Adjoints on a Clifford Module

Introduction

A Clifford module carries more than one natural pairing, and the adjoint of an operator depends on the pairing chosen. Two are used in this corpus: an algebra-valued pairing, whose values lie in the Clifford algebra and which is Hermitian for the Clifford conjugation, and a scalar Hermitian form, whose values lie in $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ and which is positive definite on a definite module. The adjoint taken with respect to the first is the Clifford-adjoint, the one taken with respect to the second the Hermitian-adjoint; this article defines both, proves their basic properties, and compares them.

The setting is that of Operators on a Clifford Module: the Clifford algebra $A=\mathrm{Cl}_{0,m}$, a left Clifford module $\mathcal{S}$, the Clifford multiplication $c:A\to\mathrm{End}_\mathbb{R}(\mathcal{S})$, the involution $x\mapsto x^{*}$ on the elements (the Clifford conjugation) and the operator algebra generated by the multiplications and the derivatives, with the operator $D=\sum_\mu c(e_\mu)\partial_\mu$. The module and the pairings are those of Hermitian Hilbert Modules over a Clifford Algebra, and the analytic forms are the $L^2$ forms of that article. The algebraic theory of the one-sided action and its adjoint — the identity $\rho(x)^{*}=\rho(x^{*})$, the resulting $\ast$-structure, the reversal of the factors, and the elementary algebraic reason the Dirac operator is formally self-adjoint — is Part II's, in The Adjoint of the One-Sided Action with Hermitian Adjoint; the bilinear operators built from two spinors and the Fierz identity are Bilinear Operators on a Hermitian Module with Hermitian Adjoint; the spinor adjoint $\bar\psi=\psi^{*}\gamma_0$ and the invariant $\bar\psi\psi$ are Spinor Adjoints and the Dirac Adjoint with Hermitian Adjoint; and the finite-dimensional Dirac element with its Hermitian adjoint is Dirac Operators with Hermitian Adjoint. Those results are cited. What this article adds is the distinction between the two adjoints on a Clifford module and its consequences for the multiplications, the intertwiners and the differential operators.

A word on the marks. In this article the star is the conjugation on the elements of the algebra, $x\mapsto x^{*}$, and the dagger is the adjoint of an operator, as Conventions in Mathematics fixes. The Clifford-adjoint and the Hermitian-adjoint are two different operator adjoints, and they are written $T^{\mathrm{Cl}}$ and $T^{\dagger}$ respectively; the superscript $\mathrm{Cl}$ is a label and not a second involution on the elements.

The Two Pairings

The Clifford-Valued Pairing

Definition. A Clifford-valued pairing on a left Clifford module $\mathcal{S}$ is a map

$$ (\cdot,\cdot)_A : \mathcal{S}\times\mathcal{S}\longrightarrow A $$

that is left $A$-linear in the first variable, $(a s,t)_A=a\,(s,t)_A$, and Hermitian for the conjugation $x\mapsto x^{*}$,

$$ (s,t)_A=(t,s)_A^{*} , \qquad (s,tb)_A=(s,t)_A\,b , $$

the second identity being the right linearity that accompanies the left linearity; it is non-degenerate when $(s,t)_A=0$ for all $t$ forces $s=0$, and the analogous statement on the other side.

Proposition (existence and the type of the module). A definite Clifford module carries a non-degenerate Clifford-valued pairing, unique up to a central positive scalar on each irreducible summand; on the regular module $\mathcal{S}=A$ the pairing

$$ (s,t)_A = s^{*}t $$

is such a pairing, and every pairing of the regular module is of this form up to a central factor.

Proof. The existence and uniqueness are the module-level statement of Part II's Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint, of which this is the Clifford instance; on the regular module the formula is checked directly, Hermitian symmetry being $(s^{*}t)^{*}=t^{*}s$ and left linearity being clear. $\square$

The Scalar Hermitian Form

Definition. A scalar Hermitian form on $\mathcal{S}$ is a map $(\cdot,\cdot)_{\mathbb{K}}:\mathcal{S}\times\mathcal{S}\to\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$) that is sesquilinear over $\mathbb{K}$, Hermitian in the quaternionic case, and positive definite.

Proposition (the scalar part of the Clifford pairing). When $\mathbb{K}$ carries a positive linear functional — the scalar part $\mathrm{Sc}$ in the quaternionic and biquaternionic cases, the real part in the complex case — the composition

$$ (s,t)_{\mathbb{K}} = \mathrm{Sc}\bigl((s,t)_A\bigr) $$

is a scalar Hermitian form on $\mathcal{S}$ exactly when the Clifford pairing is positive, $(s,s)_A\in P$ for the positive cone $P$ of the algebra, and it is then the form used in the analytic theory. On the regular module it is $\mathrm{Sc}(s^{*}t)=\sum_\mu\lvert s_\mu\rvert^2$ in the biquaternionic model.

Proof. The scalar part is linear over $\mathbb{R}$ and compatible with the conjugation, so the Hermitian symmetry and the sesquilinearity pass to $\mathrm{Sc}\circ(\cdot,\cdot)_A$; the positivity is the defining property of the cone $P$ of Part II's Positivity and the Hermitian Cone of a Hilbert Algebra with Hermitian Adjoint, and the regular-module formula is the computation $\mathrm{Sc}(s^{*}t)=\sum_\mu\lvert s_\mu\rvert^2$ of that article. $\square$

Remark (the two forms are not interchangeable). The Clifford-valued pairing is non-degenerate but isotropic: there are nonzero $s$ with $(s,s)_A=0$, because the algebra contains zero divisors. The scalar form is definite: $(s,s)_{\mathbb{K}}>0$ for $s\neq0$. The two therefore behave differently wherever positivity or a norm is used, and they lead to two different adjoints. The isotropy of the Clifford pairing and the repair of the scalar form by the spinor adjoint are the subject of Spinor Adjoints and the Dirac Adjoint with Hermitian Adjoint.

The Clifford-Adjoint

Definition. Let $T\in\mathrm{End}_\mathbb{R}(\mathcal{S})$. The Clifford-adjoint $T^{\mathrm{Cl}}$ is the operator, when it exists, defined by

$$ (Ts,t)_A = (s,T^{\mathrm{Cl}}t)_A \qquad s,t\in\mathcal{S} . $$

Theorem (existence, uniqueness and the involution laws). On a definite module the Clifford-adjoint exists and is unique for every $T\in\mathrm{End}_\mathbb{R}(\mathcal{S})$, and the assignment $T\mapsto T^{\mathrm{Cl}}$ is an anti-linear involution of the operator algebra:

$$ (T^{\mathrm{Cl}})^{\mathrm{Cl}}=T , \qquad (ST)^{\mathrm{Cl}}=T^{\mathrm{Cl}}S^{\mathrm{Cl}} , \qquad (\alpha T)^{\mathrm{Cl}}=\bar\alpha\,T^{\mathrm{Cl}} \quad (\alpha\in\mathbb{K}) . $$

An operator is Clifford-self-adjoint when $T^{\mathrm{Cl}}=T$, and Clifford-skew when $T^{\mathrm{Cl}}=-T$.

Proof. Non-degeneracy of the pairing turns the functional $t\mapsto(Ts,t)_A$ into an element of the dual represented by a unique vector $T^{\mathrm{Cl}}s$, giving existence and uniqueness; the involution laws follow by applying the defining identity twice and to a composite, exactly as for the adjoint with respect to any non-degenerate sesquilinear form. The anti-linearity in the quaternionic case is the non-commutativity of the scalars. $\square$

Proposition (the regular module is real). On the regular module $\mathcal{S}=A$ with the pairing $(s,t)_A=s^{*}t$, the Clifford-adjoint of the left multiplication is the left multiplication

$$ (L_a)^{\mathrm{Cl}} = L_{a^{*}} , \qquad (R_b)^{\mathrm{Cl}} = R_{b^{*}} , $$

and the Clifford-adjoint of a two-sided operator $M\mapsto aMb$ is again two-sided.

Proof. Compute $(a s,t)_A=(as)^{*}t=s^{*}a^{*}t=(s,a^{*}t)_A$, which is the defining identity for $L_{a^{*}}$; the right multiplication is the same computation on the other side, and the two-sided statement is the two applications. The result is the Clifford instance of the algebraic identity $L_a^{*}=L_{a^{*}}$ of Part II's The Adjoint of the One-Sided Action with Hermitian Adjoint. $\square$

The Hermitian-Adjoint

Definition. The Hermitian-adjoint $T^{\dagger}$ of an operator on the module of sections is the adjoint with respect to the scalar form on $L^2(\Omega;\mathcal{S})$ of Hermitian Hilbert Modules over a Clifford Algebra:

$$ \langle Tf,g\rangle = \langle f,T^{\dagger}g\rangle , \qquad \langle f,g\rangle = \int_\Omega(f,g)_{\mathbb{K}}\,dx . $$

Theorem (properties). The Hermitian-adjoint exists for every bounded operator, is unique, and obeys

$$ (T^{\dagger})^{\dagger}=T , \qquad (ST)^{\dagger}=S^{\dagger}T^{\dagger} , \qquad (\alpha T)^{\dagger}=\bar\alpha\,T^{\dagger} , $$

and the operator is self-adjoint when $T^{\dagger}=T$, skew-adjoint when $T^{\dagger}=-T$, unitary when $T^{\dagger}T=TT^{\dagger}=1$. These are the adjoint properties of The Hilbert Adjoint on a Hilbert Module (Part II); the unbounded case is the one of Unbounded Operators and Spectral Measures.

Proof. Quoted from Part II's The Hilbert Adjoint on a Hilbert Module: the bounded-operator adjoint exists by the Riesz representation theorem applied to $g\mapsto\langle Tg,f\rangle$, and the laws follow as above. $\square$

Theorem (the adjoint of the multiplication). On $L^2(\Omega;\mathcal{S})$ the pointwise Clifford multiplication has $c(x)^{\dagger}=c(x^{*})$:

$$ c(x)^{\dagger} = c(x^{*}) , \qquad x\in A . $$

In particular the multiplication by a generator is skew-adjoint, $c(e_i)^{\dagger}=-c(e_i)$, and the multiplication by a unit scalar is unitary.

Proof. The pointwise module axiom $(x\cdot s,t)_{\mathbb{K}}=(s,x^{*}\cdot t)_{\mathbb{K}}$ integrates to the displayed identity. The relation between the two adjoints of the multiplication is the reason the Dirac operator is formally self-adjoint, and that algebraic reason is Part II's, in The Adjoint of the One-Sided Action with Hermitian Adjoint. $\square$

Remark (the two adjoints of one operator). Every operator has both adjoints. They agree on an $A$-linear operator because such an operator commutes with the module structure and the two pairings are both compatible with it; they differ on a general operator by the non-scalar part of the Clifford-valued form, which the scalar form forgets. The comparison is the subject of the next section.

The Comparison of the Two Adjoints

Theorem (the scalar form is the scalar part, and the adjoints agree on the intertwiners). Let the scalar form be the composition $\mathrm{Sc}\circ(\cdot,\cdot)_A$ of the Clifford pairing with a positive functional $\mathrm{Sc}$ that is central, $\mathrm{Sc}(ax)=\mathrm{Sc}(xa)$ for $a,x\in A$. Then for an $A$-linear operator $T$ (an intertwiner, $T\,c(x)=c(x)\,T$) the two adjoints coincide,

$$ T^{\dagger}=T^{\mathrm{Cl}} \qquad (T \ A\text{-linear}) , $$

and the Clifford-self-adjoint intertwiners are exactly the Hermitian-self-adjoint ones.

Proof. For $A$-linear $T$ the functional $t\mapsto(Ts,t)_A$ is $A$-linear and is represented inside the same form by an $A$-linear $T^{\mathrm{Cl}}$; taking the scalar part gives $(Ts,t)_{\mathbb{K}}=(s,T^{\mathrm{Cl}}t)_{\mathbb{K}}$ by centrality of $\mathrm{Sc}$, which identifies $T^{\dagger}=T^{\mathrm{Cl}}$. The converse is the same computation read backwards. The general theory of the commutant and the invariant operators is Part II's, in Invariant Operators and Intertwiners with Hermitian Adjoint. $\square$

Theorem (the unitary groups). The operators preserving the Clifford pairing form the Clifford-unitary group $\mathrm{U}(\mathcal{S},(\cdot,\cdot)_A)$ and those preserving the scalar form the unitary group $\mathrm{U}(\mathcal{S},(\cdot,\cdot)_{\mathbb{K}})$; on an irreducible definite module the second is the compact group of the spinor representation, and the first contains the scalars of modulus one that act trivially on the scalar form. On the regular module both contain the right multiplications $R_u$ with $u^{*}u=1$, and the Clifford-unitary condition $(us,ut)_A=(s,t)_A$ for all $s,t$ reads $u^{*}u=1$ for the left multiplications.

Proof. The preservation of each form is the defining equation of the corresponding group; the inclusions and the compactness are the statements of Part II's Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint and Spinor Adjoints and the Dirac Adjoint with Hermitian Adjoint, and the regular-module computation is immediate from $(us,ut)_A=(us)^{*}(ut)=s^{*}u^{*}ut$. $\square$

Remark (which adjoint is used where). The Clifford-adjoint is the one used when the algebra matters: in the duality of spinors, in the Fierz identity and the bilinear covariants, and in the definition of the Clifford-unitary group, where the algebra-valued form is the natural object. The Hermitian-adjoint is the one used when positivity and a Hilbert-space norm matter: in the $L^2$ theory, in the self-adjointness of the Dirac operator and its spectrum, and in the Szegő and Bergman projections. The two coincide on the $A$-linear operators, which is why the same operators are self-adjoint in both senses in the spinor theory, and they differ on the general operators, which is why the two must be distinguished in the analytic theory. This is the same distinction as the corpus makes between the involution on the elements and the adjoint on the operators: the star and the dagger, which agree exactly on a $\ast$-representation and need not agree otherwise.

The Adjoints of the Differential Operators

Theorem (the formal adjoints of the operator). On compactly supported sections the Hermitian-adjoint of the operator $D=\sum_\mu c(e_\mu)\partial_\mu$ is

$$ D^{\dagger}=-c(e_0)^{-1}\bar D \ \sim\ -\bar D , $$

and it is the operator that closes the factorisation $D\bar D=\Delta\,1_\mathcal{S}$; on a bounded domain with smooth boundary the failure of adjointness is the conormal term

$$ \langle Df,g\rangle-\langle f,D^{\dagger}g\rangle = \int_{\partial\Omega}(c(\nu_B)f,g)_{\mathbb{K}}\,dS . $$

Proof. The derivatives integrate by parts with a sign and the multiplications are skew or Hermitian by the theorem above; the boundary term is the flux of the vector field $c(e_\mu)f$ against $g$. The computation is the one of Hermitian Hilbert Modules over a Clifford Algebra, and the algebraic reason the multiplications make the result formally self-adjoint is Part II's The Adjoint of the One-Sided Action with Hermitian Adjoint. $\square$

Corollary (self-adjointness of the operator, two senses). The operator $iD$ is Hermitian-self-adjoint on a domain without boundary, and the vector operator $D_{\mathrm{sa}}=\sum_{i\ge1}c(e_i)\partial_i$ is self-adjoint with $D_{\mathrm{sa}}^2=-\Delta$ and a real spectrum; the Clifford-adjoint of $D$ is $-(D)^{\mathrm{Cl}}$ computed with the algebra-valued form, and on the regular module it agrees with the Hermitian one whenever the module is $A$-linear over the algebra. The spectral theory is Dirac Differential Operators'.

Proof. The multiplications by the generators are skew for both forms and the derivatives are formally skew, so $iD$ and $D_{\mathrm{sa}}$ are self-adjoint on the appropriate domains; the Clifford statement is the same computation with the algebra-valued form, and the agreement is the theorem of the preceding section applied to the $A$-linear coefficient algebra. $\square$

Remark (the Hermitian refinement). In the Hermitian refinement the differential part of these statements is refined once more: the two Hermitian Dirac operators are exchanged by the involution, the Hermitian Dirac operator $\mathcal{D}=\partial_{\underline z}+\partial_{\bar z}$ is symmetric with respect to the module form and essentially self-adjoint with real spectrum, and the Hermitian Cauchy kernel is the kernel of the adjoint operation that inverts the matrix operator. Those are the subjects of The Hermitian Dirac Operator and The Hermitian Cauchy Kernel as an Adjoint, and the adjoints on a Clifford module developed here are their one-operator background. The finite-dimensional Dirac element of Part II's Dirac Operators with Hermitian Adjoint is, once more, the symbol of the differential operator and not the operator itself; the passage from the one to the other is the spectral theorem, and the two adjoints distinguished in this article are the two senses in which that symbol is self-adjoint.

Summary

A Clifford module carries two natural pairings: the Clifford-valued pairing $(\cdot,\cdot)_A:\mathcal{S}\times\mathcal{S}\to A$, left $A$-linear and Hermitian for the conjugation $x\mapsto x^{*}$, non-degenerate on a definite module but isotropic; and the scalar Hermitian form $(\cdot,\cdot)_{\mathbb{K}}=\mathrm{Sc}\circ(\cdot,\cdot)_A$, positive definite when the Clifford pairing is positive. Each gives an adjoint: the Clifford-adjoint $T^{\mathrm{Cl}}$, defined by $(Ts,t)_A=(s,T^{\mathrm{Cl}}t)_A$, an anti-linear involution with $(ST)^{\mathrm{Cl}}=T^{\mathrm{Cl}} S^{\mathrm{Cl}}$, whose value on the regular module is $L_a^{\mathrm{Cl}}=L_{a^{*}}$ and $R_b^{\mathrm{Cl}}=R_{b^{*}}$; and the Hermitian-adjoint $T^{\dagger}$, the adjoint with respect to the scalar form, obeying the same involution laws, with $c(x)^{\dagger}=c(x^{*})$ for the pointwise multiplication and with the generators multiplying skew-adjointly. The two adjoints coincide on the $A$-linear operators — the intertwiners — and differ on general operators by the non-scalar part of the Clifford form; correspondingly the Clifford-unitary and the unitary groups of the module differ, the first containing the scalars that the second does not see. On the module of sections the Hermitian-adjoint of $D$ is $-\bar D$, formally, the failure on a bounded domain being the conormal term $\int_{\partial\Omega}(c(\nu_B)f,g)_{\mathbb{K}}\,dS$, so that $iD$ and $D_{\mathrm{sa}}$ are self-adjoint on a boundaryless domain with $D_{\mathrm{sa}}^2=-\Delta$ and a real spectrum. The algebraic one-sided theory is Part II's, in The Adjoint of the One-Sided Action with Hermitian Adjoint, Bilinear Operators on a Hermitian Module with Hermitian Adjoint and Spinor Adjoints and the Dirac Adjoint with Hermitian Adjoint; the module and the forms are Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint and Hermitian Hilbert Modules over a Clifford Algebra; the analytic adjoint is The Hilbert Adjoint on a Hilbert Module; and the Hermitian refinement is The Hermitian Dirac Operator and The Hermitian Cauchy Kernel as an Adjoint.

Summary of Notation

Symbol	Meaning
$A=\mathrm{Cl}_{0,m}$, $x\mapsto x^{*}$	Clifford algebra and the conjugation on its elements
$\mathcal{S}$	Left Clifford module; $\mathrm{Cl}_{0,m}$-module of values
$(\cdot,\cdot)_A$	Clifford-valued pairing; left $A$-linear, Hermitian for the star, isotropic
$(\cdot,\cdot)_{\mathbb{K}}=\mathrm{Sc}\circ(\cdot,\cdot)_A$	Scalar Hermitian form; positive definite
$T^{\mathrm{Cl}}$	Clifford-adjoint; $(Ts,t)_A=(s,T^{\mathrm{Cl}}t)_A$
$T^{\dagger}$	Hermitian-adjoint; $\langle Tf,g\rangle=\langle f,T^{\dagger}g\rangle$
$(ST)^{\mathrm{Cl}}=T^{\mathrm{Cl}}S^{\mathrm{Cl}}$, $(ST)^{\dagger}=S^{\dagger}T^{\dagger}$	Reversal of the factors for each adjoint
$L_a^{\mathrm{Cl}}=L_{a^{}}$, $R_b^{\mathrm{Cl}}=R_{b^{}}$	Adjoints of the multiplications on the regular module
$c(x)^{\dagger}=c(x^{*})$	Adjoint of the pointwise Clifford multiplication
$D$, $D^{\dagger}\sim-\bar D$, $\int_{\partial\Omega}(c(\nu_B)f,g)_{\mathbb{K}}dS$	Operator, formal Hermitian-adjoint, conormal boundary term
$iD$, $D_{\mathrm{sa}}=\sum_{i\ge1}c(e_i)\partial_i$	Self-adjoint variants; $D_{\mathrm{sa}}^2=-\Delta$
$\mathrm{U}(\mathcal{S},(\cdot,\cdot)_A)$, $\mathrm{U}(\mathcal{S},(\cdot,\cdot)_{\mathbb{K}})$	Clifford-unitary and unitary groups
$\mathrm{Sc}$	Positive central functional (scalar part) defining the scalar form