Hermitian Hilbert Modules over a Clifford Algebra

Introduction

The operators of Clifford analysis act on spaces of functions whose values lie in a Clifford module, and the analytic theory needs those spaces to be Hilbert spaces and the module structure to be compatible with the inner product. This article develops the resulting structure: a Clifford module that is at the same time a Hilbert space, with a Hermitian form for which the Clifford action is adjoint to the action of the conjugate element, so that the conjugation that defines the theory becomes an adjoint operation on the module.

The setting is the one of The Dirac Operator and Operators on a Clifford Module: the Clifford algebra $A=\mathrm{Cl}_{0,m}$, a left Clifford module $\mathcal{S}$ of values, the Clifford multiplication $c:A\to\mathrm{End}_\mathbb{R}(\mathcal{S})$, and the involution $*$ — the Clifford conjugation on the algebra, combined with the conjugation of the coefficient field when the values are quaternionic or complex. A Hermitian form on $\mathcal{S}$ is a positive-definite sesquilinear form for which the action obeys the self-adjointness axiom $(x\cdot s,t)=(s,x^{*}\cdot t)$. The finite-dimensional algebraic theory of such forms — the modules, the forms, the adjoints, the positivity and the cone — is Part II's, in Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint, The Adjoint of the One-Sided Action with Hermitian Adjoint, Positivity and the Hermitian Cone of a Hilbert Algebra with Hermitian Adjoint and The Hilbert Adjoint on a Hilbert Module; those articles are the algebraic reference and their results are used here. What this article adds is the analytic layer: the Hilbert completion of the module of sections, the boundedness of the action, the adjoint of the action with respect to the $L^2$ form, the self-adjointness of the operator $D$ after the boundary terms are accounted for, and the monogenic Hilbert spaces as closed subspaces. The Hermitian refinement of these objects with the split of the operator is Adjoints on a Clifford Module, The Hermitian Dirac Operator and The Hermitian Cauchy Kernel as an Adjoint; the reproducing kernels are Positive Definite Kernels in Clifford Analysis; the operator theory on the module is Operators on a Clifford Module.

Hermitian Forms on a Clifford Module

The Form and the Self-Adjointness Axiom

Definition. Let $\mathcal{S}$ be a left $A$-module, $A$ a real or complex $\ast$-algebra with involution $x\mapsto x^{*}$. A Hermitian form on $\mathcal{S}$ is a map $(\cdot,\cdot):\mathcal{S}\times\mathcal{S}\to\mathbb{K}$ ($\mathbb{K}=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$) that is sesquilinear over the coefficient field, $\mathbb{H}$-Hermitian in the quaternionic case, and positive definite, $(s,s)>0$ for $s\neq0$.

Definition. The form is compatible with the module structure when the Clifford action is adjoint to the action of the conjugate element:

$$ (x\cdot s,\,t) = (s,\,x^{*}\cdot t) \qquad x\in A,\ s,t\in\mathcal{S} , $$

equivalently $\rho(x)^{\dagger}=\rho(x^{*})$ for the representation $\rho=c$ of the action.

Proposition (the axioms are consistent and determine the module type). A Clifford module carries a compatible Hermitian form exactly when it is of the definite type; the form is unique up to a positive scalar on each irreducible summand, and the involution $*$ under which the axiom holds is the Clifford conjugation, possibly composed with the conjugation of the coefficient field. Under it a generator $e_i$ is skew, $(e_i\cdot s,t)=-(s,e_i\cdot t)$ for $i\ge1$, so the multiplication by a vector is skew-adjoint, and the scalar unit $e_0=1$ is Hermitian.

Proof. The skewness of $e_i$ is $e_i^{*}=-e_i$, which is the Clifford conjugation; the axiom then reads $(e_i\cdot s,t)=(s,(-e_i)\cdot t)=-(s,e_i\cdot t)$, which is skew-adjointness. The existence and uniqueness of a compatible definite form on a Clifford module are the results of Part II's Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint, and the involution is the one fixed there. $\square$

Remark (definite and indefinite). The definite case is the one in which the form is positive and the axiom holds as stated; in the indefinite case, where the form has a signature, the conjugation is replaced by the Krein adjoint and the theory becomes the Krein-space theory of Part II, in * $J$-Self-Adjoint and $J$-Unitary Operators and Spectral Theory on Krein Spaces. This article is the definite case, and the word Hermitian* throughout refers to the positive form and the Clifford conjugation.

The Hilbert Module of Sections

The $L^2$ Completion

Definition. Let $\Omega\subseteq\mathbb{R}^{m+1}$ be open and let $\mathcal{S}$ be a definite Clifford module with a compatible Hermitian form. The Hilbert module of sections is the completion

$$ L^2(\Omega;\mathcal{S}) = \overline{C_c^\infty(\Omega;\mathcal{S})}^{\ \|\cdot\|} , \qquad \|f\|^2 = \int_\Omega (f(x),f(x))\,dx , $$

of the compactly supported smooth sections in the norm of the inner product

$$ \langle f,g\rangle = \int_\Omega (f(x),g(x))\,dx . $$

Proposition (the module structure survives the completion). For $a$ in the algebra of bounded functions on $\Omega$ with values in $A$, the pointwise action $f\mapsto a\cdot f$ is a bounded operator on $L^2(\Omega;\mathcal{S})$, with $\|a\|\le\sup_x\|a(x)\|$; the involution acts by $a\mapsto a^{*}$ and the action satisfies $\rho(a)^{\dagger}=\rho(a^{*})$ with respect to the $L^2$ form. Hence $L^2(\Omega;\mathcal{S})$ is a Hilbert module over the algebra of bounded functions.

Proof. The bound is the pointwise estimate $\|a\cdot f\|\le\sup\|a\|\,\|f\|$; the adjoint identity is the pointwise axiom integrated over $\Omega$, the measure being real. $\square$

The Action of the Algebra

Remark (the Hilbert-algebra structure). The algebra of bounded functions with the involution $*$ and the inner product $\langle f,g\rangle=\int(f,g)$ is a Hilbert algebra in the sense of Part II, and $L^2(\Omega;\mathcal{S})$ is a Hermitian module over it; the general theory of the one-sided action, the commutant and the standard form is Part II's, in Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint and The Hilbert Adjoint on a Hilbert Module. The realisation here is the concrete one of Clifford analysis, in which the algebra is generated by the Clifford generators and the bounded functions and the module is the space of sections.

The Adjoint of the Clifford Action

The Adjoint of the Multiplication

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

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

In particular the multiplication by a generator is skew-adjoint and the multiplication by a unit scalar is unitary.

Proof. The pointwise axiom $(x\cdot s,t)=(s,x^{*}\cdot t)$ integrates to $\langle c(x)s,t\rangle=\langle s,c(x^{*})t\rangle$. $\square$

The Adjoint of the Operator $D$

Theorem (formal adjointness and the boundary term). On $L^2(\Omega;\mathcal{S})$ the operator $D=\sum_\mu c(e_\mu)\partial_\mu$ has formal adjoint $D^{\dagger}=-\bar D$ on compactly supported sections, and for sections on a bounded domain with smooth boundary,

$$ \langle Df,g\rangle-\langle f,D^{\dagger}g\rangle = \int_{\partial\Omega}\bigl(c(\nu_B)f,g\bigr)\,dS , $$

the boundary term being the conormal pairing of The Cauchy Integral Operator.

Proof. The derivatives integrate by parts with a sign, the multiplications are skew or Hermitian as in the preceding theorem, and the boundary term is the flux of the vector field $c(e_\mu)f$ against $g$; the computation is the one of The Dirac Operator for the formal adjoint $D^{\dagger}=-\bar D$. $\square$

Corollary (self-adjointness on a closed domain, and on the Hardy space). If the boundary term vanishes — for example on the boundary values of monogenic functions, or on a domain without boundary — then $D$ is skew-adjoint up to the conjugation, and the operator $D_{\mathrm{sa}}=\sum_{i\ge1}c(e_i) \partial_i$ is self-adjoint on the appropriate domain, with $D_{\mathrm{sa}}^2=-\Delta$ and a real spectrum. The spectral theory of that operator is Dirac Differential Operators', and the passage from the formal adjoint to the unbounded self-adjoint operator is the one of Unbounded Operators and Spectral Measures in Analysis on Linear Spaces.

The Monogenic Hilbert Spaces

Definition. The monogenic Bergman space is the closed subspace

$$ \mathcal{A}^2(\Omega;\mathcal{S}) = \{f\in L^2(\Omega;\mathcal{S}) : Df=0 \text{ in the distributional sense}\} = \ker D\cap L^2(\Omega;\mathcal{S}) , $$

and the monogenic Hardy space $H^2(\Omega;\mathcal{S})$ is the closure of the boundary values of the monogenic $L^2$ functions, as in The Cauchy Integral Operator.

Theorem (the monogenic spaces are Hilbert modules). $\mathcal{A}^2(\Omega;\mathcal{S})$ is a closed subspace of $L^2(\Omega;\mathcal{S})$, hence a Hilbert space and a Hermitian module over the algebra of bounded functions; the same holds for $H^2(\Omega;\mathcal{S})$. The orthogonal projection onto $\mathcal{A}^2$ is the Bergman projection, and onto $H^2$ the Szegő projection $P^+=\tfrac12(I+\mathcal{S})$ of The Cauchy Integral Operator; both are self-adjoint idempotents with respect to the module form.

Proof. $\ker D$ is closed in the distributional sense because $D$ is elliptic, hence $\mathcal{A}^2$ is a closed subspace; the module action preserves $\ker D$ because the coefficients are constant, so the subspace is a submodule; the projections are the orthogonal projections onto closed subspaces and are self-adjoint idempotents. The Szegő description is the boundary form of the same statement. $\square$

Remark (reproducing kernels). A monogenic Hilbert space with continuous point evaluations has a reproducing kernel, which for $\mathcal{A}^2$ and $H^2$ is a positive definite kernel in the sense of Positive Definite Kernels in Clifford Analysis; the Bergman kernel and the Szegő kernel are the two classical instances, and their relation to the Cauchy kernel is the integral theory of The Cauchy Integral Operator. The kernel is Hermitian in the sense of the module form, and this is the point at which the module structure and the Hilbert structure meet.

Summary

A Hermitian Hilbert module over a Clifford algebra is a definite Clifford module $\mathcal{S}$ with a positive Hermitian form for which the action obeys the self-adjointness axiom $(x\cdot s,t)=(s,x^{*}\cdot t)$, completed to a Hilbert space of sections $L^2(\Omega;\mathcal{S})=\overline{C_c^\infty(\Omega;\mathcal{S})}$. The algebra acts by bounded operators with $c(x)^{\dagger}=c(x^{*})$; a generator is skew-adjoint and the scalar unit is unitary. The operator $D$ has formal adjoint $D^{\dagger}=-\bar D$, the failure on a bounded domain being the conormal boundary term $\int_{\partial\Omega}(c(\nu_B)f,g)dS$; when that term vanishes, $D$ is skew and the vector operator $D_{\mathrm{sa}}$ is self-adjoint with $D_{\mathrm{sa}}^2=-\Delta$ and a real spectrum, whose spectral theory is Dirac Differential Operators'. The spaces of monogenic $L^2$ functions — the Bergman space $\mathcal{A}^2=\ker D\cap L^2$ and the Hardy space $H^2$ — are closed subspaces, hence Hilbert modules, and their orthogonal projections are the Bergman and Szegő projections, the second being the boundary projection of The Cauchy Integral Operator. The algebraic theory of the forms, the adjoints, the positivity and the cone is Part II's; the Hermitian refinement of the operator is Adjoints on a Clifford Module, The Hermitian Dirac Operator and The Hermitian Cauchy Kernel as an Adjoint; the reproducing kernels are Positive Definite Kernels in Clifford Analysis.

Summary of Notation

Symbol	Meaning
$A=\mathrm{Cl}_{0,m}$, $*$	Clifford algebra and its conjugation
$\mathcal{S}$	Definite Clifford module of values
$(\cdot,\cdot)$, $(x\cdot s,t)=(s,x^{*}\cdot t)$	Hermitian form and the self-adjointness axiom
$c(x)^{\dagger}=c(x^{*})$	Adjoint of the multiplication
$L^2(\Omega;\mathcal{S})$	Hilbert module of sections; $\langle f,g\rangle=\int(f,g)$
$D$, $D^{\dagger}=-\bar D$	Operator and its formal adjoint; conormal boundary term
$D_{\mathrm{sa}}=\sum_{i\ge1}c(e_i)\partial_i$	Self-adjoint vector operator; $D_{\mathrm{sa}}^2=-\Delta$
$\mathcal{A}^2(\Omega;\mathcal{S})=\ker D\cap L^2$	Monogenic Bergman space; Bergman projection
$H^2(\Omega;\mathcal{S})$, $P^+=\tfrac12(I+\mathcal{S})$	Monogenic Hardy space; Szegő projection
$c(\nu_B)$	Conormal multiplication in the boundary term