# Bicomplex Matrices and Operators: Jordan Forms, Invariant Subspace Lattice
Diagrams, and Compact Operators

William Johnston and Rebecca G. Wahl

###### Abstract

This paper extends topics in linear algebra and operator theory for linear
transformations on complex vector spaces to those on bicomplex Hilbert and
Banach spaces. For example, Definition 3 for the first time defines a
bicomplex vector space, its dimension, and its basis in terms of a
corresponding vectorial idempotent representation, and the paper shows how an
$n\times n$ bicomplex matrix’s idempotent representation leads to its
bicomplex Jordan form and a description of its bicomplex invariant subspace
lattice diagram. Similarly, the paper rigorously defines for the first time
“bicomplex Banach and Hilbert” spaces, and then it expands the theory of
compact operators on complex Banach and Hilbert spaces to those on bicomplex
Banach and Hilbert spaces. In these ways, the paper shows that complex linear
algebra and operator theory are not necessarily built upon the broadest and
most natural set of scalars to study. Instead, when using the idempotent
representation, such results surprisingly generalize in a straightforward way
to the higher-dimensional case of bicomplex values.

##  1 Preliminaries.

Bicomplex numbers111This introduction is adapted from [9]. are of the form
$\zeta=z_{1}+jz_{2}$, where $z_{1}=x_{1}+iy_{1}$ and $z_{2}=x_{2}+iy_{2}$ are
complex numbers, $i^{2}={-1}$, and $j^{2}={-1}$. Corrado Segre first described
them in 1892 [20]. A host of compelling analytic function properties extend
outward to bicomplex functional analysis in often straightforward ways. These
include generalizations of Euler’s formula, the representation of complex-
analytic functions as power series, and Cauchy’s integral formula. See [1, 4,
9, 12, 13, 14, 15, 19] for such results. The theory is also useful in applied
settings (see [2, 16, 17, 18]). The set of bicomplex numbers
$\mathbb{B}\mathbb{C}$ is a four real dimensional extension of the complex
numbers $\mathbb{C}$ in the sense that
$\mathbb{B}\mathbb{C}\big{|}_{z_{2}=0}=\mathbb{C}$. Here
$\zeta=x_{1}+iy_{1}+jx_{2}+ijy_{2}$, where $ij=ji$ so that
$(ij)^{2}=(-1)^{2}=1$. Given $\zeta$ and $\omega=w_{1}+jw_{2}$ in
$\mathbb{B}\mathbb{C}$,

$\zeta+\omega=(z_{1}+w_{1})+j(z_{2}+w_{2})$ and

$\zeta\cdot\omega=(z_{1}w_{1}-z_{2}w_{2})+j(z_{1}w_{2}+w_{1}z_{2})$.

These equations agree with complex addition and multiplication when
$\zeta,\omega\in\mathbb{C}$, and $\mathbb{B}\mathbb{C}$ is a commutative
algebraic ring, as not all bicomplex numbers have a multiplicative inverse.
But a so-called idempotent representation

${\zeta=(z_{1}-iz_{2})\mathbf{\boldsymbol{\mathrm{e}}_{1}}+(z_{1}+iz_{2})\mathbf{\boldsymbol{\mathrm{e}}_{2}}\equiv\zeta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}}\,$

importantly gives a better way to think about bicomplex numbers and associated
mathematical structures such as analytic functions (see [9]) and, in
particular, linear transformations, as this paper shows. Here
$\mathbf{\boldsymbol{\mathrm{e}}_{1}}=\displaystyle{(1+ij)/}{2}$ and
$\displaystyle\mathbf{\boldsymbol{\mathrm{e}}_{2}}={(1-ij)/}{2}$, and
$\zeta_{1},\zeta_{2}\in\mathbb{C}$ are the idempotent components. The
equalities
$\mathbf{\boldsymbol{\mathrm{e}}_{1}}^{2}=\mathbf{\boldsymbol{\mathrm{e}}_{1}}$,
$\mathbf{\boldsymbol{\mathrm{e}}_{2}}^{2}=\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
and
$\mathbf{\boldsymbol{\mathrm{e}}_{1}}\mathbf{\boldsymbol{\mathrm{e}}_{2}}=\mathbf{\boldsymbol{\mathrm{e}}_{2}}\mathbf{\boldsymbol{\mathrm{e}}_{1}}=0$
result in the following property: For
$\zeta=\zeta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
and
$\omega=\omega_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\omega_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
the product

$\zeta\omega=\zeta_{1}\omega_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}\omega_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$

computes idempotent componentwise! Hence
$\zeta^{n}=\zeta_{1}^{n}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}^{n}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
and any (multi-branched) $n$th root translates into complex idempotent
components (say for $n\in\mathbb{N}$) as

$\zeta^{1/n}=\zeta_{1}^{1/n}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}^{1/n}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.

Simple algebra also proves two extremely important special items. First,
$\zeta$ is complex if and only if $\zeta_{1}=\zeta_{2}=\zeta$. Second, the
multiplicative noninvertible elements $\zeta$ are exactly the values that have
either $\zeta_{1}=0$ or $\zeta_{2}=0$.

A powerful so-called hyperbolic norm evolves from consideration of bicomplex
numbers with real idempotent components. This norm equals the Euclidean norm
when applied to complex numbers, but it outputs moduli for noncomplex numbers
that are in general not real! Hence its properties form a generalization of
the standard defining properties of a norm, obtained by replacing the
nonnegative real numbers by a partially ordered set of bicomplex numbers
called the nonnegative hyperbolic numbers $\mathbb{H}^{+}$, which the
following definition describes. To distinguish it from a norm that has the
standard properties, mathematicians sometimes call the hyperbolic norm a
poset-valued norm (cf. [12]).

######  Definition 1 (Hyperbolic Numbers).

The hyperbolic numbers

$\mathbb{H}=\\{x+ijy:\ x,y\in\mathbb{R}\\}$

are a strict subset of $\mathbb{B}\mathbb{C}$. The subset of nonnegative
hyperbolic numbers is

$\mathbb{H}^{+}=\\{\eta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\eta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}:\eta_{1},\eta_{2}\geq
0\\}$.

For any
$\zeta=\zeta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\in\mathbb{B}\mathbb{C}$,
the hyperbolic norm is defined as

$|\zeta|_{{}_{\mathbb{H}}}=|\zeta_{1}|\mathbf{\boldsymbol{\mathrm{e}}_{1}}+|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,

which is an element of $\mathbb{H}^{+}$ and satisfies
$|\zeta\cdot\omega|_{{}_{\mathbb{H}}}=|\zeta|_{{}_{\mathbb{H}}}\cdot|\omega|_{{}_{\mathbb{H}}}$
for any $\zeta,\omega\in\mathbb{B}\mathbb{C}$.

A partial ordering on $\mathbb{H}^{+}$ now follows. For two elements
$\zeta=\eta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\eta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
and
$\omega=\psi_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\psi_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
in $\mathbb{H}^{+}$, define the hyperbolic inequality
$\zeta<_{{}_{\mathbb{H}}}\omega$ when both $\eta_{1}<\psi_{1}$ and
$\eta_{2}<\psi_{2}$. A less-than-or-equal-to partial ordering is similarly
defined. The hyperbolic norm of any bicomplex number is a hyperbolic number.
That norm can therefore be used via this hyperbolic inequality to define open
balls of bicomplex numbers that have hyperbolic radii. The balls are of the
form

$B_{{}_{\mathbb{H}}}(c,R)=\\{\zeta:|\zeta-c|_{{}_{\mathbb{H}}}<_{{}_{\mathbb{H}}}R\\}$,

where the center $c\in\mathbb{B}\mathbb{C}$ and the hyperbolic radius
$R\in\mathbb{H}^{+}$. The collection of balls $B_{{}_{\mathbb{H}}}(c,R)$ can
also be thought of as the neighborhood basis open sets in a topology induced
by the hyperbolic norm.

A bicomplex number’s Euclidean form $\eta=z_{1}+jz_{2}$ produces a choice of
three natural conjugations (cf. [11, p. 562] or [10, p. 593]):

  1. 1.

$\overline{\zeta}=\overline{z}_{1}+j\overline{z}_{2}$;

  2. 2.

$\overline{\zeta}={z}_{1}-j{z}_{2}$; or

  3. 3.

$\overline{\zeta}=\overline{z}_{1}-j\overline{z}_{2}$

The first and third choices both generalize conjugation for complex numbers
but the third is most natural because it corresponds with the only natural
conjugations in terms of the idempotent component; namely, for
$\zeta=\zeta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,

$\overline{\zeta}=\overline{z}_{1}-j\overline{z}_{2}=\overline{\zeta}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\overline{\zeta}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$

This calculation follows from the formulas for $\zeta_{1}$ and $\zeta_{2}$ in
terms of $z_{1}$ and $z_{2}$, along with the simple facts that
$\overline{\mathbf{\boldsymbol{\mathrm{e}}_{1}}}=\mathbf{\boldsymbol{\mathrm{e}}_{1}}$
and
$\overline{\mathbf{\boldsymbol{\mathrm{e}}_{2}}}=\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.
Furthermore,

$\zeta\cdot\overline{\zeta}=(\zeta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(\overline{\zeta}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\overline{\zeta}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})=|\zeta_{1}|^{2}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+|\zeta_{2}|^{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}=|\zeta|^{2}_{{}_{\mathbb{H}}}$,

which says this conjugation in its relationship with the hyperbolic norm
generalizes complex conjugation in its relationship with the complex modulus.
In all that follows in this paper, this third choice is always used as the
definition of the bicomplex conjugate, which we formalize here.

######  Definition 2.

The bicomplex conjugate of
$\zeta=\zeta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
is
$\overline{\zeta}=\overline{\zeta}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\overline{\zeta}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.

Finally, we introduce the notion of a bicomplex vector space, expanding the
definition of a vector space on complex scalars. The expansion to the
bicomplex setting makes special use of the idempotent representation.

######  Definition 3.

A bicomplex vector space, taken over bicomplex scalars, is a set of well-
defined elements, called “bicomplex vectors,” that may be decomposed using the
bicomplex idempotent structure into complex idempotent components as
$v=v_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+v_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
where the resulting corresponding sets of idempotent “vectors” $\\{v_{1}\\}$
and $\\{v_{2}\\}$, here taken over complex scalars unless noted otherwise, are
each vector spaces with the same dimension. The bicomplex vector space
$V=\\{v\\}$ has finite dimension $n^{2}$ when $\\{v_{1}\\}$ and $\\{v_{2}\\}$
both have dimension $n$, where a basis for $V$ is formed from all $n^{2}$
combinations of the bases for the two idempotent components
$\\{b_{1,k}\\}_{k=1}^{n}$ and $\\{b_{2,j}\\}_{j=1}^{n}$, respectively, as
$\\{b_{k,j}\\}=\\{b_{1,k}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+b_{2,j}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\}$.

This paper examines such vector spaces with well-defined complete norms
defined on each idempotent component, so that each component vector space
forms a Banach space, calling such vector spaces “bicomplex Banach spaces.” In
addition, it examines such vector spaces with well-defined inner products, so
that each component vector space forms a Hilbert space, calling such vector
spaces “bicomplex Hilbert spaces.” Examining the bicomplex Banach and Hilbert
spaces’ idempotent structure, it will also define and examine types of linear
transformations on such spaces.

##  2 Bicomplex Matrices

We first consider the extension of complex linear algebra to linear
transformations on finite $n$-dimensional bicomplex vector spaces. In what
follows in this section, each matrix is $n\times n$ unless otherwise noted.
Writing each entry of a bicomplex matrix $A$ in terms of its idempotent
representation produces an idempotent representation for the matrix
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.
Multiplication of bicomplex matrices is defined via row times column products
in the same way as for complex matrices. A delightful fact is that matrix
multiplication computes idempotent componentwise, just as multiplication
computes componentwise for scalars! See, e.g., [1, Chapter 5]. This fact is so
important that we call it the Fundamental Theorem for Bicomplex Matrices.

######  Theorem 1.

The Fundamental Theorem for Bicomplex Matrices: Given bicomplex matrices
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
of size $m\times k$ and
$B=B_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+B_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
of size $k\times n$, then $A\cdot
B=A_{1}B_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}B_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
is $m\times n$.

###### Proof.

Since
$\mathbf{\boldsymbol{\mathrm{e}}_{1}}^{2}=\mathbf{\boldsymbol{\mathrm{e}}_{1}}$,
$\mathbf{\boldsymbol{\mathrm{e}}_{2}}^{2}=\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
and
$\mathbf{\boldsymbol{\mathrm{e}}_{1}}\mathbf{\boldsymbol{\mathrm{e}}_{2}}=\mathbf{\boldsymbol{\mathrm{e}}_{2}}\mathbf{\boldsymbol{\mathrm{e}}_{1}}=0$,
$A\cdot
B=(A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(B_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+B_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})\\\
\hskip
56.0pt=A_{1}B_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}^{2}+A_{1}B_{2}\mathbf{\boldsymbol{\mathrm{e}}_{1}}\mathbf{\boldsymbol{\mathrm{e}}_{2}}+A_{2}B_{1}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}B_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}^{2}\\\
\hskip
56.0pt=A_{1}B_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}B_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.

For an equivalent proof, denote $AB=[(ab)_{ij}]$, with $A=[a_{ik}]$ and
$B=[b_{kj}]$. Each term is
$(ab)_{ij}=\sum\limits_{k=1}^{m}a_{ik}b_{kj}=\sum\limits_{k=1}^{m}(\hat{a}_{ik}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\tilde{a}_{ik}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(\hat{b}_{kj}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\tilde{b}_{kj}\mathbf{\boldsymbol{\mathrm{e}}_{2}})\\\
\hskip
60.5pt=\sum\limits_{k=1}^{m}(\hat{a}_{ik}\hat{b}_{kj}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\tilde{a}_{ik}\tilde{b}_{kj}\mathbf{\boldsymbol{\mathrm{e}}_{2}})=\sum\limits_{k=1}^{m}(\hat{a}_{ik}\hat{b}_{kj})\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\sum\limits_{k=1}^{m}(\tilde{a}_{ik}\tilde{b}_{kj})\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.
∎

The following three corollaries for $m\times k$ matrices immediately result.
For the third corollary, we define a bicomplex number $\lambda$ as an
eigenvalue for a bicomplex matrix $A$ when there exists a column vector
$\vec{v}=\vec{v}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\vec{v}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
(the corresponding eigenvector) with both idempotent components $\vec{v}_{1}$
and $\vec{v}_{2}$ nonzero so that $A\vec{v}=\lambda\vec{v}$.222We note that
for a matrix
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
a column vector $\vec{v}$ with $A\vec{v}=\lambda\vec{v}$ could have been
described as an eigenvector so long as $\vec{v}$ was nonzero, but the specific
case of $\vec{v}$ having exactly one nonzero idempotent component reduces to
the eigenvalue/eigenvector study of the corresponding matrix component. For
example, if $\vec{v}_{1}$ is nonzero but $\vec{v}_{2}$ is the zero vector,
then the examination reduces to the study of the complex matrix $A_{1}$. Our
eigenvalue definition matches other authors’; for example, see [11, Definition
2.3]. A related commentary is [1, pp. 27–28].

######  Corollary 1.

For a bicomplex matrix
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
the inverse $A^{-1}$ exists exactly when both $A_{1}^{-1}$ and $A_{2}^{-1}$
exist, and then
$A^{-1}=A_{1}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.

######  Corollary 2.

Define the determinant of a bicomplex matrix $A$ in terms of its idempotent
component determinants; i.e., $\det A\equiv\det
A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\det
A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$. Then $A^{-1}$ exists exactly when
$\det A$ is bicomplex invertible; i.e., when $\det A_{1}$ and $\det A_{2}$ are
both nonzero.

######  Corollary 3.

A bicomplex number $\lambda$ is an eigenvalue for a bicomplex matrix $A$
exactly when $\det(A-\lambda I)=0$.

The first main result of this section now follows for $n\times n$ bicomplex
matrices.

######  Theorem 2.

Bicomplex Jordan Form: Write
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
and put $A_{1}$ and $A_{2}$ in their (complex) Jordan forms as
$A_{1}=P_{1}J_{1}P_{1}^{-1}$ and $A_{2}=P_{2}J_{2}P_{2}^{-1}$. Then a Jordan
form of $A$ is $A=PJP^{-1}$, where
$P=P_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
and
$J=J_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+J_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.
Because the Jordan forms of $A_{1}$ and $A_{2}$ are not unique (for example,
the diagonal blocks for $J_{1}$ and $J_{2}$ may appear in any order), this
Jordan form representation of $A$ can take on many different forms.

###### Proof.

Express the matrix in its idempotent form:
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}=P_{1}J_{1}P_{1}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}J_{2}P_{2}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\\
\hskip
39.6pt=(P_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(J_{1}P_{1}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+J_{2}P_{2}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{2}})\\\
\hskip
39.6pt=(P_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(J_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+J_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(P_{1}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{2}})=PJP^{-1}$.
∎

It is important to realize the Jordan form $J$ is “almost diagonal.” Similar
to the complex Jordan form, $J$ has zeros in each entry except for possibly
the diagonal and the immediate first off-diagonal in the upper-triangular
portion. The proof of Theorem 2 shows the upper off-diagonal can consist of
0’s, 1’s, $\mathbf{\boldsymbol{\mathrm{e}}_{1}}$’s, and
$\mathbf{\boldsymbol{\mathrm{e}}_{2}}$’s. Also, $A_{1}$ and $A_{2}$ are
diagonalizable if and only if $J$ is diagonal. This paper will soon show an
immediate application of the Bicomplex Jordan form: it helps determine the
invariant subspaces for any bicomplex matrix. To clarify, when
$\mathbb{B}\mathbb{C}^{n}$ is the collection of $n$-dimensional column vectors
with bicomplex entries, a subspace $S\subseteq\mathbb{B}\mathbb{C}^{n}$ is
invariant under an $n\times n$ matrix $A$ when $A\vec{\zeta}\in S$ for every
$\vec{\zeta}\in S$. The following fact, the second main result of the section,
results as a direct consequence of the Fundamental Theorem of Bicomplex
Matrices.

######  Corollary 4.

Writing any vector $\vec{\zeta}\in\mathbb{B}\mathbb{C}^{n}$ as
$\vec{\zeta}=\vec{\zeta}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\vec{\zeta}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
where $\vec{\zeta}_{1},\vec{\zeta}_{2}\in\mathbb{C}^{n}$, decompose any given
subspace $S\subseteq\mathbb{B}\mathbb{C}^{n}$ as
$S=S_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}\oplus
S_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$, where $S_{1}$ and $S_{2}$ are
subspaces of $\mathbb{C}^{n}$. Then $S$ is invariant under any given $n\times
n$ bicomplex matrix
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
exactly when $S_{1}$ is invariant under $A_{1}$ and $S_{2}$ is invariant under
$A_{2}$.

###### Proof.

The subspace $S$ is invariant under $A$ exactly when

$A\vec{\zeta}=A_{1}\vec{\zeta}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\vec{\zeta}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\in
S_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}\oplus
S_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$ for every
$\vec{\zeta}=\vec{\zeta}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\vec{\zeta}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\in
S$.

But that fact is equivalent to both

$A_{1}\vec{\zeta}_{1}\in S_{1}$ for every $\vec{\zeta}_{1}\in S_{1}$, and
$A_{2}\vec{\zeta}_{2}\in S_{2}$ for every $\vec{\zeta}_{2}\in S_{2}$,

which means exactly that $S_{1}$ is invariant under $A_{1}$ and $S_{2}$ is
invariant under $A_{2}$. ∎

Corollary 4 is extremely useful, especially since a constructive method to
determine the invariant subspace lattice diagram for any complex $n\times n$
matrix is known. See [6]. From Corollary 4, this constructive method extends
to find the invariant subspace lattice diagram for any bicomplex $n\times n$
matrix $A$: it says simply to form each possible invariant subspace
$S_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}\oplus
S_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$, where $S_{1}$ is invariant for
$A_{1}$ and $S_{2}$ is invariant for $A_{2}$, and then to realize each of
these invariant subspaces for $A$ “locks into place naturally” into the
corresponding lattice diagram. Example 2 illustrates that process for a simple
case, below.

An analysis that is similar to the development of the Jordan form also applies
to other representations of bicomplex matrices, for example, to form a
diagonalization of a self-adjoint bicomplex matrix. Begin with the following
definitions.

######  Definition 4.

The adjoint of a bicomplex matrix $A=[a_{ij}]$ is its conjugate-transpose
$A^{*}=[\overline{a}_{ji}]$.

Note this definition generalizes the adjoint of a complex matrix. Furthermore,
if $\vec{\zeta},\vec{\omega}\in\mathbb{B}\mathbb{C}^{n}$, we define as in, for
example, [11], a bicomplex inner product as

$\langle\vec{\zeta},\vec{\omega}\rangle=\sum\limits_{i=1}^{n}\zeta_{i}\overline{\omega}_{i}=\sum\limits_{i=1}^{n}(\zeta_{i1}\overline{\omega}_{i1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\zeta_{i2}\overline{\omega}_{i2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})=\langle\vec{\zeta}_{1},\vec{\omega}_{1}\rangle\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\langle\vec{\zeta}_{2},\vec{\omega}_{2}\rangle\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.

The following theorem results.

######  Theorem 3.

For any $n\times n$ bicomplex matrix $A$ and
$\vec{\zeta},\vec{\omega}\in\mathbb{B}\mathbb{C}^{n}$,

$\langle
A\vec{\zeta},\vec{\omega}\rangle=\langle\vec{\zeta},A^{*}\vec{\omega}\rangle$.

###### Proof.

By Theorem 1, the definition of the bicomplex inner product, and the adjoint
properties of complex matrices,
$\langle A\vec{\zeta},\vec{\omega}\rangle=\langle
A_{1}\vec{\zeta}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\vec{\zeta}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}},\vec{\omega}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\vec{\omega}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\rangle=\langle
A_{1}\vec{\zeta}_{1},\vec{\omega}_{1}\rangle\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\langle
A_{2}\vec{\zeta}_{2},\vec{\omega}_{2}\rangle\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\\
\indent\hskip
37.5pt=\langle\vec{\zeta}_{1},A_{1}^{*}\vec{\omega}_{1}\rangle\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\langle\vec{\zeta}_{2},A_{2}^{*}\vec{\omega}_{2}\rangle\mathbf{\boldsymbol{\mathrm{e}}_{2}}=\langle\vec{\zeta},A^{*}\vec{\omega}\rangle$
∎

Similar to the complex situation, a bicomplex matrix is self-adjoint when
$A=A^{*}$. A bicomplex unitary matrix $U$ satisfies $U^{-1}=U^{*}$. We note,
as in [11], the following well-known result.

######  Theorem 4.

The Spectral Theorem for Self-adjoint Matrices: Any self-adjoint bicomplex
matrix is unitarily diagonalizable, in the sense that

$A=PDP^{-1}$

for a (bicomplex) diagonal matrix $D$ and a unitary matrix $P$.

###### Proof.

When
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
by the definition of a self-adjoint bicomplex matrix, both $A_{1}$ and $A_{2}$
are complex self-adjoint, and then the spectral theorem implies
$A_{1}=P_{1}D_{1}P_{1}^{-1}$ and $A_{2}=P_{2}D_{2}P_{2}^{-1}$. Then
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}=P_{1}D_{1}P_{1}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}D_{2}P_{2}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\\
\hskip
40.0pt=(P_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(D_{1}P_{1}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+D_{2}P_{2}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{2}})\\\
\indent\hskip
28.0pt=(P_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(D_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+D_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}})(P_{1}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{2}})=PDP^{-1}$,
with each matrix having the properties required in the theorem statement. For
example, $P$ may be chosen unitary because $P_{1}$ and $P_{2}$ may be chosen
as (complex) unitary, and then

$P^{*}=P_{1}^{*}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}^{*}\mathbf{\boldsymbol{\mathrm{e}}_{2}}=P_{1}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+P_{2}^{-1}\mathbf{\boldsymbol{\mathrm{e}}_{2}}=P^{-1}$.

∎

##  3 Examples

Example 1: Let $A=\left[\begin{matrix}0&\zeta\\\
\overline{\zeta}&0\end{matrix}\right]=\left[\begin{matrix}0&\zeta_{1}\\\
\overline{\zeta_{1}}&0\end{matrix}\right]\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\left[\begin{matrix}0&\zeta_{2}\\\
\overline{\zeta_{2}}&0\end{matrix}\right]\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
with $\zeta\neq 0$. Then $\det(A-\lambda I)=0$ implies
$\lambda_{1}^{2}-|\zeta_{1}|^{2}=0$, or $\lambda_{1}=\pm|\zeta_{1}|$, and
similarly $\lambda_{2}=\pm|\zeta_{2}|$. Therefore, $A$ can be analyzed in
terms of the pair of bicomplex eigenvalues
$\lambda=|\zeta_{1}|\mathbf{\boldsymbol{\mathrm{e}}_{1}}+|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}},-|\zeta_{1}|\mathbf{\boldsymbol{\mathrm{e}}_{1}}-|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
more concisely written as
$\lambda=|\zeta|_{{}_{\mathbb{H}}},-|\zeta|_{{}_{\mathbb{H}}}$. The
corresponding eigenvectors (which come as the nonzero solutions to
$A\vec{\zeta}=\lambda\vec{\zeta}$) are

$\left[\begin{matrix}|\zeta|_{{}_{\mathbb{H}}}\\\
\overline{\zeta}\end{matrix}\right]$ and
$\left[\begin{matrix}-|\zeta|_{{}_{\mathbb{H}}}\\\
\overline{\zeta}\end{matrix}\right]$, respectively.

The Spectral Theorem for Self-adjoint Matrices then implies $A=PDP^{-1}$,
where

$P=\left[\begin{matrix}|\zeta|_{{}_{\mathbb{H}}}&-|\zeta|_{{}_{\mathbb{H}}}\\\
\overline{\zeta}&\overline{\zeta}\end{matrix}\right]$ and
$D=\left[\begin{matrix}|\zeta|_{{}_{\mathbb{H}}}&0\\\
0&-|\zeta|_{{}_{\mathbb{H}}}\end{matrix}\right]$.

An important note is that this is the standard complex Jordan form for $A$
when $\zeta\in\mathbb{C}$.

But even when $A$ is complex, these bicomplex diagonalizations are generally
not unique (nor even “essentially” unique). In this example, a second
diagonalization results, realizing $\det(A-\lambda I)=0$ also for the pair of
eigenvalues
$\lambda=|\zeta_{1}|\mathbf{\boldsymbol{\mathrm{e}}_{1}}-|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}},-|\zeta_{1}|\mathbf{\boldsymbol{\mathrm{e}}_{1}}+|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.
The corresponding (nonzero) eigenvectors are

$\left[\begin{matrix}|\zeta|_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}-|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\\
\overline{\zeta}\end{matrix}\right]$ and
$\left[\begin{matrix}-|\zeta|_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\\
\overline{\zeta}\end{matrix}\right]$, respectively.

A second diagonalization provided by the Spectral Theorem for Self-adjoint
Matrices results, even when $A$ is complex. We get $A=PDP^{-1}$, where

$P=\left[\begin{matrix}|\zeta|_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}-|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}&-|\zeta|_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\\
\overline{\zeta}&\overline{\zeta}\end{matrix}\right]$ and
$D=\left[\begin{matrix}|\zeta_{1}|\mathbf{\boldsymbol{\mathrm{e}}_{1}}-|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}&0\\\
0&-|\zeta_{1}|\mathbf{\boldsymbol{\mathrm{e}}_{1}}+|\zeta_{2}|\mathbf{\boldsymbol{\mathrm{e}}_{2}}\end{matrix}\right]$.

In general, an $n\times n$ self-adjoint matrix
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
will have $n!$ different bicomplex diagonalizations when $A_{1}$ and $A_{2}$
have $n$ distinct eigenvalues, since these eigenvalues can be paired in $n!$
ways.

Example 2: An Illustration of the Invariant Subspace Lattice Diagram for
Bicomplex Matrices. Let $A=\left[\begin{matrix}0&0\\\
0&0\end{matrix}\right]\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\left[\begin{matrix}0&1\\\
0&0\end{matrix}\right]\mathbf{\boldsymbol{\mathrm{e}}_{2}}$. This matrix $A$
is a bicomplex Jordan form matrix as Theorem 2 describes. As developed in [6],
the invariant subspace lattice diagram for a complex matrix can be determined
in a fairly simple manner from a complex matrix’s Jordan form structure. For
example, in the simple case for a matrix with invariant subspaces that are all
marked333An invariant subspace $\mathcal{N}$ is “marked” when there exists a
Jordan basis for $\mathcal{N}$ that can be extended (by adjoining new vectors)
to form a Jordan basis for the entire space $\mathbb{C}^{n}$ (see [3, p.
210]). and whose Jordan form has a single eigenvalue (see [6, Theorem A1]),
the invariant subspace lattice diagram is found by forming diagonal block
matrices $Z$ with the same block structure as the Jordan form $J$ but with
exponential powers of the backward shift matrix forming the diagonal blocks.
Choosing a set of exponential powers corresponds to a particular invariant
subspace (in a pairwise manner). Creating a lattice diagram with all the
different $Z$’s formed in this manner thereby creates a lattice diagram
equivalent to the invariant subspace’s lattice diagram.

Corollary 4 implies the invariant subspace lattice diagram for a bicomplex
matrices
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
can be found from combining the invariant subspace lattices, applying the
techniques in [6] on the complex matrices $A_{1}$ and $A_{2}$. In this
example, $A_{1}=\left[\begin{matrix}0&0\\\ 0&0\end{matrix}\right]$ has two
Jordan blocks with eigenvalues 0, and $A_{2}=\left[\begin{matrix}0&1\\\
0&0\end{matrix}\right]$ has a single Jordan block with eigenvalues 0. Their
two invariant subspace lattice diagrams, found using the techniques in [6],
are in Figure 1.

Figure 1: Invariant subspace lattices for complex matrices $A_{1}$ (on left)
and $A_{2}$ (on right)

Combining the invariant subspaces from each idempotent portion creates an
invariant subspace lattice diagram for the matrix $A$. Our diagram uses a
notational convenience: the symbol $[Z_{1}|S]$, for example, means that the
invariant subspace for $A$ is formed from the invariant subspaces that come
from each matrix in the idempotent component. Here, since $Z_{1}$ corresponds
to the invariant subspace span$\\{\vec{e}_{1}\\}$, where $\vec{e}_{1}$ is the
eigenvector that corresponds to the upper left Jordan block of $A_{1}$ (see
[6] for this formulation), and $S$ corresponds to the invariant subspace
span$\\{\vec{e}_{2}\\}$, where $\vec{e}_{2}$ is the eigenvector for the single
Jordan block of $A_{2}$, the symbol $[Z_{1}|S]$ corresponds to the invariant
subspace
span$\\{\vec{e}_{1}\\}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\mbox{span}\\{\vec{e}_{2}\\}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
of $A$. With this symbolism in mind, the invariant subspace lattice diagram
for $A$ is formed from all such combinations as follows in Figure 2.

Figure 2: Invariant Subspace Lattice for the bicomplex matrix $A$

The same methodology, using the techniques of [6] on each idempotent component
of any $n\times n$ bicomplex matrix
$A=A_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+A_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
routinely develops the bicomplex matrix’s complete invariant subspace lattice
diagram.

##  4 Bicomplex Compact Operators on Banach Spaces

This section shows how the theory of operators on complex Banach spaces
extends, via an operator’s idempotent representation, to a corresponding
theory of so-called bicomplex compact operators on bicomplex Banach spaces.

######  Definition 5.

A bicomplex Hilbert space is of the form
$\mathcal{H}=\mathcal{H}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\mathcal{H}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
where $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are complex Hilbert spaces with
the same dimension. The bicomplex inner product on any elements
$\vec{\zeta},\vec{\omega}\in\mathcal{H}$ is defined via complex inner products
on the idempotent components of $\vec{\zeta}$ and $\vec{\omega}$ as

$\langle\vec{\zeta},\vec{\omega}\rangle=\langle\vec{\zeta}_{1},\vec{\omega}_{1}\rangle\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\langle\vec{\zeta}_{2},\vec{\omega}_{2}\rangle\mathbf{\boldsymbol{\mathrm{e}}_{2}}$

A set of bicomplex vectors $\\{\vec{\zeta}_{k}\\}$ is orthogonal in
$\mathcal{H}$ when $\langle\vec{\zeta}_{n},\vec{\zeta}_{m}\rangle=0$ for
$n\neq m$ and orthonormal when in addition the hyperbolic norm of each vector
in the set, defined via idempotent component norms for $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$ according to
$\|\vec{\zeta}_{n}\|_{{}_{\mathbb{H}}}=\|\vec{\zeta}_{n,1}\|\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\|\vec{\zeta}_{n,2}\|\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
equals 1.

Similarly, a bicomplex Banach space is of the form
$\mathcal{B}=\mathcal{B}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\mathcal{B}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
where $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are complex Banach spaces with
the same dimension.

Such spaces are often called “$\mathbb{B}\mathbb{C}$-modules” in the
literature (see, for example, [10]), where they have been studied in detail.
Definition 5 appropriately names such a space as a Hilbert and/or Banach
space, as the idempotent arrangement is uniquely structured to produce broad
general theories for linear operators on such spaces. This and the next
section develops such theory for bicomplex compact operators, defined here:

######  Definition 6.

An operator $K:\ \mathcal{B}\to\mathcal{B}$ mapping a bicomplex Banach space
$\mathcal{B}$ to itself is compact if for every bounded set
$S\subseteq\mathcal{B}$, the set $K(S)$ is relatively compact in
$\mathcal{B}$, which means that every infinite subset of
$K(S)\equiv\\{K\vec{s}:\vec{s}\in S\\}$ has at least one limit point in its
(hyperbolic) norm closure. This is equivalent to every bounded sequence
$\\{x_{n}\\}$ of vectors in $\mathcal{B}$ producing the sequence
$\\{Kx_{n}\\}$ with a hyperbolic norm-convergent subsequence.

Note importantly that hyperbolic norm convergence in $\mathcal{B}$ happens
exactly when norm convergence of the idempotent components happen in both
complex spaces $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$. In such a way, the
actions of any operator in $\mathcal{B}$ follows from actions of the
corresponding operator elements acting on $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$. Here, $T$ acts on
$v=v_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+v_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\in\mathcal{B}$
according to
$Tv=T_{1}v_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+T_{2}v_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.
The following lemma notes the impact of those facts on compact operators.

######  Lemma 1.

For any operator
$T=T_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+T_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
on a bicomplex Banach space
$\mathcal{B}=\mathcal{B}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\mathcal{B}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
the operators $T_{1}$ and $T_{2}$ are compact on $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$ respectively iff $T$ is compact.

###### Proof.

For $i=1,2$, the compactness of $T_{i}$ implies for every bounded set
$S_{i}\subseteq\mathcal{B}_{i}$, the set $T_{i}(S_{i})$ is relatively compact
in $\mathcal{B}_{i}$; i.e., every infinite subset of $T_{i}(S_{i})$ has at
least one limit point $x_{i}$ in its (complex Banach space) norm closure. But
then every hyperbolic bounded set
$S=S_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+S_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
which must have $S_{1}$ and $S_{2}$ bounded in the corresponding complex
Banach space norm, has at least one limit point
$x=x_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+x_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
in its (hyperbolic) norm closure. Hence $T$ is compact by definition.

Conversely, assume
$T=T_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+T_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
is compact and take any bounded sequences $\\{x_{1,n}\\}\in\mathcal{B}_{1}$
and $\\{x_{2,n}\\}\in\mathcal{B}_{2}$. Then
$\\{x_{n}\\}=\\{x_{1,n}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+x_{2,n}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\}$
is a bounded set of vectors in $\mathcal{B}$ (in the hyperbolic norm sense),
producing the sequence $\\{Tx_{n}\\}$ with a hyperbolic norm-convergent
subsequence $\\{Tx_{{n}_{k}}\equiv
T_{1}x_{{1,n}_{k}}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+T_{2}x_{{2,n}_{k}}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\}$.
But the hyperbolic norm structure then implies that each idempotent component
$T_{1}x_{{1,n}_{k}}$ and $T_{2}x_{{2,n}_{k}}$ is norm-convergent in
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, respectively. Hence $T_{1}$ and
$T_{2}$ each satisfy the definition of a compact operator. ∎

The (hyperbolic) operator norm of any (bicomplex) bounded operator $T$ is
naturally defined as
$\|T\|_{{}_{\mathbb{H}}}=\|T_{1}\|\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\|T_{2}\|\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.
Also, for example, it is easy to see that the identity operator $I$ is not
compact on any infinite-dimensional bicomplex Hilbert space $\mathcal{H}$. For
we can choose an infinite sequence $x_{n}$ of $\mathcal{H}$ to be the
orthonormal set of basis elements for the Hilbert space, which are each
distance (in hyperbolic norm) 1 apart. Then $I(x_{n})=x_{n}$ has no convergent
subsequence. A similar result is seen true for the identity operator on any
infinite-dimensional bicomplex Banach space by applying Lemma 2 below.

The spectrum of a bicomplex operator is defined similarly to the spectrum of a
complex operator.

######  Definition 7.

For a bounded operator $T$:

  1. 1.

The spectrum of $T$ is $\sigma(T)=\\{\zeta\in\mathbb{B}\mathbb{C}\ \colon\zeta
I-T\mbox{ is not invertible }\\}$.

  2. 2.

The point spectrum of $T$ is the set of eigenvalues $\sigma_{p}(T)$; in other
words, the set of $\lambda\in\mathbb{B}\mathbb{C}$ that satisfy

$(\lambda I-T)v=0$

for some
$v=v_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+v_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\in\mathcal{B}$
with both $v_{1}$ and $v_{2}$ nonzero, where $v$ is called an eigenvector for
$\lambda$. Any eigenvalue
$\lambda=\lambda_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\lambda_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
produces for
$T=T_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+T_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
the eigenvalue $\lambda_{1}$ for $T_{1}$ and $\lambda_{2}$ for $T_{2}$ on the
associated spaces $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, respectively, with
associated eigenvectors $v_{1}$ and $v_{2}$, respectively.

A spectral theory for bicomplex compact operators holds on bicomplex Banach
spaces. The following theorem describes that fact, and is well known in the
case of compact operators on complex Banach spaces (see, for example [8, pp.
377–378]).

######  Theorem 5.

For a bicomplex compact operator $K:\mathcal{B}\to\mathcal{B}$ with
$\mathcal{B}$ a bicomplex Banach space,

  1. 1.

Every invertible $\lambda\in\sigma(K)$ is an eigenvalue of $K$.

  2. 2.

The eigenvalues can only accumulate at noninvertible values. If the dimension
of $\mathcal{B}$ is not finite, then $\sigma(K)$ must contain 0, even if 0 is
not an eigenvalue.

  3. 3.

$\sigma(K)$ is countable; in other words, its elements can be listed as either
a finite list or as $\sigma(K)=\\{\lambda_{1},\lambda_{2},\ldots\\}$.

The theorem’s proof follows from the fact that $K$ on $\mathcal{B}$ decomposes
into
$K=K_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+K_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
where $K_{i}$ is compact on $\mathcal{B}_{i}$, $i=1,2$. Applying the fact that
the theorem is well known and true on complex Banach spaces, the proof of each
part follows immediately.

Furthermore, the next two lemmas, which are historically important and true
for operators on complex Banach spaces, are true for bicomplex Banach spaces
as well. The first uses the concept of an infimum of a set of hyperbolic
numbers
$\eta=\eta_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\eta_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
for which $\eta_{1},\eta_{2}\in\mathbb{R}$, defined as
$\inf\\{\eta\\}=\inf\\{\eta_{1}\\}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\inf\\{\eta_{2}\\}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.

######  Lemma 2.

Riesz’s Lemma. For a non-dense subspace
$X=X_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+X_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
of a bicomplex Banach space
$\mathcal{B}=\mathcal{B}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\mathcal{B}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
, given $0<r<1$, there is an element $y\in\mathcal{B}$ with
$\|y\|_{\mathbb{H}}=1$ and $\inf\limits_{x\in X}\|x-y\|_{\mathbb{H}}=r$.

###### Proof.

As mentioned, the lemma holds for each of the associated operators on the
complex Banach spaces: $X_{1}$ in $\mathcal{B}_{1}$ and $X_{2}$ in
$\mathcal{B}_{2}$, in the sense that it produces a vector
$y_{1}\in\mathcal{B}_{1}$ and $y_{2}\in\mathcal{B}_{2}$ with $\|y_{i}\|=1$ and
$\inf\limits_{x\in X_{i}}\|x-y_{i}\|=r$ for $i=1,2$. Then the vector
$y=y_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+y_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
satisfies the lemma, since
$\|y\|_{\mathbb{H}}=1\mathbf{\boldsymbol{\mathrm{e}}_{1}}+1\mathbf{\boldsymbol{\mathrm{e}}_{2}}=1$
and $\inf\limits_{x\in
X}\|x-y\|_{{}_{\mathbb{H}}}=r\mathbf{\boldsymbol{\mathrm{e}}_{1}}+r\mathbf{\boldsymbol{\mathrm{e}}_{2}}=r$.
∎

######  Lemma 3.

For a compact operator
$K=K_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+K_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
on a bicomplex Banach space
$\mathcal{B}=\mathcal{B}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\mathcal{B}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
and for which $I-K$ is one-to-one, the range of $I-K$ is closed in the
hyperbolic norm.

###### Proof.

The lemma holds for operators on a complex Banach space; see, for example,
[7]. Apply that fact to each of $K_{1}$ on
$\mathcal{B}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}$ and $K_{2}$ on
$\mathcal{B}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$, producing, since
$I-K_{1}$ must be one-to-one on
$\mathcal{B}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}$ and $I-K_{2}$ must be
one-to-one on $\mathcal{B}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$, closed
ranges for $I-K_{1}$ and $I-K_{2}$. But then the range of $I-K$ is closed as
desired. ∎

##  5 Bicomplex Compact Operators on Hilbert Spaces

Similar as for the material in the last section, important theorems describe
the structure of a bicomplex compact operator $K$ on a bicomplex Hilbert space
$\mathcal{H}=\mathcal{H}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\mathcal{H}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$.
The theorems result from corresponding theory for each complex compact
operator $K_{1}$ and $K_{2}$ that form the idempotent components of $K$ and
act on the complex Hilbert spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$,
respectively.

######  Theorem 6.

For any operator
$T=T_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+T_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
on a bicomplex Hilbert space
$\mathcal{H}=\mathcal{H}_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\mathcal{H}_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
if there is a sequence $\\{K_{n}\\}_{n=1}^{\infty}$ of bicomplex compact
operators with $\|T-K_{n}\|_{{}_{\mathbb{H}}}\to 0$, then $T$ is compact.

###### Proof.

Since the (hyperbolic) operator norm of any bicomplex operator $Y$ is
calculated via idempotent complex operator norms as
$\|Y\|_{{}_{\mathbb{H}}}=\|Y_{1}\|\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\|Y_{2}\|\mathbf{\boldsymbol{\mathrm{e}}_{2}}$,
there are two (idempotent component) sequences of complex compact operators
$K_{n,i}$ such that $\|T_{i}-K_{n,i}\|\to 0$ for $i=1,2$. Then, since the
theorem is true for any operator on a complex Hilbert space (cf. [8, p. 367]),
it must be the case that the idempotent components $T_{1}$ and $T_{2}$ are
both compact. But then $T$ is compact by Lemma 1. ∎

Since every finite-rank bicomplex operator in $\mathcal{B}(\mathcal{H})$ (of
the form

$T(f)=\sum\limits_{i=1}^{m}\sigma_{i}\langle f,g_{i}\rangle g_{i}$

for hyperbolic nonnegative bicomplex $\sigma_{i}$ and $g_{i}$ chosen as
orthogonal in $\mathcal{H}$) is compact (as it splits into compact operators
on $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$), Theorem 6 says any bicomplex
operator norm limit of a bicomplex finite-rank operator is compact. But just
as for complex compact operators on complex Hilbert spaces, the converse is
also true, as the next theorem points out. See [5, p. 11] for a parallel
statement of the next theorem.

######  Theorem 7.

Every compact operator $K$ on a bicomplex Hilbert space $\mathcal{H}$ is an
operator norm limit of finite-rank bicomplex operators. In fact, $K$ takes the
form

$K(f)=\sum\limits_{i=1}^{\infty}\sigma_{i}\langle f,g_{i}\rangle g_{i}$

for hyperbolic nonnegative bicomplex $\sigma_{i}$, and with $g_{i}$ chosen as
orthogonal in $\mathcal{H}$.

###### Proof.

Write
$K=K_{1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+K_{2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
in its idempotent complex Hilbert space compact operator components, and
express both $K_{1}$ and $K_{2}$ as operator norm limits (in each idempotent
complex Hilbert space) of finite-rank operators:

$K_{j}(f)=\sum\limits_{i=1}^{\infty}\sigma_{i,j}\langle f,g_{i,j}\rangle
g_{i,j}$, for $j=1,2$.

Note $\sigma_{i,j}$ may be chosen as nonnegative and each set of vectors
$\\{g_{i,j}\\}_{i=1}^{\infty}$ may be chosen as an orthogonal family of
vectors for $j=1,2$. Recombining the idempotent structures gives

$K(f)=\sum\limits_{i=1}^{\infty}\sigma_{i}\langle f,g_{i}\rangle g_{i}$,

where
$\sigma_{i}=\sigma_{i,1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+\sigma_{i,2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}$
are nonnegative hyperbolic numbers and the set of vectors
$\\{g_{i}=g_{i,1}\mathbf{\boldsymbol{\mathrm{e}}_{1}}+g_{i,2}\mathbf{\boldsymbol{\mathrm{e}}_{2}}\\}_{i=1}^{\infty}$
forms an orthogonal family in the bicomplex inner product structure. ∎

##  6 Conclusion

This paper’s results continue analysis found already in many other articles
that show how natural it is to use the idempotent structure on both the vector
elements of a Hilbert or Banach space as well as the idempotent structure of
operators. The idempotent approach, as opposed to misguided efforts that
continue to examine the Euclidean rectangular coordinates, easily extends
results in linear algebra and operator theory for complex Hilbert or Banach
spaces to similar results on bicomplex spaces. This paper’s theory decidedly
confirms that a full study of bicomplex operators is possible as a complete
generalization of operator theory on complex operators when using the
idempotent structure. The authors urge continued investigations into bicomplex
spaces and bicomplex function theory, using the idempotent structures and
hyperbolic norms to extend known complex-valued results.

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