The Adjoint of the One-Sided Action with Hermitian Adjoint

Introduction

A Hermitian Clifford module is a Clifford module with a Hermitian form for which the Clifford action is self-adjoint, $(x\cdot s,t) = (s,x^{\dagger}\cdot t)$. That single axiom is a statement about the adjoint of the action: the adjoint of the operator "multiply on the left by $x$" is the operator "multiply on the left by $x^{\dagger}$", so that the action of the algebra and the action of its dagger are adjoint to one another. This article is about that adjoint: the identity $\rho(x)^{*} = \rho(x^{\dagger})$, the resulting $*$-structure on the algebra of operators, the rule for the adjoint of a composite operator built from the one-sided action, and the special case that is the algebraic reason a Dirac operator is formally self-adjoint.

The topic is algebraic. The analytic consequences of the self-adjointness — domains, closures, essential self-adjointness, spectrum, compactness of the resolvent — are the content of Dirac Differential Operators and are cited, not repeated. What is established here is the algebra that makes those theorems apply: the coefficients of a Dirac operator are elements of the Clifford algebra acting one-sidedly, and they are skew-adjoint exactly when the elements are; from that, and the skew-adjointness of the derivative, the formal self-adjointness of the operator follows in one line.

The Clifford action and the module form are Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint; the one-sided operators on the algebra and their adjoints are One-Sided Operators on a Hilbert Algebra with Hermitian Adjoint; the two-sided family is Two-Sided Operators on a Clifford Algebra; the Hermitian member of that family, whose adjoint is the theorem at the end, is The Hermitian Sandwich on a Hilbert Algebra with Hermitian Adjoint; the forms of the dagger and the adjoint of multiplication are The Blade Form and the Hilbert Structure with Hermitian Adjoint; the unitary slice is The Unitary Slice and the Compact Real Form with Hermitian Adjoint; the spinor module is Spinors as Minimal Left Ideals with Inner Conjugation; and the analysis of the Dirac operator is Dirac Differential Operators.

The One-Sided Action on a Hermitian Module

The Action and Its Adjoint

Setting. Let $S$ be a Hermitian Clifford module over $\mathrm{Cl}(V,q)$ with form $(\cdot,\cdot)$ and scalar field $A$ with involution $\sigma$, and let

$$ \rho : \mathrm{Cl}(V,q)\longrightarrow \mathrm{End}_A(S), \qquad \rho(x)(s) = x\cdot s $$

be the action. The adjoint action is the assignment $\rho^{*}(x) = \rho(x)^{*}$, the adjoint of $\rho(x)$ in $\mathrm{End}_A(S)$ with respect to the module form.

Theorem (the adjoint of the action is the action of the dagger). For every $x \in \mathrm{Cl}(V,q)$,

$$ \rho(x)^{*} = \rho\bigl(x^{\dagger}\bigr), $$

and consequently $\rho$ is a $*$-homomorphism of the daggered algebra into the $*$-algebra of operators on $S$:

$$ \rho(xy) = \rho(x)\rho(y), \qquad \rho(x^{\dagger}) = \rho(x)^{*} . $$

Proof. The adjointness axiom of Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint is precisely $(x\cdot s,t) = (s,x^{\dagger}\cdot t)$, that is $(\rho(x)s,t) = (s,\rho(x^{\dagger})t)$; comparing with the defining property of the adjoint, $(\rho(x)s,t) = (s,\rho(x)^{*}t)$, and using the non-degeneracy of the form to cancel $s$ and $t$, gives $\rho(x)^{*} = \rho(x^{\dagger})$. The multiplicativity of $\rho$ is the module axiom, and the $*$-property is what has just been shown.

Corollary (the adjoint action is multiplicative). The adjoint action is a homomorphism of the daggered algebra, $\rho(x)^{*}\rho(y)^{*} = \rho(x^{\dagger})\rho(y^{\dagger}) = \rho(x^{\dagger}y^{\dagger})$, and it agrees with the action on the daggered element, so $\rho(x^{*}) = \rho(x)^{*}$ when ${}^{*}$ is the dagger of the coefficient field composed with the algebra anti-involution. Consequently the image of $\rho$ is a $*$-subalgebra of $\mathrm{End}_A(S)$, isomorphic to the quotient of $\mathrm{Cl}(V,q)$ by the kernel of the action.

The Adjoint on the Algebra

Corollary (the adjoint of a one-sided operator on the algebra). With $S = \mathrm{Cl}(V,q)$ the regular module and the form $(x,y) = \mathrm{Sc}(x^{\dagger}y)$,

$$ L_a^{*} = L_{a^{\dagger}}, \qquad R_b^{*} = R_{b^{\dagger}}, $$

the statement of One-Sided Operators on a Hilbert Algebra with Hermitian Adjoint; so the module theorem above is the general form of that identity, and the algebra case is its regular instance.

Corollary (vectors act by skew-adjoint operators). For $v \in V$, $\rho(v)^{*} = \rho(v^{\dagger}) = -\rho(v)$, because the dagger negates the vectors; so every vector acts by a skew-adjoint operator, $\rho(v)^{*} = -\rho(v)$, and the Clifford relation reads $\rho(v)^{2} = q(v)\cdot\mathrm{id}$ with $\rho(v)$ skew-adjoint. This is the single fact from which the formal self-adjointness of the Dirac operator is read off below.

The Adjoint as an Involution on Operators

Proposition (the adjoint is an involutive anti-automorphism of the operator algebra). For all $S, T \in \mathrm{End}_A(S)$ and $\lambda \in A$,

$$ (T^{*})^{*} = T, \qquad (ST)^{*} = T^{*}S^{*}, \qquad (\lambda T)^{*} = \sigma(\lambda)T^{*}, $$

so that $T\mapsto T^{*}$ is an involutive $\sigma$-antilinear anti-automorphism of $\mathrm{End}_A(S)$; the image of $\rho$ is closed under it, by the theorem.

Proof. The first two are the standard properties of the Hilbert-space adjoint with respect to a $\sigma$-sesquilinear form, and the third records the semilinearity: $(\lambda Ts,t) = \lambda(Ts,t)$ while $(s, T^{*}\sigma(\lambda)t) = \sigma(\lambda)(s,T^{*}t)$.

Corollary (self-adjoint, skew-adjoint and normal elements). For $a \in \mathrm{Cl}(V,q)$,

$$ \rho(a) \text{ is self-adjoint} \iff a^{\dagger} = a, \qquad \rho(a) \text{ is skew-adjoint} \iff a^{\dagger} = -a, \qquad \rho(a) \text{ is normal} \iff [a^{\dagger},a] \text{ acts as } 0 , $$

and $\rho(a)$ is unitary iff $a^{\dagger}a = 1$, that is iff $a$ lies in the unitary slice $U$. The slice is therefore exactly the set of elements whose one-sided action is a unitary operator on $S$, which is the operator-theoretic content of the slice.

The Adjoint of a Composite Operator

The General Rule

Theorem (adjoint of a composite one-sided action). Let $T_1,\dots,T_k$ be $A$-linear operators on $S$ that admit adjoints, and let $a_1,\dots,a_k \in \mathrm{Cl}(V,q)$. Then

$$ \Bigl(\sum_{i} \rho(a_i)\,T_i\Bigr)^{*} = \sum_{i} T_i^{*}\,\rho\bigl(a_i^{\dagger}\bigr) . $$

Proof. By the anti-multiplicativity and the involution property, $(\rho(a_i)T_i)^{*} = T_i^{*}\rho(a_i)^{*} = T_i^{*}\rho(a_i^{\dagger})$, and the adjoint of a sum is the sum of the adjoints.

Corollary (formal self-adjointness of the Dirac operator). Let $D = \sum_j \rho(e_j)\,\partial_j$ be the flat Dirac-type operator built from an orthonormal frame $e_j$ and first-order operators $\partial_j$ that are skew-adjoint, $\partial_j^{*} = -\partial_j$. Then

$$ D^{*} = \sum_j \partial_j^{*}\rho(e_j^{\dagger}) = \sum_j (-\partial_j)(-\rho(e_j)) = \sum_j \rho(e_j)\partial_j = D, $$

so $D$ is formally self-adjoint. The two sign flips — the coefficient is skew because it is a vector, and the derivative is skew because integration by parts introduces a minus — cancel, and the operator is its own adjoint. This is the algebraic core of the self-adjointness that Dirac Differential Operators develops analytically: the ellipticity of the symbol, the domain, the closure, essential self-adjointness and the spectrum are treated there, and what is shown here is only why the formal adjoint of the operator coincides with the operator.

Remark (why the scalar direction breaks it, algebraically). The corpus's Cauchy–Riemann operator has one coefficient equal to $1$, which is self-adjoint and not skew, $\rho(1) = \mathrm{id}$ with $\rho(1)^{*} = \rho(1)$, so the corresponding term acquires no minus from the coefficient and the operator is not symmetric; its formal adjoint is the conjugate operator, and the two split into a self-adjoint part and a skew-adjoint part. The algebra of the split is exactly the failure of one coefficient to be skew, and the analysis of the two operators is Dirac Differential Operators.

The Adjoint Action of the Slice, and the Two-Sided Case

Theorem (the slice acts by inner $*$-automorphisms). For $u$ in the unitary slice $U$ let $\mathrm{Ad}_u(T) = \rho(u)\,T\,\rho(u)^{-1}$. Then $\rho(u)$ is unitary, $\mathrm{Ad}_u$ is an automorphism of the operator algebra, and it preserves the adjoint:

$$ \mathrm{Ad}_u(T)^{*} = \mathrm{Ad}_u\bigl(T^{*}\bigr) \qquad \text{for all } T . $$

Proof. Unitarity of $\rho(u)$ is the slice theorem of Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint; then $\mathrm{Ad}_u(T)^{*} = \rho(u)^{-*}\,T^{*}\,\rho(u)^{*} = \rho(u)\,T^{*}\,\rho(u)^{-1}$ because $\rho(u)^{*} = \rho(u)^{-1}$ for a unitary.

Remark. So the slice $U$ acts on the one-sided operators by $*$-automorphisms, and this action extends to the whole operator algebra; on the image of $\rho$ it is the adjoint action inside the algebra, and on the module it is the unitary action of the slice. This is the operator-algebraic form of the statement that the slice acts by operators and not by isometries of the quadratic space.

Theorem (the adjoint of a two-sided operator is the operator at the dagger). The standard maps $\mathrm{id}$, $\alpha$, $r$, $\bar\cdot$ and ${}^{\dagger}$ commute pairwise and each commutes with the dagger, so for the two-sided operator $\Phi^{\theta,c}_x = L_{\theta(x)}R_{c(x)}$ with $\theta$ and $c$ among them one has

$$ \bigl(\Phi^{\theta,c}_x\bigr)^{*} = \Phi^{\theta,c}_{x^{\dagger}} . $$

In particular the Hermitian sandwich $\Phi^{\mathrm{id},\dagger}_x = L_xR_{x^{\dagger}}$ has adjoint $\Phi^{\mathrm{id},\dagger}_{x^{\dagger}} = L_{x^{\dagger}}R_x$, and on the slice $U$ it is unitary, $(\Phi^{\mathrm{id},\dagger}_x)^{*}\Phi^{\mathrm{id},\dagger}_x = \mathrm{id}$, which is the operator form of $u^{\dagger}u = 1$ that The Hermitian Sandwich on a Hilbert Algebra with Hermitian Adjoint uses.

Proof. $(\Phi^{\theta,c}_x)^{*} = (L_{\theta(x)}R_{c(x)})^{*} = R_{c(x)}^{*}L_{\theta(x)}^{*} = R_{c(x)^{\dagger}}L_{\theta(x)^{\dagger}} = L_{\theta(x)^{\dagger}}R_{c(x)^{\dagger}}$; and because each of the maps commutes with the dagger, $\theta(x)^{\dagger} = \theta(x^{\dagger})$ and $c(x)^{\dagger} = c(x^{\dagger})$, giving $\Phi^{\theta,c}_{x^{\dagger}}$.

Summary

The adjoint of the one-sided action is the one-sided action of the dagger: for the action $\rho$ of a Clifford algebra on a Hermitian Clifford module, $\rho(x)^{*} = \rho(x^{\dagger})$, so $\rho$ is a $*$-homomorphism of the daggered algebra into the $*$-algebra of operators on the module. On the regular module this is the identity $L_a^{*} = L_{a^{\dagger}}$, $R_b^{*} = R_{b^{\dagger}}$ of the one-sided operators; for the vectors it is the skew-adjointness $\rho(v)^{*} = -\rho(v)$, since the dagger negates the vectors; and for an element it gives the criteria: self-adjoint, skew-adjoint, normal or unitary exactly as the element is self-adjoint, skew-adjoint, normal or in the unitary slice $U$.

The adjoint is an involutive, $\sigma$-antilinear, anti-multiplicative map on the operator algebra. For a composite operator it gives $(\sum_i\rho(a_i)T_i)^{*} = \sum_i T_i^{*}\rho(a_i^{\dagger})$, and applying it to the flat Dirac-type operator $D = \sum_j\rho(e_j)\partial_j$ with skew coefficients and skew derivatives produces $D^{*} = D$: the two sign flips cancel and the operator is formally self-adjoint, which is the algebraic core of the analysis of Dirac Differential Operators, the failure of the corpus's Cauchy–Riemann operator being exactly its one self-adjoint scalar coefficient. Finally the unitary slice acts by inner $*$-automorphisms $\mathrm{Ad}_u(T) = \rho(u)T\rho(u)^{-1}$ preserving the adjoint, and a two-sided operator has its adjoint at the dagger, $(\Phi^{\theta,c}_x)^{*} = \Phi^{\theta,c}_{x^{\dagger}}$, because the standard involutions commute pairwise; on the slice the Hermitian sandwich is unitary.

Summary of Notation

Symbol	Meaning
$\rho(x)(s) = x\cdot s$	One-sided action of the algebra on the module
$\rho(x)^{*} = \rho(x^{\dagger})$	Adjoint of the action
$\rho(xy) = \rho(x)\rho(y)$, $\rho(x^{\dagger}) = \rho(x)^{*}$	$\rho$ is a $*$-homomorphism
$\rho(v)^{*} = -\rho(v)$	Vectors act by skew-adjoint operators
$U$	Unitary slice, $\rho(u)$ unitary iff $u \in U$
$\bigl(\sum_i\rho(a_i)T_i\bigr)^{} = \sum_iT_i^{}\rho(a_i^{\dagger})$	Adjoint of a composite operator
$D = \sum_j\rho(e_j)\partial_j$, $\partial_j^{*} = -\partial_j$	Flat Dirac-type operator, formally self-adjoint
$\mathrm{Ad}_u(T) = \rho(u)T\rho(u)^{-1}$	Inner $*$-automorphism by the slice
$\bigl(\Phi^{\theta,c}_x\bigr)^{*} = \Phi^{\theta,c}_{x^{\dagger}}$	Adjoint of a two-sided operator