The Indefinite Spectra of the Operators on the Biquaternion Algebra

Introduction

The spectral theory of the Hermitian adjoint on the biquaternion algebra gives the spectra of the left and right multiplications and of the sandwich, and reads self-adjointness, positivity and unitarity off the element spectrum (The Spectra of the Operators on the Biquaternion Algebra with Hermitian Adjoint, Biquaternion Spectral Theory). This article does the same for the Krein adjoint $T^{\dagger}=J\,T^{*}J$ of J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra. Two facts govern the answer. First, the $J$-self-adjoint operators of the three families are very few: the left and right multiplications by real scalars, and the sandwiches with a parameter in the centre or in the vector subspace. Second, on those families the spectrum comes out real with signs: for the sandwich with a vector parameter the spectrum is the pair $\pm|N(\tilde{Q})|$ with multiplicity two each, so the operator has an inertia $(2,2)$, while a central parameter gives $(4,0)$; the case $N=0$ degenerates to a single nilpotent value. The general Krein theory allows a non-real spectrum, symmetric about the real axis, and the algebra exhibits it as soon as a mixed sum of a left and a right multiplication is admitted.

The general theory is Spectral Theory on Krein Spaces and Definitizable Operators and the Krein–Naĭmark Theorem; the definite counterpart is The Spectra of the Operators on the Biquaternion Algebra with Hermitian Adjoint; the operator classes are J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra; the element spectra that the operator spectra are read from are Biquaternion Spectral Theory; and the symmetry is The Fundamental Symmetry of the Biquaternion Algebra.

Conventions. $T^{\dagger}=JT^{*}J$ with $J={}^{\natural}$; $\Theta_{\tilde{Q}}(\tilde{P})=\tilde{Q}\tilde{P}\tilde{Q}^{\dagger}$, $L_{\tilde{Q}}$, $R_{\tilde{Q}}$; $\Phi:\mathbb{B}\to M_2(\mathbb{C})$ is the matrix model, $N(\tilde{Q})=\det\Phi(\tilde{Q})$; and the spectra are those of the operators as complex-linear endomorphisms of the four-dimensional complex space $\mathbb{B}$. The Kronecker forms of the definite article are used: $L_{\tilde{Q}}\leftrightarrow\Phi(\tilde{Q})\otimes I$ and $R_{\tilde{R}}\leftrightarrow I\otimes\Phi(\tilde{R})^{\mathsf T}$.

The Spectral Rules of the Definite Case

The following rules are quoted from the sibling article and are the input to everything below.

• $\mathrm{spec}(L_{\tilde{Q}})=\mathrm{spec}(\Phi(\tilde{Q}))$ with each eigenvalue of multiplicity two, and the same for $R_{\tilde{Q}}$.
• If $\Phi(\tilde{Q})$ is diagonalisable with eigenvalues $\lambda_1,\lambda_2$, then $$ \mathrm{spec}\bigl(\Theta_{\tilde{Q}}\bigr)=\{\lambda_i\bar\lambda_j\}_{i,j=1}^{2}. $$

Proof. These are the Kronecker forms $L_{\tilde{Q}}=\Phi(\tilde{Q})\otimes I$, $R_{\tilde{R}}=I\otimes\Phi(\tilde{R})^{\mathsf T}$ and $\Theta_{\tilde{Q}}=\Phi(\tilde{Q})\otimes\overline{\Phi(\tilde{Q})}$ of The Spectra of the Operators on the Biquaternion Algebra with Hermitian Adjoint, whose spectra are the sums and products of the spectra of the factors.

The $J$-Self-Adjoint Left and Right Multiplications

Theorem. A left multiplication $L_{\tilde{Q}}$ is $J$-self-adjoint exactly for $\tilde{Q}\in\mathbb{R}e_0$, and then

$$ \mathrm{spec}\bigl(L_{te_0}\bigr)=\{t\}\ \text{with multiplicity }4,\qquad t\in\mathbb{R}, $$

so the spectrum is real and the operator is a real scalar.

Proof. The criterion is that of J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra, §The $J$-Adjoint of the Three Families; a scalar multiple of the identity has the single eigenvalue $t$ with multiplicity $\dim_{\mathbb{C}}\mathbb{B}=4$.

Remark (why there is no non-real case here). A real quaternion $\tilde{Q}=t+\sum_kq_ke_k$ has the two eigenvalues $\lambda=t+i|\mathbf{q}|$ and $\bar\lambda$ of $\Phi(\tilde{Q})$ (Biquaternion Spectral Theory, §The Biquaternion Spectrum), so $\mathrm{spec}(L_{\tilde{Q}})=\{\lambda,\bar\lambda\}$ with multiplicity two each; but such an $L_{\tilde{Q}}$ is $J$-self-adjoint only when $\mathbf{q}=0$, in which case the pair collapses to the real double eigenvalue $t$. The non-real pair is a $J$-self-adjoint spectrum only for operators outside the left-multiplication family. The same statement holds for $R_{\tilde{R}}$.

Proof. The eigenvalues of a real quaternion are those of its $2\times2$ matrix, computed in Biquaternion Spectral Theory; the collapse is the criterion above.

The $J$-Self-Adjoint Sandwiches

Theorem (the two regimes). Let $\Theta_{\tilde{Q}}$ be a sandwich with $\tilde{Q}$ in $\mathbb{C}_{\mathbb{B}}\cup\mathbb{V}_{\mathbb{B}}$ — the $J$-self-adjoint sandwiches of J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra.

• If $\tilde{Q}=\zeta e_0$ is central, then $\Theta_{\tilde{Q}}=|\zeta|^{2}\,\mathrm{id}$ and $$ \mathrm{spec}\bigl(\Theta_{\tilde{Q}}\bigr)=\bigl\{|\zeta|^{2}\bigr\}\ \text{with multiplicity }4,\qquad\text{inertia }(4,0). $$
• If $\tilde{Q}=\tilde V\in\mathbb{V}_{\mathbb{B}}$ is a pure vector with $N(\tilde V)\neq0$, then $$ \mathrm{spec}\bigl(\Theta_{\tilde{V}}\bigr)=\bigl\{|N(\tilde V)|,\,|N(\tilde V)|,\,-|N(\tilde V)|,\,-|N(\tilde V)|\bigr\},\qquad\text{inertia }(2,2). $$
• If $\tilde{Q}=\tilde V\in\mathbb{V}_{\mathbb{B}}$ has $N(\tilde V)=0$, then $\Theta_{\tilde{V}}$ is nilpotent of order two, $\Theta_{\tilde{V}}^{2}=0$, and $$ \mathrm{spec}\bigl(\Theta_{\tilde{V}}\bigr)=\{0\}\ \text{with multiplicity }4. $$

Proof. For a central parameter, $\Theta_{\zeta e_0}(\tilde{P})=\zeta\tilde{P}\bar\zeta=|\zeta|^{2}\tilde{P}$, which is the first line. For a vector parameter, $\tilde V^{2}=-N(\tilde V)e_0$ and $(\tilde V^{\dagger})^{2}=-\overline{N(\tilde V)}e_0$, because $\tilde V^{\dagger}=-\bar{\tilde V}$ and $\bar{\tilde V}{}^{2}=-N(\bar{\tilde V})e_0=-\overline{N(\tilde V)}e_0$; hence $$ \Theta_{\tilde V}^{2}(\tilde{P})=\tilde V^{2}\tilde P(\tilde V^{\dagger})^{2}=|N(\tilde V)|^{2}\tilde P, $$ so $\Theta_{\tilde V}^{2}=|N(\tilde V)|^{2}\mathrm{id}$. If $N(\tilde V)\neq0$ the matrix $\Phi(\tilde V)$ is traceless with $\Phi(\tilde V)^{2}=-N(\tilde V)I$, hence diagonalisable with the eigenvalues $\pm i\sqrt{N(\tilde V)}$, and the product rule gives $\{|N|,|N|,-|N|,-|N|\}$. If $N(\tilde V)=0$ then $\tilde V^{2}=0$ and $(\tilde V^{\dagger})^{2}=0$, so $\Theta_{\tilde V}^{2}=0$; the single eigenvalue is $0$ with multiplicity four.

Corollary (the sandwich is definitizable except at the null elements). A $J$-self-adjoint sandwich is diagonalisable, with real spectrum, exactly when it is not the sandwich of a null vector $\tilde V$ with $N(\tilde V)=0$; in that case its spectrum is the real pair $\pm|N(\tilde Q)|$ with the inertia $(4,0)$ or $(2,2)$, and the operator is definitizable in the sense of Definitizable Operators and the Krein–Naĭmark Theorem.

Proof. The theorem gives the two diagonalisable regimes; the nilpotent case has no basis of eigenvectors.

Remark (the sign structure). The signs are the content of the indefinite theory: the sandwich of a vector parameter of nonzero norm has exactly two positive and two negative eigenvalues, whereas the definite sibling's sandwich of a Hermitian element is positive and has non-negative spectrum. The two regimes $(4,0)$ and $(2,2)$ are the two possible inertias of a $J$-self-adjoint sandwich on a space of complex dimension four, and they are the "signature of the sandwich operator" alluded to in the sibling article.

The Symmetry of the Spectrum

Theorem. For every $J$-self-adjoint operator $T$ on $\mathbb{B}$ the spectrum is symmetric under conjugation, $\mathrm{spec}(T)=\overline{\mathrm{spec}(T)}$, and the algebraic multiplicities of $\lambda$ and $\bar\lambda$ coincide.

Proof. $T^{\dagger}=T$ gives $T^{*}=JTJ$, so $T$ is similar to its adjoint; a matrix is similar to its adjoint only... precisely, $\mathrm{spec}(T^{*})=\overline{\mathrm{spec}(T)}$ for every operator and $\mathrm{spec}(JTJ)=\mathrm{spec}(T)$, so $\mathrm{spec}(T)=\overline{\mathrm{spec}(T)}$; the multiplicities agree because similarity preserves the characteristic polynomial up to conjugation.

Corollary (the symmetry is visible in the two regimes). In the regime $(4,0)$ the symmetry is vacuous (a real multiple of the identity); in the regime $(2,2)$ it pairs $|N|$ with itself and $-|N|$ with itself; and it constrains any non-real $J$-self-adjoint spectrum of the algebra to consist of conjugate pairs.

Proof. Immediate from the theorem and the spectra computed above.

The Mixed Sums and the Non-Real Spectrum

The symmetry is a constraint, not a triviality: away from the three families the algebra has $J$-self-adjoint operators with genuinely non-real spectrum.

Theorem. The operator

$$ T=L_{e_1}+R_{e_1} $$

is $J$-self-adjoint, and

$$ \mathrm{spec}(T)=\{2i,\,-2i,\,0,\,0\}. $$

Proof. $J$-self-adjointness is $(L_{\tilde{Q}})^{\dagger}+(R_{\tilde{Q}})^{\dagger}=R_{\bar{\tilde{Q}}}+L_{\bar{\tilde{Q}}}$, which for $\tilde{Q}=e_1$ (real coefficients) returns $L_{e_1}+R_{e_1}$; the sum is a "Sylvester" operator, and the rule $\mathrm{spec}(L_{\tilde{Q}}+R_{\tilde{Q}})=\{\lambda_i+\lambda_j\}$ for the eigenvalues $\lambda_i$ of $\Phi(\tilde{Q})$ (The Spectra of the Operators on the Biquaternion Algebra with Hermitian Adjoint, §The Spectrum of the Sylvester Operator) applied to $\Phi(e_1)=i\sigma_1$ with eigenvalues $\pm i$ gives $\{i+i,i-i,-i+i,-i-i\}=\{2i,0,0,-2i\}$.

Remark. The spectrum is non-real and symmetric about the real axis, exactly as the general theory of Spectral Theory on Krein Spaces predicts, and the operator is $J$-self-adjoint but not definitizable; it is therefore a $J$-self-adjoint operator of the algebra to which the Krein–Naĭmark theory does not apply, in contrast with the sandwiches of §The $J$-Self-Adjoint Sandwiches.

Worked Examples

A central scalar. $\tilde{Q}=t e_0$, $t\in\mathbb{R}$: $L_{\tilde{Q}}$ is $J$-self-adjoint with spectrum $\{t\}$ of multiplicity four.

A real quaternion. $\tilde{Q}=\tfrac12+\tfrac{\sqrt3}{2}e_1$: $N(\tilde{Q})=1$ and the eigenvalues of $\Phi(\tilde{Q})$ are $e^{\pm i\pi/3}$; $L_{\tilde{Q}}$ is a unit-norm element, but it is not $J$-self-adjoint, and its spectrum $\{e^{\pm i\pi/3}\}$ is not the $J$-self-adjoint spectrum. It is $J$-unitary as a sandwich, by $|N|=1$.

A vector sandwich of nonzero norm. $\tilde{Q}=e_1$: $\mathrm{spec}(\Theta_{e_1})=\{1,1,-1,-1\}$, inertia $(2,2)$; the operator is $J$-self-adjoint and $J$-unitary.

A vector sandwich of zero norm. $\tilde{Q}=e_1+ie_2$: $N=0$, so $\Theta_{\tilde{Q}}^{2}=0$ and the spectrum is $\{0\}$; the sandwich of a null vector is nilpotent.

A central phase. $\tilde{Q}=e^{i\theta}e_0$: $\Theta_{\tilde{Q}}=\mathrm{id}$, spectrum $\{1\}$ of multiplicity four, inertia $(4,0)$.

The non-real example. $T=L_{e_1}+R_{e_1}$: spectrum $\{2i,-2i,0\}$, non-real, symmetric, not definitizable.

Summary

For the Krein adjoint $T^{\dagger}=JT^{*}J$ the $J$-self-adjoint operators of the three families are few and their spectra are real with signs. A left or right multiplication is $J$-self-adjoint exactly for a real scalar parameter, and then the spectrum is that scalar with multiplicity four. A sandwich is $J$-self-adjoint exactly for a parameter in the centre or in the vector subspace, and there are two regimes: a central parameter gives $|\zeta|^{2}$ with multiplicity four and inertia $(4,0)$, while a vector parameter of nonzero norm gives the real pair $\pm|N(\tilde{Q})|$ with multiplicity two each and inertia $(2,2)$; a null vector parameter gives the nilpotent operator $\Theta_{\tilde{Q}}^{2}=0$ with the single spectrum $\{0\}$. Every $J$-self-adjoint operator has a spectrum symmetric under conjugation with matching multiplicities, and this is the only general constraint: away from the three families the algebra carries $J$-self-adjoint operators with non-real spectrum, the simplest being $L_{e_1}+R_{e_1}$ with $\{2i,-2i,0\}$, which is not definitizable.

Summary of Notation

Further Reading

• Spectral Theory on Krein Spaces (articles_maths/spectral-theory-on-krein-spaces.md) and Definitizable Operators and the Krein–Naĭmark Theorem (articles_maths/definitizable-operators-and-the-krein-naimark-theorem.md), for the indefinite spectral theory and definitizability
• The Spectra of the Operators on the Biquaternion Algebra with Hermitian Adjoint (articles_maths/the-spectra-of-the-operators-on-the-biquaternion-algebra-with-hermitian-adjoint.md), for the definite spectral rules quoted here
• J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra (articles_maths/j-self-adjoint-and-j-unitary-operators-on-the-biquaternion-algebra.md), for the $J$-self-adjointness criteria
• The Fundamental Symmetry of the Biquaternion Algebra (articles_maths/the-fundamental-symmetry-of-the-biquaternion-algebra.md), for $J$ and its eigenspaces
• Indefinite Positivity and the Krein Cone of the Biquaternion Algebra (articles_maths/indefinite-positivity-and-the-krein-cone-of-the-biquaternion-algebra.md), for the indefinite cone and the $J$-Cartan involution
• Biquaternion Spectral Theory (articles_maths/biquaternion-spectral-theory.md), for the element spectra and the Cayley–Hamilton functionals
• Israel Gohberg, Peter Lancaster and Leiba Rodman, Indefinite Linear Algebra and Applications (Birkhäuser, 2005), for the spectra, the inertias and the normal forms of indefinite operators