-unbroken for all
invertible antilinear symmetries.

###### Proof.

First, I show that matrices with real eigenvalues have an unbroken antilinear
symmetry satisfying
$\Theta^{2}=\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}$,
and that every antilinear symmetry of such a matrix is unbroken. Every matrix
is similar to its Jordan normal form [133, §7.3]. In the case where a matrix
has a real spectrum, its Jordan normal form is real, so by theorem 2.4.5, a
matrix with real eigenvalues has an antilinear symmetry satisfying
$\Theta^{2}=\lx@glossaries@gls@link{symbols}{id}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{1}}}}$.

For my next trick, I prove that a matrix with real eigenvalues, $A$, is
$\Theta$-unbroken for every invertible antilinear $\Theta$ satisfying $\Theta
A=A\Theta$. This is because for every eigenvalue $\lambda\in\sigma_{p}(A)$,
since $\lambda$ is real by supposition,

$\displaystyle\\{0\\}$
$\displaystyle=(\lambda\mathbb{1}-A)\ker(\lambda\mathbb{1}-A)$ (A.85)
$\displaystyle=\Theta(\lambda\mathbb{1}-A)\ker(\lambda\mathbb{1}-A)$ (A.86)
$\displaystyle=(\lambda\mathbb{1}-A)\Theta\ker(\lambda\mathbb{1}-A).$ (A.87)

Thus, $\ker(\lambda\mathbb{1}-A)$ is an invariant subspace of $\Theta$ for all
$\lambda\in\sigma_{p}(A)$, so $\Theta$ is unbroken by definition.

The second direction of the proof in the case where all eigenvalues of $A$ are
simple is given in [24]. The general case follows from the definition of
unbroken symmetry and item 2 of theorem 2.3.5. ∎

Figure A.1: This graph summarizes the mathematical spaces studied in this
appendix and their relationships. An arrow pointing from one space to another
indicates that all elements of the latter space are included in the former.