Idempotent vector spaces and their linear transformations

William Johnston; Rebecca G. Wahl

doi:10.1515/conop-2024-0003

Article Open Access

Idempotent vector spaces and their linear transformations

William Johnston and Rebecca G. Wahl

Published/Copyright: August 19, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Concrete Operators Volume 11 Issue 1

Abstract

This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n × n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.

Keywords: bicomplex; linear algebra; compact operators

MSC 2010: 15A20; 15A66; 47A15; 47B07

1 Preliminaries

Bicomplex numbers^[1] are of the form ζ = z 1 + j z 2 , where z 1 = x 1 + i y 1 and z 2 = x 2 + i y 2 are complex numbers, i 2 = − 1 , and j 2 = − 1 . The set of bicomplex numbers B C is a four real-dimensional extension of the complex numbers C in the sense that B C ∣ z 2 = 0 = C . 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 [1,4,8,12–14,18,19] for such results. The theory is also useful in applied settings (see [2,15–17]). Here, ζ = x 1 + i y 1 + j x 2 + i j y 2 , where i j = j i so that ( i j ) 2 = ( − 1 ) 2 = 1 . Given ζ and ω = w 1 + j w 2 in B C ,

ζ + ω = ( z 1 + w 1 ) + j ( z 2 + w 2 ) and ζ ⋅ ω = ( 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 ζ , ω ∈ C . A so-called idempotent representation

ζ = ( z 1 − i z 2 ) e 1 + ( z 1 + i z 2 ) e 2 ≡ ζ 1 e 1 + ζ 2 e 2 ,

importantly gives a better way to think about bicomplex numbers and associated mathematical structures such as analytic functions [8] and, in particular, linear transformations, as this article shows. Here, e 1 = ( 1 + i j ) ∕ 2 and e 2 = ( 1 − i j ) ∕ 2 , and ζ 1 , ζ 2 ∈ C are the idempotent components. The equalities e 1 2 = e 1 , e 2 2 = e 2 , and e 1 e 2 = e 2 e 1 = 0 result in the following property: For ζ = ζ 1 e 1 + ζ 2 e 2 and ω = ω 1 e 1 + ω 2 e 2 , the product

ζ ω = ζ 1 ω 1 e 1 + ζ 2 ω 2 e 2

computes idempotent componentwise. Hence, ζ n = ζ 1 n e 1 + ζ 2 n e 2 , and any (multibranched) n th root translates into complex idempotent components (say for n ∈ N ) as follows:

ζ 1 ∕ n = ζ 1 1 ∕ n e 1 + ζ 2 1 ∕ n e 2 .

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

A 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 H + , which the following definition describes. We call it the idempotent norm; it is a poset-valued norm (cf. [1, pp. 5, 11]).

Definition 1

(Hyperbolic numbers) The hyperbolic numbers

H = { x + i j y : x , y ∈ R }

are a strict subset of B C . The subset of nonnegative hyperbolic numbers is

H + = { η 1 e 1 + η 2 e 2 : η 1 , η 2 ≥ 0 } .

For any ζ = ζ 1 e 1 + ζ 2 e 2 ∈ B C , the idempotent norm is defined as follows:

∣ ζ ∣ H = ∣ ζ 1 ∣ e 1 + ∣ ζ 2 ∣ e 2 ,

which is an element of H + and satisfies ∣ ζ ⋅ ω ∣ H = ∣ ζ ∣ H ⋅ ∣ ω ∣ H for any ζ , ω ∈ B C .

A partial ordering on H + now follows. For two elements ζ = η 1 e 1 + η 2 e 2 and ω = ψ 1 e 1 + ψ 2 e 2 in H + , define the idempotent norm inequality ζ < H ω when both η 1 < ψ 1 and η 2 < ψ 2 . A less-than-or-equal-to partial ordering is similarly defined. The idempotent norm of any bicomplex number is a nonnegative hyperbolic number. That norm can therefore be used via this idempotent norm inequality to define open balls of bicomplex numbers that have nonnegative hyperbolic radii. The balls are of the form

B H ( c , R ) = { ζ : ∣ ζ − c ∣ H < H R } ,

where the center c ∈ B C and the hyperbolic radius R ∈ H + . The collection of balls B H ( c , R ) can also be thought of as the neighborhood basis open sets in a topology induced by the idempotent norm.

A bicomplex number’s Euclidean form η = z 1 + j z 2 produces a choice of three natural conjugations (cf. [11, p. 562] or [10, p. 593]):

ζ ¯ = z ¯ 1 + j z ¯ 2 ;
ζ ¯ = z 1 − j z 2 ; or
ζ ¯ = z ¯ 1 − j z ¯ 2

Both the first and third choices generalize conjugation for complex numbers but the third is most natural because it corresponds with the only natural conjugation in terms of the idempotent component; namely, for ζ = ζ 1 e 1 + ζ 2 e 2 ,

ζ ¯ = z ¯ 1 − j z ¯ 2 = ζ ¯ 1 e 1 + ζ ¯ 2 e 2 .

This calculation follows from the formulas for ζ 1 and ζ 2 in terms of z 1 and z 2 , along with the simple facts that e 1 ¯ = e 1 and e 2 ¯ = e 2 . Furthermore,

ζ ⋅ ζ ¯ = ( ζ 1 e 1 + ζ 2 e 2 ) ( ζ ¯ 1 e 1 + ζ ¯ 2 e 2 ) = ∣ ζ 1 ∣ 2 e 1 + ∣ ζ 2 ∣ 2 e 2 = ∣ ζ ∣ H 2 ,

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

Definition 2

The bicomplex conjugate of ζ = ζ 1 e 1 + ζ 2 e 2 is ζ ¯ = ζ ¯ 1 e 1 + ζ ¯ 2 e 2 .

Just as bicomplex numbers evolve as four real-dimensional expansions of the two-dimensional complex numbers, it is well known that 2 n -dimensional multicomplex numbers evolve from recursive expansions of bicomplex numbers for n = 3 , 4 , 5 , … (cf. [14, ch. 5]). For example, the set of eight-dimensional multicomplex numbers is { ζ = ζ 1 + k ζ 2 : ζ 1 , ζ 2 ∈ B C and k 2 = − 1 } , and any such element has an idempotent representation

(1) ζ = η 1 ε 1 + η 2 ε 2 : η 1 , η 2 ∈ B C ,

where in this case ε 1 = 1 + j k 2 and ε 2 = 1 − j k 2 . Higher dimensions evolve similarly.

2 Idempotent vector spaces

The following definition newly defines an idempotent vector space, generalizing the standard definition of a vector space on complex scalars. The presentation describes the algebraic structure needed for the generalization, and indicates how the bicomplex and multicomplex settings each give a special case.

Definition 3

Part 1: A rank 0 idempotent vector space is a vector space taken over a field of scalars S 0 . A rank 1 idempotent vector space has idempotent vectors of the form v 1 → ε 1 + v 2 → ε 2 , where v 1 → and v 2 → are vectors from idempotent vector spaces with the same rank 0, with scalars S 1 of the form ζ 1 ε 1 + ζ 2 ε 2 for ζ 1 , ζ 2 ∈ S 0 , and where ε 1 and ε 2 are called elementary idempotents that are required simply to satisfy algebraically ε 1 2 = ε 1 , ε 2 2 = ε 2 , and ε 1 ε 2 = 0 . In a recursive manner, for any n ∈ N , a rank n idempotent vector space has idempotent vectors of the form v 1 → ε 1 + v 2 → ε 2 , where v 1 → and v 2 → are vectors from idempotent vector spaces that have the same rank n − 1 , with scalars S n of the form ζ 1 ε 1 + ζ 2 ε 2 for ζ 1 , ζ 2 ∈ S n − 1 , and where ε 1 and ε 2 are elementary idempotents that satisfy ε 1 2 = ε 1 , ε 2 2 = ε 2 , and ε 1 ε 2 = 0 .

Part 2: A bicomplex vector space V is any rank 1 idempotent vector space that evolves from a rank 0 complex vector space with complex scalars S 0 and elementary idempotents ε 1 = e 1 and ε 2 = e 2 . In other words, a bicomplex vector space is taken over bicomplex scalars and is any set of well-defined elements { v → } , called “bicomplex vectors,” that may be decomposed, using the bicomplex idempotent structure, into complex idempotent components

v → = v 1 → e 1 + v 2 → e 2 ,

where the resulting corresponding sets of idempotent “vectors” { v 1 → } and { v 2 → } are taken over complex scalars, and each forms a vector space with the same dimension. The bicomplex vector space V = { v → } , considered over bicomplex scalars, has finite dimension n when both { v 1 → } and { v 2 → } have dimension n . In some applications, such as the study of general relativity, it may be natural to consider V over complex scalars, and then its dimension is 2 n , as a basis for V is formed from all 2 n combinations of the bases for the two idempotent components { b 1 , k } k = 1 n and { b 2 , j } j = 1 n , respectively, as follows:

{ b k , j } = { b 1 , k e 1 , b 2 , j e 2 } .

In the same way, a multicomplex vector space V is any rank n > 1 idempotent vector space that evolves from a rank n − 1 bi- or multicomplex vector space with 2 k − 1 -dimensional ( k > 2 ) bi- or multicomplex scalars S n − 1 and elementary idempotents ε 1 and ε 2 that formulate the idempotent expansion of the multicomplex scalars as ζ = η 1 ε 1 + η 2 ε 2 : η 1 , η 2 ∈ S n − 1 .

In any situation, when the idempotent vector spaces are separable and infinite dimensional, then V is separable and infinite dimensional.

Examples. 1. A simple illustration is the four-dimensional vector space of length 2 column vectors with bicomplex entries taken over bicomplex scalars. The bicomplex vector space V of column vectors

v 11 e 1 + v 12 e 2 v 21 e 1 + v 22 e 2 , with v i j ∈ C ,

then has dimension 2 with (elementary) basis

1 0 , 0 1 .

When considered over complex scalars, an elementary basis for V is

e 1 0 , e 2 0 , 0 e 1 , 0 e 2 .

2. It is also important to illustrate a multicomplex vector space as Definition 3 Part 2 also newly describes. Define the multicomplex vector space V of linear functions taken over the eight-dimensional multicomplex scalars defined in equation (1), which appears just after Definition 2. Then V is the collection of vectors

v 11 1 + j k 2 + v 12 1 − j k 2 η + v 21 1 + j k 2 + v 22 1 − j k 2 ,

with v i j ∈ B C and η an eight-dimensional multicomplex variable. The vector field set of such polynomials has dimension two over these multicomplex scalars, with basis set { 1 , η } , and has vector field dimension eight when considered over the complex scalars, with an elementary basis for V as follows:

1 + i j 2 1 + j k 2 , 1 − i j 2 1 + j k 2 , 1 + i j 2 1 − j k 2 , 1 − i j 2 1 − j k 2 , 1 + i j 2 1 + j k 2 η , 1 − i j 2 1 + j k 2 η , 1 + i j 2 1 − j k 2 η , 1 − i j 2 1 − j k 2 η .

This article describes theory for bicomplex 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 bicomplex 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. Every concept presented here for bicomplex vector spaces and linear transformations defined on them has a straightforward and obvious generalization to any general rank n idempotent vector space, such as any multicomplex vector space (as in Example 2), along with the linear transformations defined on them, but our article limits itself, both for brevity and for simplicity of presentation, to the bicomplex vector space case.

3 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 × 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 e 1 + A 2 e 2 . Multiplication of bicomplex matrices is defined via row times column products in the same way as for complex matrices. A delightful well-known fact is that matrix multiplication computes idempotent componentwise, just as multiplication computes componentwise for scalars. See [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 e 1 + A 2 e 2 of size m × k and B = B 1 e 1 + B 2 e 2 of size k × n , then A ⋅ B = A 1 B 1 e 1 + A 2 B 2 e 2 is m × n .

Proof

Since e 1 2 = e 1 , e 2 2 = e 2 , and e 1 e 2 = e 2 e 1 = 0 ,

A ⋅ B = ( A 1 e 1 + A 2 e 2 ) ( B 1 e 1 + B 2 e 2 ) = A 1 B 1 e 1 2 + A 1 B 2 e 1 e 2 + A 2 B 1 e 2 e 1 + A 2 B 2 e 2 2 = A 1 B 1 e 1 + A 2 B 2 e 2 .

For an equivalent known proof, denote A B = [ ( a b ) i j ] , with A = [ a i k ] and B = [ b k j ] having bicomplex entries with idempotent representation a i k = a ˆ i k e 1 + a ˜ i k e 2 and b k j = b ˆ k j e 1 + b ˜ k j e 2 . Each term is

□ ( a b ) i j = ∑ k = 1 m a i k b k j = ∑ k = 1 m ( a ˆ i k e 1 + a ˜ i k e 2 ) ( b ˆ k j e 1 + b ˜ k j e 2 ) = ∑ k = 1 m ( a ˆ i k b ˆ k j e 1 + a ˜ i k b ˜ k j e 2 ) = ∑ k = 1 m ( a ˆ i k b ˆ k j ) e 1 + ∑ k = 1 m ( a ˜ i k b ˜ k j ) e 2 .

The following three corollaries for m × k matrices immediately result. For the third corollary, we define a bicomplex number λ as an eigenvalue for a bicomplex matrix A when there exists a column vector v → = v → 1 e 1 + v → 2 e 2 (the corresponding eigenvector) with both idempotent components v → 1 and v → 2 nonzero so that A v → = λ v → .^[2]

Corollary 1

For a bicomplex matrix A = A 1 e 1 + A 2 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 e 1 + A 2 − 1 e 2 .

Corollary 2

Define the determinant of a bicomplex matrix A in terms of its idempotent component determinants; i.e., det A ≡ det A 1 e 1 + det A 2 e 2 . Then A − 1 exists exactly when det A is bicomplex invertible; i.e., when both det A 1 and det A 2 are nonzero.

Corollary 3

A bicomplex number λ is an eigenvalue for a bicomplex matrix A exactly when det ( A − λ I ) = 0 .

The section’s first main result, for n × n bicomplex matrices, now follows and is new.

Theorem 2

Bicomplex Jordan form: Write A = A 1 e 1 + A 2 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 = P J P − 1 , where P = P 1 e 1 + P 2 e 2 and J = J 1 e 1 + J 2 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 e 1 + A 2 e 2 = P 1 J 1 P 1 − 1 e 1 + P 2 J 2 P 2 − 1 e 2 = ( P 1 e 1 + P 2 e 2 ) ( J 1 P 1 − 1 e 1 + J 2 P 2 − 1 e 2 ) = ( P 1 e 1 + P 2 e 2 ) ( J 1 e 1 + J 2 e 2 ) ( P 1 − 1 e 1 + P 2 − 1 e 2 ) = P J P − 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, e 1 ’s, and e 2 ’s. Also, A 1 and A 2 are diagonalizable if and only if J is diagonal. This article will soon show an immediate application of the Bicomplex Jordan form: it helps determine the invariant subspaces for any bicomplex matrix. To clarify, when B C n is the collection of n -dimensional column vectors with bicomplex entries, a subspace S ⊆ B C n is invariant under an n × n matrix A when A ζ → ∈ S for every ζ → ∈ S . The following fact, the second main result of the section, results as a direct consequence of the fundamental theorem for bicomplex matrices.

Corollary 4

Writing any vector ζ → ∈ B C n as ζ → = ζ → 1 e 1 + ζ → 2 e 2 , where ζ → 1 , ζ → 2 ∈ C n , decompose any given subspace S ⊆ B C n as S = S 1 e 1 ⊕ S 2 e 2 , where S 1 and S 2 are subspaces of C n . Then S is invariant under any given n × n bicomplex matrix A = A 1 e 1 + A 2 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 ζ → = A 1 ζ → 1 e 1 + A 2 ζ → 2 e 2 ∈ S 1 e 1 ⊕ S 2 e 2 for every ζ → = ζ → 1 e 1 + ζ → 2 e 2 ∈ S .

But that fact is equivalent to both

A 1 ζ → 1 ∈ S 1 for every ζ → 1 ∈ S 1 , and A 2 ζ → 2 ∈ S 2 for every ζ → 2 ∈ S 2 ,

which means exactly that S 1 is invariant under A 1 and S 2 is invariant under A 2 .□

Corollary 4 is useful since a constructive method to determine the invariant subspace lattice diagram for any complex n × n matrix is known. See [6]. Using Corollary 4, this constructive method extends to find the invariant subspace lattice diagram for any bicomplex n × n matrix A : it forms each possible invariant subspace as S 1 e 1 ⊕ S 2 e 2 , where S 1 is invariant for A 1 and S 2 is invariant for A 2 . Each of these invariant subspaces for A “locks into place naturally” into A ’s corresponding invariant subspace 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 i j ] is its conjugate-transpose A * = [ a ¯ j i ] .

Note this definition generalizes the adjoint of a complex matrix. Furthermore, if ζ → , ω → ∈ B C n , we define as in, for example, [11], a bicomplex inner product as follows:

⟨ ζ → , ω → ⟩ = ∑ i = 1 n ζ i ω ¯ i = ∑ i = 1 n ( ζ i 1 ω ¯ i 1 e 1 + ζ i 2 ω ¯ i 2 e 2 ) = ⟨ ζ → 1 , ω → 1 ⟩ e 1 + ⟨ ζ → 2 , ω → 2 ⟩ e 2 .

The following theorem results.

Theorem 3

For any n × n bicomplex matrix A and ζ → , ω → ∈ B C n ,

⟨ A ζ → , ω → ⟩ = ⟨ ζ → , A * ω → ⟩ .

Proof

By Theorem 1, the definition of the bicomplex inner product, and the adjoint properties of complex matrices,

□ ⟨ A ζ → , ω → ⟩ = ⟨ A 1 ζ → 1 e 1 + A 2 ζ → 2 e 2 , ω → 1 e 1 + ω → 2 e 2 ⟩ = ⟨ A 1 ζ → 1 , ω → 1 ⟩ e 1 + ⟨ A 2 ζ → 2 , ω → 2 ⟩ e 2 = ⟨ ζ → 1 , A 1 * ω → 1 ⟩ e 1 + ⟨ ζ → 2 , A 2 * ω → 2 ⟩ e 2 = ⟨ ζ → , A * ω → ⟩ .

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 = P D P − 1

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

Proof

When A = A 1 e 1 + A 2 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 e 1 + A 2 e 2 = P 1 D 1 P 1 − 1 e 1 + P 2 D 2 P 2 − 1 e 2 = ( P 1 e 1 + P 2 e 2 ) ( D 1 P 1 − 1 e 1 + D 2 P 2 − 1 e 2 ) = ( P 1 e 1 + P 2 e 2 ) ( D 1 e 1 + D 2 e 2 ) ( P 1 − 1 e 1 + P 2 − 1 e 2 ) = P D P − 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 * e 1 + P 2 * e 2 = P 1 − 1 e 1 + P 2 − 1 e 2 = P − 1 .

4 Examples

Example 1

Let A = 0 ζ ζ ¯ 0 = 0 ζ 1 ζ 1 ¯ 0 e 1 + 0 ζ 2 ζ 2 ¯ 0 e 2 with ζ ≠ 0 . Then det ( A − λ I ) = 0 implies λ 1 2 − ∣ ζ 1 ∣ 2 = 0 , or λ 1 = ± ∣ ζ 1 ∣ , and similarly λ 2 = ± ∣ ζ 2 ∣ . Therefore, A can be analyzed in terms of a first pair of bicomplex eigenvalues λ = ∣ ζ 1 ∣ e 1 + ∣ ζ 2 ∣ e 2 , − ∣ ζ 1 ∣ e 1 − ∣ ζ 2 ∣ e 2 , more concisely written as λ = ∣ ζ ∣ H , − ∣ ζ ∣ H . Corresponding eigenvectors (nonunique, and which come as the nonzero solutions to A ζ → = λ ζ → ) are

∣ ζ ∣ H ζ ¯ and − ∣ ζ ∣ H ζ ¯ , respectively .

The spectral theorem for self-adjoint matrices then implies A satisfies A = P D P − 1 , where

P = ∣ ζ ∣ H − ∣ ζ ∣ H ζ ¯ ζ ¯ and D = ∣ ζ ∣ H 0 0 − ∣ ζ ∣ H .

An important note is that this is the standard complex Jordan form for A when ζ ∈ 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 − λ I ) = 0 also for the pair of eigenvalues λ = ∣ ζ 1 ∣ e 1 − ∣ ζ 2 ∣ e 2 , − ∣ ζ 1 ∣ e 1 + ∣ ζ 2 ∣ e 2 . The corresponding (nonzero) eigenvectors are as follows:

∣ ζ ∣ 1 e 1 − ∣ ζ 2 ∣ e 2 ζ ¯ and − ∣ ζ ∣ 1 e 1 + ∣ ζ 2 ∣ e 2 ζ ¯ , respectively .

A second diagonalization provided by the spectral theorem for self-adjoint matrices results, even when A is complex. We obtain A = P D P − 1 , where

P = ∣ ζ ∣ 1 e 1 − ∣ ζ 2 ∣ e 2 − ∣ ζ ∣ 1 e 1 + ∣ ζ 2 ∣ e 2 ζ ¯ ζ ¯ and D = ∣ ζ 1 ∣ e 1 − ∣ ζ 2 ∣ e 2 0 0 − ∣ ζ 1 ∣ e 1 + ∣ ζ 2 ∣ e 2 .

In general, an n × n self-adjoint matrix A = A 1 e 1 + A 2 e 2 will have n ! different bicomplex diagonalizations when A 1 and A 2 each 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 = 0 0 0 0 e 1 + 0 1 0 0 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 marked^[3] 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 matrix A = A 1 e 1 + A 2 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 = 0 0 0 0 has two Jordan blocks with eigenvalues 0, and A 2 = 0 1 0 0 has a single Jordan block with eigenvalues 0. Their two invariant subspace lattice diagrams, found using the techniques in [6], are shown in Figure 1.

$Figure 1 Invariant subspace lattices for complex matrices A 1 {A}_{1} (on left) and A 2 {A}_{2} (on right).$

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 { e → 1 } , where 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 { e → 2 } , where e → 2 is the eigenvector for the single Jordan block of A 2 , the symbol [ Z 1 ∣ S ] corresponds to the invariant subspace span { e → 1 } e 1 + span { e → 2 } 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 × n bicomplex matrix A = A 1 e 1 + A 2 e 2 , routinely develops the bicomplex matrix’s complete invariant subspace lattice diagram.

5 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 the so-called bicomplex compact operators on bicomplex Banach spaces. The theorems and proofs in this section are presented here for the first time.

Definition 5

A bicomplex Hilbert space is of the form ℋ = ℋ 1 e 1 + ℋ 2 e 2 , where ℋ 1 and ℋ 2 are complex Hilbert spaces with the same dimension. The bicomplex inner product on elements ζ → , ω → ∈ ℋ is defined via complex inner products on the idempotent components of ζ → = ζ → 1 e 1 + ζ → 2 e 2 and ω → = ω → 1 e 1 + ω → 2 e 2 as follows:

⟨ ζ → , ω → ⟩ = ⟨ ζ → 1 , ω → 1 ⟩ e 1 + ⟨ ζ → 2 , ω → 2 ⟩ e 2 .

A set of bicomplex vectors { ζ → k } is orthogonal in ℋ when ⟨ ζ → n , ζ → m ⟩ = 0 for n ≠ m and orthonormal when in addition the idempotent norm of each vector in the set, defined via idempotent component norms for ℋ 1 and ℋ 2 according to ‖ ζ → n ‖ H = ‖ ζ → n , 1 ‖ e 1 + ‖ ζ → n , 2 ‖ e 2 , equals 1.

Similarly, a bicomplex Banach space is of the form ℬ = ℬ 1 e 1 + ℬ 2 e 2 , where ℬ 1 and ℬ 2 are complex Banach spaces with the same dimension.

Such spaces are often called “ B C -modules” in the literature [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. We also mention, but only in passing as any detailed treatment would require an amount of substance beyond what this article intends, a broader generalization: that multicomplex Hilbert and/or Banach spaces are also naturally defined in a manner exactly parallel to the way Definition 3 Part 2 defines multicomplex vector spaces. In other words, multicomplex Hilbert and/or Banach spaces are collections of elements that can foundationally be decomposed into 2 n Hilbert and/or Banach spaces in the same way a bicomplex Hilbert and/or Banach space is decomposed into two Hilbert and/or Banach spaces. In fact, the vector fields that create such spaces may also be defined over multicomplex scalars as described in Definition 3. In short, such a multicomplex Hilbert and/or Banach space has idempotent decomposition ζ → = ζ → 1 ε 1 + ζ → 2 ε 2 with ζ → 1 and ζ → 2 taken from bi- or multicomplex Hilbert and/or Banach spaces with the same dimension and that can be both decomposed into 2 n − 1 Hilbert and/or Banach spaces.

This and the next section develop theory for bicomplex compact operators, defined here:

Definition 6

An operator K : ℬ → ℬ mapping a bicomplex Banach space ℬ to itself is compact if for every bounded set S ⊆ ℬ , the set K ( S ) is relatively compact in ℬ , which means that every infinite subset of K ( S ) ≡ { K s → : s → ∈ S } has at least one limit point in its (hyperbolic) norm closure. This is equivalent to every bounded sequence { x n } of vectors in ℬ producing the sequence { K x n } with an idempotent norm-convergent subsequence.

Note importantly that idempotent norm convergence in ℬ happens exactly when norm convergence of the idempotent components happen in both complex spaces ℬ 1 and ℬ 2 . In such a way, the actions of any operator in ℬ follows from actions of the corresponding operator elements acting on ℬ 1 and ℬ 2 . Here, T acts on v → = v 1 → e 1 + v 2 → e 2 ∈ ℬ according to T v = T 1 v 1 → e 1 + T 2 v 2 → e 2 . The following lemma notes the impact of those facts on compact operators.

Lemma 1

For any operator T = T 1 e 1 + T 2 e 2 on a bicomplex Banach space ℬ = ℬ 1 e 1 + ℬ 2 e 2 , the operators T 1 and T 2 are compact on ℬ 1 and ℬ 2 , respectively, iff T is compact.

Proof

For i = 1 , 2 , the compactness of T i implies for every bounded set S i ⊆ ℬ i , the set T i ( S i ) is relatively compact in 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 e 1 + S 2 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 e 1 + x 2 e 2 in its (idempotent) norm closure. Hence, T is compact by definition.

Conversely, assume T = T 1 e 1 + T 2 e 2 is compact and take any bounded sequences { x 1 , n } ∈ ℬ 1 and { x 2 , n } ∈ ℬ 2 . Then { x n } = { x 1 , n e 1 + x 2 , n e 2 } is a bounded set of vectors in ℬ (in the idempotent norm sense), producing the sequence { T x n } with a idempotent norm-convergent subsequence { T x n k ≡ T 1 x 1 , n k e 1 + T 2 x 2 , n k e 2 } . But the idempotent 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 ℬ 1 and 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 ‖ H = ‖ T 1 ‖ e 1 + ‖ T 2 ‖ 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 ℋ . For we can choose an infinite sequence x n of ℋ to be the orthonormal set of basis elements for the Hilbert space, which are each distance (in idempotent 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, stated below.

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

Definition 7

For a bounded operator T :

The spectrum of T is σ ( T ) = { ζ ∈ B C : ζ I − T is not invertible } .
The point spectrum of T is the set of eigenvalues σ p ( T ) ; in other words, the set of λ ∈ B C that satisfy
( λ I − T ) v = 0
for some v → = v 1 → e 1 + v 2 → e 2 ∈ ℬ with both v 1 → and v 2 → nonzero, where v is called an eigenvector for λ . Any eigenvalue λ = λ 1 e 1 + λ 2 e 2 produces for T = T 1 e 1 + T 2 e 2 the eigenvalue λ 1 for T 1 and λ 2 for T 2 on the associated spaces ℬ 1 and 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 [9, pp. 377–378].

Theorem 5

For a bicomplex compact operator K : ℬ → ℬ with ℬ a bicomplex Banach space,

Every invertible λ ∈ σ ( K ) is an eigenvalue of K .
The eigenvalues can only accumulate at noninvertible values. If the dimension of ℬ is not finite, then σ ( K ) must contain 0, even if 0 is not an eigenvalue.
σ ( K ) is countable; in other words, its elements can be listed as either a finite list or as σ ( K ) = { λ 1 , λ 2 , … } .

The theorem’s proof follows from the fact that K on ℬ decomposes into K = K 1 e 1 + K 2 e 2 , where K i is compact on 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 η = η 1 e 1 + η 2 e 2 for which η 1 , η 2 ∈ R , defined as inf { η } = inf { η 1 } e 1 + inf { η 2 } e 2 .

Lemma 2

Riesz’s lemma. For a nondense subspace X = X 1 e 1 + X 2 e 2 of a bicomplex Banach space ℬ = ℬ 1 e 1 + ℬ 2 e 2 , given 0 < r < 1 , there is an element y ∈ ℬ with ‖ y ‖ H = 1 and inf x ∈ X ‖ x − y ‖ H = r .

Proof

As mentioned, the lemma holds for each of the associated operators on the complex Banach spaces: X 1 in ℬ 1 and X 2 in ℬ 2 , in the sense that it produces a vector y 1 ∈ ℬ 1 and y 2 ∈ ℬ 2 with ‖ y i ‖ = 1 and inf x ∈ X i ‖ x − y i ‖ = r for i = 1 , 2 . Then the vector y = y 1 e 1 + y 2 e 2 satisfies the lemma, since ‖ y ‖ H = 1 e 1 + 1 e 2 = 1 and inf x ∈ X ‖ x − y ‖ H = r e 1 + r e 2 = r .□

Lemma 3

For a compact operator K = K 1 e 1 + K 2 e 2 on a bicomplex Banach space ℬ = ℬ 1 e 1 + ℬ 2 e 2 and for which I − K is one-to-one, the range of I − K is closed in the idempotent norm.

Proof

The lemma holds for operators on a complex Banach space [7]. Apply that fact to each of K 1 on ℬ 1 e 1 and K 2 on ℬ 2 e 2 , producing, since I − K 1 must be one-to-one on ℬ 1 e 1 and I − K 2 must be one-to-one on ℬ 2 e 2 , closed ranges for I − K 1 and I − K 2 . But then the range of I − K is closed as desired.□

6 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 ℋ = ℋ 1 e 1 + ℋ 2 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 ℋ 1 and ℋ 2 , respectively. The next theorem and its proof are new.

Theorem 6

For any operator T = T 1 e 1 + T 2 e 2 on a bicomplex Hilbert space ℋ = ℋ 1 e 1 + ℋ 2 e 2 , if there is a sequence { K n } n = 1 ∞ of bicomplex compact operators with ‖ T − K n ‖ H → 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 ‖ H = ‖ Y 1 ‖ e 1 + ‖ Y 2 ‖ e 2 , there are two (idempotent component) sequences of complex compact operators K n , i such that ‖ T i − K n , i ‖ → 0 for i = 1 , 2 . Then, since the theorem is true for any operator on a complex Hilbert space (cf. [9, 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.□

A finite-rank bicomplex operator in ℬ ( ℋ ) is defined to be of the form

T ( f ) = ∑ i = 1 m σ i ⟨ f , g i ⟩ g i

for hyperbolic nonnegative bicomplex σ i and g i chosen as orthogonal in ℋ . It is compact, as it splits into compact operators on ℋ 1 and ℋ 2 ; namely, writing f = f ˆ e 1 + f ˜ e 2 , σ i = σ ˆ i e 1 + σ ˜ i e 2 and g i = g ˆ i e 1 + g ˜ i e 2 ,

T ( f ) = ∑ i = 1 m σ ˆ i ⟨ f ˆ , g ˆ i ⟩ g ˆ i e 1 + ∑ i = 1 m σ ˜ i ⟨ f ˜ , g ˜ i ⟩ g ˜ i e 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 ℋ is an operator norm limit of finite-rank bicomplex operators. In fact, K takes the form

K ( f ) = ∑ i = 1 ∞ σ i ⟨ f , g i ⟩ g i

for hyperbolic nonnegative bicomplex σ i , and with g i chosen as orthogonal in ℋ .

Proof

Write K = K 1 e 1 + K 2 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 ) = ∑ i = 1 ∞ σ i , j ⟨ f , g i , j ⟩ g i , j , for j = 1 , 2 .

Note σ i , j may be chosen as nonnegative, and each set of vectors { g i , j } i = 1 ∞ may be chosen as an orthogonal family of vectors for j = 1 , 2 . Recombining the idempotent structures gives

K ( f ) = ∑ i = 1 ∞ σ i ⟨ f , g i ⟩ g i ,

where σ i = σ i , 1 e 1 + σ i , 2 e 2 are nonnegative hyperbolic numbers, and the set of vectors { g i = g i , 1 e 1 + g i , 2 e 2 } i = 1 ∞ forms an orthogonal family in the bicomplex inner product structure.□

7 Conclusion

This article’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 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 and multicomplex spaces and indeed, to any rank n idempotent vector space. This article’s theory decidedly confirms that a full study of bicomplex and multicomplex 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 and multicomplex spaces, as well as corresponding bicomplex and multicomplex function theory, using the idempotent structures and idempotent norms to extend known complex-valued results.

Acknowledgements

The authors sincerely thank the reviewers for their valuable comments that greatly improved the manuscript.

Funding information: They formally state that no funding was involved in the research or authorship of this manuscript.
Author contributions: Johnston and Wahl have contributed equally to this article, accept the responsibility for the content of the manuscript, and consent to its submission in this journal. They have reviewed all the results, and they approve the final version of the manuscript.
Conflict of interest: Authors state no conflict of interest.

References

[1] D. Alpay, M. E. Luna-Elizarrarás, M. Shapiro, and D. C. Struppa, Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis, SpringerBriefs in Mathematics, Cham, Switzerland, 2014. 10.1007/978-3-319-05110-9Search in Google Scholar

[2] H. T. Anastassiu, P. E. Atlamazoglou, and D. I. Kaklamani, Application of bicomplex (quaternion) algebra to fundamental electromagnetics: a lower order alternative to the Helmholtz equation. IEEE Trans. Antennas Propagat. 51 (2003), no. 8, 2130–2136. 10.1109/TAP.2003.810231Search in Google Scholar

[3] R. Bru, L. Rodman, and H. Schneider, Extensions of Jordan bases for invariant subspaces of a matrix, Linear Algebra Appl. 150 (1991), 209–225. 10.1016/0024-3795(91)90170-2Search in Google Scholar

[4] K. S. Charak and R. Kumar, Bicomplex Riesz-Fisher Theorem, 2011, arXiv:1109.3429v1. Search in Google Scholar

[5] K. S. Charak, R. Kumar, and D. Rochon, Infinite dimensional bicomplex spectral decomposition theorem, Adv. Appl. Clifford Algebras 23 (2013), 593–605. 10.1007/s00006-013-0385-5Search in Google Scholar

[6] C. C. Cowen, W. Johnston, and R. Wahl, Constructing invariant subspaces as Kernels of commuting matrices, Linear Algebra Appl. 583 (2019), no. 15, 46–62. 10.1016/j.laa.2019.08.014Search in Google Scholar

[7] P. Garrett, Riesz’ lemma, 2022, http://www-users.math.umn.edu/garrett/m/fun/riesz_lemma.pdf. Search in Google Scholar

[8] W. Johnston and C. Makdad, A comparison of norms: Bicomplex root and ratio test and an extension theorem. Amer. Math. Monthly 128 (2021), no. 6, 525–533. 10.1080/00029890.2021.1898871Search in Google Scholar

[9] W. Johnston, The Calculus of Complex Functions, Vol. 71, AMS/MAA Textbooks, American Mathematical Society, Providence, Rhode Island, 2022. ISBN: 978-1-4704-6565-0. Search in Google Scholar

[10] R. Kumar and K. Singh, Bicomplex linear operators on bicomplex Hilbert spaces and Littlewoodas subordination theorem, Adv. Appl. Clifford Algebras 25 (2015), 591–610, https://doi.org/10.1007/s00006-015-0531-3. Search in Google Scholar

[11] R. G. Lavoie, L. Marchildon, and D. Rochon, Finite-dimensional bicomplex Hilbert spaces, Adv. Appl. Clifford Algebras 21 (2011), 561–581, https://doi.org/10.1007/s00006-010-0274-0. Search in Google Scholar

[12] M. E. Luna-Elizarrarás, M. Shapiro, D. C. Struppa, and A. Vajiac, Bicomplex numbers and their elementary functions. CUBO (2012), 61–80, https://doi.org/10.4067/S0719-06462012000200004. Search in Google Scholar

[13] M. E. Luna-Elizarrarás, M. Shapiro, D. C. Struppa, and A. Vajiac, Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers, Birkhäser, Boston, 2015. 10.1007/978-3-319-24868-4Search in Google Scholar

[14] G. B. Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker, New York, 1991. Search in Google Scholar

[15] D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, Anal. Univ. Oradea. fasc. Math. 11 (2004), 71–110. Search in Google Scholar

[16] D. Rochon and S. Tremblay, Bicomplex quantum mechanics I. The generalized Schrödinger equation, Adv. Appl. Clifford Algebr. 14 (2004), no. 2, 231–248. 10.1007/s00006-004-0015-3Search in Google Scholar

[17] D. Rochon and S. Tremblay, Bicomplex quantum mechanics II. The Hilbert space, Adv. Appl. Clifford Algebr. 16 (2006), no. 2, 135–157. 10.1007/s00006-006-0008-5Search in Google Scholar

[18] S. Rönn, Bicomplex Algebra and Function Theory, 2001, arXiv.math/0101200v1. Search in Google Scholar

[19] D. Rochon, A bicomplex Riemann zeta function. Tokyo J. Math. 27 (2004), no. 2, 357–369. 10.3836/tjm/1244208394Search in Google Scholar

[20] C. Segre, Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici, Math. Ann. 40 (1892), 413–467. 10.1007/BF01443559Search in Google Scholar

Received: 2024-03-05

Revised: 2024-06-29

Accepted: 2024-07-08

Published Online: 2024-08-19

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/conop-2024-0003

Keywords for this article

bicomplex; linear algebra; compact operators

Creative Commons

BY 4.0