Comparison of Spectral Theory

Introduction

This article compares the spectral theory of the eight algebras $\mathbb{R}, \mathbb{C}, \mathbb{D}, \mathbb{D}', \mathbb{H}, \mathbb{H}_{\mathrm{s}}, \mathbb{B}, \mathbb{H}_{\mathbb{D}}$: the spectrum, the resolvent and the eigenvalues; the left and right spectra of an element and their failure to agree; the Cayley–Hamilton identity and its failure; and the norm of an element as a spectral quantity. It states the situation of the eight algebras on each of these subjects in tables with the eight algebras as columns in the fixed order of The Eight Algebras Compared. Every entry restates a result of the spectral theory articles cited in the explanations.

The organising thread is the field of scalars and its centrality. In the four commutative algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{D}$, $\mathbb{D}'$ the spectrum of a single element reduces to the element itself and there is no operator theory to compare: the multiplicities of the scalar field make the eigenvector problem trivial, and the four columns of the spectral table are empty from the operator rows on. The spectral theory proper begins at $\mathbb{H}$, where the scalars are non-central and the left and right spectra of the left-multiplication operator cease to agree — the right spectrum of a non-real quaternion is a whole two-sphere, not a point. The last three columns carry the same dichotomy under different forms of the norm: the split quaternions by the sign of an indefinite vector norm, the biquaternions by the two roots of the reduced characteristic polynomial, and the split biquaternions by the pair of quaternion spectra of their two idempotent halves.

The Spectral Notions

The following table compares the spectral notions of the eight algebras: the scalar system, the spectrum of a single element, the left and right spectra, the S-spectrum, the characteristic polynomial, the Cayley–Hamilton identity, the eigenspaces, the resolvent, the spectral radius and the eigenvalue dichotomy. The eight algebras are the columns, in the fixed order, with the marker — where the notion is not carried.

datum	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{B}$	$\mathbb{H}_{\mathbb{D}}$
scalar system	field, central	field, central	ring, central, not a field	ring, central, nilpotent	division ring, non-central	ring, non-central	algebra, centre $\mathbb{C}$	algebra, centre $\mathbb{D}$
spectrum of a single element	$\{a\}$	$\{A\}$	$\{A\}$	$\{Z\}$	$\{\tilde q\}$	$\{a\pm\sqrt{-N(\mathbf v)}\}$	$\{Q_0\pm iB\}$	$\{q_0^\pm\pm i\lvert\mathbf Q_\pm\rvert\}$
left spectrum	—	—	—	—	$\{\tilde q\}$	the real spectrum	agrees with $\sigma(\tilde Q)$ as a set	$\{\tilde Q_+,\tilde Q_-\}$
right spectrum	—	—	—	—	the class $[\tilde q]$, point or two-sphere	a union of classes	agrees, multiplicities doubled	$[\tilde Q_+]\cup[\tilde Q_-]$
S-spectrum	—	—	—	—	equals the right spectrum	—	—	equals the right spectrum
characteristic polynomial	—	—	—	—	$\lambda^2-2q_0\lambda+N(\tilde q)$	$\lambda^2-2q_0\lambda+N(\tilde q)$	$\lambda^2-2Q_0\lambda+D(\tilde Q)$	$\prod_\pm(\lambda^2-2q_0^\pm\lambda+\lvert\tilde Q_\pm\rvert^2)^2$
Cayley–Hamilton	—	—	—	—	yes	yes	yes, from the reduced trace and norm	yes, componentwise
eigenspaces	—	—	—	—	right: $1$-dimensional over $\mathbb{H}$; left: all of $\mathbb{H}$	lines, or two-dimensional for null	two-dimensional over $\mathbb{C}$	$1$-dimensional over $\mathbb{H}$ generically
resolvent	—	—	—	—	closed form	closed form, poles on the spectrum	closed form in $\mathbb{C}[\tilde Q]$	real-analytic off the two classes
spectral radius	—	—	—	—	$\lvert\tilde q\rvert$	$\lvert q_0\rvert+\sqrt{\lvert N(\mathbf v)\rvert}$	$\max_i\lvert\lambda_i\rvert$	$\max(\lvert\tilde Q_+\rvert,\lvert\tilde Q_-\rvert)$
eigenvalue dichotomy	—	—	—	—	real point or two-sphere	spacelike, timelike or null	$B=0$ or $B\neq0$	decided in each half

The table records where the spectral theory begins. In the four commutative columns the spectrum of a single element is the element itself and the operator rows are empty: for a central scalar there is no distinction between the left and right eigenvector problems, no non-trivial similarity class, and no resolvent with interesting poles, so the subject of the article is trivial there and the empty cells are stated rather than omitted. At $\mathbb{H}$ the theory becomes non-trivial: the algebra spectrum of a single quaternion is still the point $\{\tilde q\}$, but the right spectrum of the left-multiplication operator is the similarity class $[\tilde q]$, a point for real $\tilde q$ and a two-sphere of fixed modulus for non-real $\tilde q$, while the left spectrum is not invariant under similarity and need not be a finite union of classes; the two spectra are in general incomparable and agree when the representative is complex (Quaternion Spectral Theory). The Cayley–Hamilton identity $\tilde q^2-T(\tilde q)\tilde q+D(\tilde q)=0$ holds with the reduced trace $T(\tilde q)=2q_0$ and the reduced norm $D(\tilde q)=N(\tilde q)$, both similarity-invariant, and the eigenspace attached to a right class is a right $\mathbb{H}$-subspace of dimension one over $\mathbb{H}$, whereas the eigenspace of a fixed non-real representative is only a complex line. In $\mathbb{H}_{\mathrm{s}}$ the spectrum of $\tilde q=q_0+\mathbf v$ is the zero set of $p_{\tilde q}(\lambda)=\lambda^2-2q_0\lambda+N(\tilde q)$, namely $\{a\pm\sqrt{-N(\mathbf v)}\}$, and the eigenvalue dichotomy is decided by the sign of the indefinite norm $N(\mathbf v)$: a distinct real pair for spacelike $\mathbf v$, a complex-conjugate pair for timelike $\mathbf v$, and a real double root for null $\mathbf v$, so the spectrum over $\mathbb{R}$ may be empty; left and right eigenvalues with central scalars coincide with the real spectrum, while the right spectrum in the full algebra is a union of conjugacy classes, and every element is normal, so there is no definite spectral theorem and the Peirce decomposition replaces it (Split-Quaternion Spectral Theory). In $\mathbb{B}$ the spectrum of a biquaternion is the root pair $\{Q_0+iB,Q_0-iB\}$ with $B=\sqrt{Q_1^2+Q_2^2+Q_3^2}$ of the reduced characteristic polynomial $\lambda^2-2Q_0\lambda+D(\tilde Q)$; the set contains $0$ exactly when $\tilde Q$ is a zero divisor, the left and right spectra agree with it as a set with the multiplicities of the left spectrum doubled, and the spectral theorem takes the biquaternion form $\tilde Q=\lambda_1\tilde\Pi_1+\lambda_2\tilde\Pi_2$ with Hermitian idempotents (Biquaternion Spectral Theory). In $\mathbb{H}_{\mathbb{D}}$ the intrinsic spectrum with scalars in the centre $\mathbb{D}$ is inadequate, because $\mathbb{D}$ is not a field, and it is empty unless a component is real; in the idempotent decomposition, where left multiplication is block diagonal, the characteristic polynomial factors as $\prod_\pm(\lambda^2-2q_0^\pm\lambda+\lvert\tilde Q_\pm\rvert^2)^2$, the complex spectrum is the four values $\{q_0^+\pm i\lvert\mathbf Q_+\rvert\}\cup\{q_0^-\pm i\lvert\mathbf Q_-\rvert\}$ with doubled multiplicities, and with quaternion scalars the left spectrum is $\{\tilde Q_+,\tilde Q_-\}$ and the right spectrum, equal to the S-spectrum, is the union of the two conjugacy classes; the reduced trace $T=(2q_0^+,2q_0^-)$ and reduced norm $D=(\lvert\tilde Q_+\rvert^2,\lvert\tilde Q_-\rvert^2)$ are $\mathbb{D}$-valued and give the componentwise Cayley–Hamilton identity $\tilde Q^2-T\tilde Q+De_0=0$ (Split-Biquaternion Spectral Theory). The whole $\mathbb{H}_{\mathbb{D}}$ theory is the pair of quaternionic spectral theories of the two halves, and the $\mathbb{D}$-valued trace and norm replace the complex-valued ones of $\mathbb{B}$. The S-spectrum row is carried by the two columns whose scalars are quaternionic, $\mathbb{H}$ and $\mathbb{H}_{\mathbb{D}}$, where it coincides with the right spectrum; in $\mathbb{B}$ the scalars are central and the ordinary complex spectrum plays the role the S-spectrum plays for quaternionic matrices, and in $\mathbb{H}_{\mathrm{s}}$ the spectrum is computed from the algebraic resolvent $(\tilde q-\lambda)^{-1}$ with its poles on the spectrum, so neither of these two columns carries an S-spectrum (Quaternion Spectral Theory; Biquaternion Spectral Theory).

The Norm and the Spectral Theorem

The following table compares the norm of an element as a spectral quantity, the spectral theorem and the behaviour of the Hermitian and unitary elements. The eight algebras are the columns, in the fixed order, with the marker — where the notion is not carried.

datum	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{B}$	$\mathbb{H}_{\mathbb{D}}$
norm as a spectral quantity	$\lvert a\rvert$	$\lvert A\rvert$	$\sqrt{\lvert N(A)\rvert}$	$\lvert a\rvert$	$\lvert\tilde q\rvert$, the spectral radius	$\lvert q_0\rvert+\sqrt{\lvert N(\mathbf v)\rvert}$	$\lvert N(\tilde Q)\rvert=\prod_i\lvert\lambda_i\rvert$	$\max(\lvert\tilde Q_+\rvert,\lvert\tilde Q_-\rvert)$
equals the spectral radius	yes	yes	yes	yes	yes for a single quaternion	yes	for normal elements	yes, the operator norm
spectral theorem	—	—	—	—	quaternionic, for normal matrices	— , Peirce replaces it	$\tilde Q=\lambda_1\tilde\Pi_1+\lambda_2\tilde\Pi_2$, Hermitian idempotents	pair of quaternion theorems
Hermitian, unitary	—	—	—	—	real S-spectrum, spectrum on $S^3$	symmetric elements are the scalars	Hermitian real, unitary on the circle	decided in each half

The table records the norm as a spectral quantity. For a single element the norm is the spectral radius in every one of the eight columns, and it is the modulus of the element in the definite cases, the split modulus or the count of the two pieces in the indefinite ones; at $\mathbb{H}$ the spectral radius of a single quaternion is exactly its modulus, and the Gelfand-type limit formula holds for matrices, the spectral radius agreeing with the operator norm for normal matrices (Quaternion Spectral Theory). At $\mathbb{H}_{\mathrm{s}}$ the spectral radius is $\lvert q_0\rvert+\sqrt{\lvert N(\mathbf v)\rvert}$ for spacelike vectors, $\sqrt{a^2+N(\mathbf v)}$ for the timelike case and $\lvert q_0\rvert$ for the null case, and null elements are square-zero by Cayley–Hamilton (Split-Quaternion Spectral Theory). At $\mathbb{B}$ the modulus of the norm is the product of the moduli of the eigenvalues, and the spectral theorem takes the biquaternion form with Hermitian idempotents $\tilde\Pi_1,\tilde\Pi_2$ summing to $e_0$; a Hermitian element has real spectrum and a unitary element has spectrum on the unit circle (Biquaternion Spectral Theory). At $\mathbb{H}_{\mathbb{D}}$ the spectral radius is $\max(\lvert\tilde Q_+\rvert,\lvert\tilde Q_-\rvert)$, equal to the operator norm of left multiplication, and it vanishes only at $\tilde Q=0$, since the algebra has no non-zero nilpotents; similarity is conjugation in each half, the class is the ordered pair of conjugacy classes with no Jordan pathology, and the exponential acts componentwise (Split-Biquaternion Spectral Theory). The spectral theorem row is empty for $\mathbb{H}_{\mathrm{s}}$, where every element is normal and the Peirce decomposition, not a diagonalisation, is the structural result, and it is empty for the four commutative columns, where the eigenvector problem is trivial.

Summary

The spectrum of a single element of a commutative algebra is the element itself, so the four columns $\mathbb{R}$, $\mathbb{C}$, $\mathbb{D}$, $\mathbb{D}'$ carry no operator theory and their cells are empty from the left-spectrum row on; the spectral theory proper begins at $\mathbb{H}$, where the scalars are non-central and the left and right spectra of left multiplication differ — the right spectrum of a non-real quaternion is a two-sphere, the left spectrum a point, and for matrices the right spectrum is a finite union of at most $n$ similarity classes while the left spectrum is not similarity-invariant. Cayley–Hamilton holds in the four higher columns, with the reduced trace and norm, and governs the eigenvalue dichotomy: a real point or a two-sphere in $\mathbb{H}$, a spacelike, timelike or null alternative in $\mathbb{H}_{\mathrm{s}}$, the vanishing or non-vanishing of $B$ in $\mathbb{B}$, and the pair of the two halves in $\mathbb{H}_{\mathbb{D}}$. The norm of an element is the spectral radius in every column: the modulus in the definite cases, the split modulus or the maximum of the two half-moduli in the indefinite ones, and the modulus of the norm in $\mathbb{B}$. The spectral theorem exists only for $\mathbb{H}$ and $\mathbb{B}$ and, as a pair, for $\mathbb{H}_{\mathbb{D}}$; in $\mathbb{H}_{\mathrm{s}}$ every element is normal and the Peirce decomposition replaces it.

Summary of Notation