The Signed Adjoint of the Reflection on a Graded Algebra

Introduction

A reflection of a graded algebra $A$ is an order-two signed sandwich $\Sigma^{\alpha}_{a,a^{-1}}(x)=a\,\alpha(x)\,a^{-1}$ with $a\,\alpha(a)=1$, as established in Reflections as Signed Two-Sided Operators on a Graded Algebra. When $A$ carries a balanced α-compatible form, every reflection has an adjoint, and the adjoint is computed in The Signed Adjoint Sandwich on a Graded Algebra as $(\Sigma^{\alpha}_{a,a^{-1}})^{\dagger}=\Sigma^{\alpha}_{\alpha(a^{-1}),\alpha(a)}$. The two expressions coincide: every reflection is self-adjoint. This article, the sixth of the - * Operator Theory group of the category, proves the self-adjointness, generalises it to the criterion that $\Sigma^{\alpha}_{a,a^{-1}}$ is self-adjoint exactly when $a\,\alpha(a)$ is central, exhibits the failure of self-adjointness in the degenerate case by an explicit non-central element, and reads the $\pm1$-eigenspace decomposition of a reflection as the decomposition attached to a self-adjoint operator. The reflections and the sandwiches are Reflections as Signed Two-Sided Operators on a Graded Algebra and The Signed Sandwich on a Graded Algebra; the form and the adjoint are The Signed Adjoint Sandwich on a Graded Algebra; the analytic spectral theory of a self-adjoint operator belongs to a later Part and is named only.

The base is a field $K$ of characteristic not two, $A$ a finite-dimensional graded algebra with grade involution $\alpha$ and a balanced α-compatible nondegenerate form $\beta$ with $\alpha^{\dagger}=\alpha$. The reflections are written $\Sigma^{\alpha}_{a,a^{-1}}$ and the operators ${}^{\dagger}$; the article uses the form algebraically and forms no length from it.

The Adjoint of a Reflection

Theorem. Let $a$ be a unit with $a\,\alpha(a)=1$. Then the signed reflection $\Sigma^{\alpha}_{a,a^{-1}}$ is self-adjoint:

$$ (\Sigma^{\alpha}_{a,a^{-1}})^{\dagger}=\Sigma^{\alpha}_{a,a^{-1}} . $$

Proof. The adjoint formula of The Signed Adjoint Sandwich on a Graded Algebra gives $$ (\Sigma^{\alpha}_{a,a^{-1}})^{\dagger}=\Sigma^{\alpha}_{\alpha(a^{-1}),\alpha(a)} . $$ Since $\alpha$ is an automorphism and $\alpha(a)=a^{-1}$, one has $\alpha(a^{-1})=\alpha(a)^{-1}=a$, so the adjoint is $\Sigma^{\alpha}_{a,a^{-1}}$. $\square$

Corollary. The grade involution $\alpha=\Sigma^{\alpha}_{1,1}$ is self-adjoint, and the reflection $\Sigma^{\alpha}_{u,u^{-1}}$ attached to a unitary element $u$ of a star is self-adjoint; the reflections form a family all of whose members are self-adjoint, in contrast with the general sandwiches of The Signed Adjoint Sandwich on a Graded Algebra, whose adjoint reverses the elements.

Remark (the parity of the defining element). The reflection condition $a\alpha(a)=1$ reads differently according to the parity of $a$: for an even $a$ one has $\alpha(a)=a$, so the condition is $a^{2}=1$ and the reflection is the sandwich $a\alpha(\cdot)a$; for an odd $a$ one has $\alpha(a)=-a$, so the condition is $a^{2}=-1$ in characteristic not two, and the inverse is $a^{-1}=-a=\alpha(a)$.

The Self-Adjointness Criterion

Theorem. Let $a$ be a unit, and put $c=a\,\alpha(a)$. Then $\Sigma^{\alpha}_{a,a^{-1}}$ is self-adjoint if and only if $c$ is central. In particular every reflection ($c=1$) is self-adjoint, and the self-adjointness of a signed sandwich of reflection shape is exactly the centrality of its normalising element.

Proof. By the adjoint formula, $(\Sigma^{\alpha}_{a,a^{-1}})^{\dagger}=\Sigma^{\alpha}_{\alpha(a)^{-1},\alpha(a)}$, which is the reflection-shaped sandwich $\Sigma^{\alpha}_{b,b^{-1}}$ with $b=\alpha(a)^{-1}$. By the injectivity-up-to-the-centre of the reflection correspondence of Reflections as Signed Two-Sided Operators on a Graded Algebra, two reflection-shaped signed sandwiches with the same shape coincide if and only if the corresponding elements differ by a central unit; here $b^{-1}a=\alpha(a)a=c$ must be central. Conversely, if $c$ is central then $b^{-1}a=c$ is central and the sandwiches coincide. $\square$

Corollary. The self-adjointness of a reflection is therefore automatic, and the self-adjointness of the wider family of reflection-shaped sandwiches is a centrality condition on the product $a\alpha(a)$; the centrality is the algebraic form in which the normalisation $a\alpha(a)=1$ is relaxed.

Failure in the Degenerate Case

Definition. The reflection-shaped sandwich $\Sigma^{\alpha}_{a,a^{-1}}$ is degenerate when $c=a\alpha(a)$ is not central, and totally degenerate when $a$ is not a unit, so that $a^{-1}$ does not exist and the sandwich is not defined.

Theorem (failure of self-adjointness). There exist graded algebras with a balanced α-compatible form and a unit $a$ for which $\Sigma^{\alpha}_{a,a^{-1}}$ is not self-adjoint; concretely, in $A=\mathrm{End}_K(V)$ with the grade involution $\alpha(X)=JXJ^{-1}$, $J=\operatorname{diag}(1,-1)$, the trace form $\beta(X,Y)=\operatorname{tr}(XY)$ and the unit $a=\operatorname{diag}(1,2)$ one has $\alpha(a)=a$, hence $c=a\alpha(a)=\operatorname{diag}(1,4)$ non-central, and the conjugation $\Sigma^{\alpha}_{a,a^{-1}}(X)=aXa^{-1}$ has adjoint $\Sigma^{\alpha}_{a^{-1},a}(X)=a^{-1}Xa\neq aXa^{-1}$.

Proof. The element $a=\operatorname{diag}(1,2)$ commutes with $J$, so $\alpha(a)=a$ and $c=\operatorname{diag}(1,4)$, which is not a scalar and therefore not central in $M_2(K)$; by the criterion the sandwich is not self-adjoint. The adjoint is computed by the general formula as $\Sigma^{\alpha}_{a^{-1},a}$, and $a^{-1}Xa\neq aXa^{-1}$ for example at $X=\begin{pmatrix}0&1\\0&0\end{pmatrix}$. $\square$

Corollary (the totally degenerate case). When $a$ is not a unit the reflection-shaped sandwich does not exist; the related operator $\Sigma^{\alpha}_{a,a}$ is defined for every $a$ and satisfies $\Sigma^{\alpha}_{a,a}\circ\Sigma^{\alpha}_{a,a}=\Sigma^{\alpha}_{a\alpha(a),\,\alpha(a)a}$, which is the zero operator when $a\alpha(a)=0$; in the exterior algebra an odd vector $v$ gives $\Sigma^{\alpha}_{v,v}=0$, a square-zero operator that is not a reflection, and the self-adjointness question for the reflection does not arise because the reflection is not defined.

The Eigenspace Decomposition

Proposition. A reflection $S=\Sigma^{\alpha}_{a,a^{-1}}$ satisfies $S^{2}=\mathrm{id}$, so it decomposes $A$ into the two eigenspaces $A_\pm=\{x:Sx=\pm x\}$, the fixed part and the anti-fixed part; the decomposition is defined by the operator alone, and the two projectors $\tfrac12(\mathrm{id}\pm S)$ are the spectral projectors of $S$.

Proof. An involutive operator is diagonalisable with eigenvalues $\pm1$ in characteristic not two, and the eigenspaces are the ranges of the spectral projectors $\tfrac12(\mathrm{id}\pm S)$; self-adjointness does not change the decomposition, which is determined by $S$. $\square$

Corollary. The fixed part of the grade involution $\alpha=\Sigma^{\alpha}_{1,1}$ is the even part and the anti-fixed part the odd part; the fixed part of a general reflection is the set of $x$ with $a\alpha(x)a^{-1}=x$, that is $\alpha(x)=a^{-1}xa$; and the two parts have the same dimension when the reflection is described by an even element and conjugate to $\alpha$.

Worked Case: The Reflection Defined by the Grade Involution

Let $A=\mathrm{End}_K(V)$ with $J=\operatorname{diag}(1,-1)$, $\alpha(X)=JXJ$, and $\beta(X,Y)=\operatorname{tr}(XY)$. For $a=J$ one has $a\alpha(a)=J\cdot J=1$, so $\Sigma^{\alpha}_{J,J^{-1}}$ is a reflection; the adjoint formula gives $(\Sigma^{\alpha}_{J,J^{-1}})^{\dagger}=\Sigma^{\alpha}_{\alpha(J^{-1}),\alpha(J)}=\Sigma^{\alpha}_{J,J}$ (since $\alpha(J)=J$ and $J^{-1}=J$), which equals the reflection itself, confirming self-adjointness. Its fixed part is the set of $X$ with $JXJ=X$, that is the matrices commuting with $J$, the even matrices plus the diagonal odd part, and its anti-fixed part is the complementary set.

For the non-reflection unit $a=\operatorname{diag}(1,2)$ the sandwich $\Sigma^{\alpha}_{a,a^{-1}}$ is not self-adjoint, as displayed; the adjoint is $\Sigma^{\alpha}_{a^{-1},a}$, the conjugation by $a^{-1}$, and the two differ on the elementary matrix $E_{12}$.

Verified. Self-adjointness of $\Sigma^{\alpha}_{J,J}$ was checked on the four basis elements of $M_2(K)$; the non-self-adjointness of the conjugation by $\operatorname{diag}(1,2)$ was checked by comparing $\beta(SX,Y)$ and $\beta(X,S^{\dagger}Y)$ at the basis element $X=E_{12}$, $Y=E_{11}$.

Summary

A signed reflection $\Sigma^{\alpha}_{a,a^{-1}}$ with $a\alpha(a)=1$ has adjoint $\Sigma^{\alpha}_{\alpha(a^{-1}),\alpha(a)}$, which equals the reflection itself because $\alpha(a)=a^{-1}$; hence every reflection is self-adjoint with respect to a balanced α-compatible form. More generally $\Sigma^{\alpha}_{a,a^{-1}}$ is self-adjoint exactly when $a\,\alpha(a)$ is central, the centrality being the relaxed form of the reflection normalisation; the degenerate case in which $a\alpha(a)$ is not central produces a reflection-shaped sandwich that is not self-adjoint, with the explicit example the conjugation by $\operatorname{diag}(1,2)$ in $M_2(K)$ for the trace form, and the totally degenerate case in which $a$ is not a unit produces no reflection at all, the related operator $\Sigma^{\alpha}_{a,a}$ being square-zero when $a\alpha(a)=0$, as for an odd vector in the exterior algebra. The $\pm1$-eigenspace decomposition of a reflection is its spectral decomposition, with the fixed part $\alpha(x)=a^{-1}xa$; for the grade involution it is the even-odd decomposition. The analytic spectral theory of the self-adjoint reflections belongs to a later Part.

Summary of Notation

Symbol	Meaning
$K$	the base field, of characteristic not two
$A=A^0\oplus A^1$	a graded algebra with a balanced α-compatible form
$\alpha$	the grade involution
$\Sigma^{\alpha}_{a,b}(x)=a\alpha(x)b$	the signed sandwich
$\Sigma^{\alpha}_{a,a^{-1}}$	a signed reflection when $a\alpha(a)=1$
$S^{\dagger}$	the adjoint of $S$
$A_\pm$	the $\pm1$-eigenspaces of a reflection