The Hermitian Cauchy Kernel as an Adjoint

Introduction

An integral kernel is not only a function of two variables but the matrix of an operator, and the kernel of the adjoint operator is the conjugate-reverse of the kernel: if $T$ has kernel $K(x,y)$, then $T^{\dagger}$ has kernel $K(y,x)^{*}$. The Hermitian Cauchy kernel $E_r(Z)=Z_r^{*}/(a_{4n}|Z|^{4n})$ is, literally, built from the adjoint variable $Z_r^{*}$ of the Hermitian theory, and this article reads the kernel in that way: as the kernel of the adjoint, as the object implementing the adjoint pairing between the Hermitian monogenic functions and the boundary densities, and as the entry of the matrix that is the inverse of the adjoint matrix operator. In this reading the integral theory of The Hermitian Cauchy Integral and the Boundary Values becomes a statement about adjoints, and the boundary projections of the monogenic Hardy space appear as self-adjoint idempotents.

The setting is that of Hermitian Quaternionic Analysis and the Conjugate Cauchy–Riemann Operator: the algebra $\mathbb{H}_N$, the four Hermitian operators $\partial_{Z_r}$ and their conjugates $\partial_{Z_r}^{\dagger}$, the Hermitian variables $Z_r$ and their adjoints $Z_r^{*}$, and the kernels $E_r(Z)=Z_r^{*}/(a_{4n}|Z|^{4n})$. The module and form background is Hermitian Hilbert Modules over a Clifford Algebra; the operator-theoretic statements about the Hermitian Dirac operator are The Hermitian Dirac Operator; the integral formulae and the matrix device are The Hermitian Cauchy Integral and the Boundary Values; the one-operator statements — the Cauchy transform, the Hilbert transform, the self-adjointness of the projection — are The Cauchy Integral Operator and The Hilbert Adjoint on a Hilbert Module (Part II), and the general kernel theory of the adjoint is Analysis on Linear Spaces', in Hermitian Kernels and the Integral Operator.

Kernels and their Adjoints

The General Principle

Theorem (the adjoint kernel). Let $K$ be a distributional kernel on $X\times X$ and let $T$ be the operator $Tf(x)=\int_XK(x,y)f(y)\,dy$. Then the adjoint with respect to the module form has kernel

$$ K^{\dagger}(x,y)=K(y,x)^{*} , $$

the conjugate-reverse of $K$. In particular the kernel is the kernel of a self-adjoint operator exactly when $K(y,x)^{*}=K(x,y)$ — the Hermitian symmetry of Positive Definite Kernels in Clifford Analysis — and the kernel is that of an antiself-adjoint operator exactly when $K(y,x)^{*}=-K(x,y)$.

Proof. Compute $\langle Tf,g\rangle=\int\!\!\int(K(x,y)f(y),g(x))\,dy\,dx$ and move the form to the other factor using its sesquilinearity and the module axiom; the resulting expression is $\int\!\!\int(f(y),K(x,y)^{*}g(x))\,dy\,dx$, which is $\langle f,T^{\dagger}g\rangle$ with the kernel $K(y,x)^{*}$ after the interchange of the variables. The statement is the Clifford-module instance of the general kernel theorem of Hermitian Kernels and the Integral Operator. $\square$

Remark (why the conjugate-reverse is the object of this article). The Euclidean and Hermitian Cauchy kernels are built from the conjugate variable: $F_r(X)=-X_r/(a_{4n}|X|^{4n})$ is odd, $F_r(-X)=-F_r(X)$, and $E_r(Z)=Z_r^{*}/(a_{4n}|Z|^{4n})$ carries the conjugated variable in the numerator and the conjugated value in the kernel. The recursion $K\mapsto K^{\dagger}$ is therefore not an afterthought: the kernels of the Hermitian theory are constructed from the adjoint, and the whole integral theory is a theory of adjoint pairings. This is the reason the Hermitian Cauchy kernel is naturally read as an adjoint, and the reason the boundary projections of the theory come out self-adjoint.

The Hermitian Cauchy Kernel

Proposition (the Hermitian Cauchy kernel and the adjoint variable). The Hermitian kernel $E_r(Z)=Z_r^{*}/(a_{4n}|Z|^{4n})$ is odd, $E_r(-Z)=-E_r(Z)$, and is built from the adjoint Hermitian variable $Z_r^{*}$; under the involution $*$ the kernel is carried to the kernel of the conjugate operator,

$$ (E_r)^{*}(Z) \sim Z_r/(a_{4n}|Z|^{4n}) , $$

so the four Hermitian Cauchy kernels pair into the four conjugate Hermitian Dirac operators, and the matrix $E$ is correspondingly the adjoint partner of the matrix $D$ of the operators.

Proof. The oddness is immediate from $(Z_r)^{*}=-Z_r$-type skewness of the variables and the evenness of $|Z|$; the involution statement is the conjugation of the numerator and of the coefficients, with the same normalisation; the matrix statement is that $D^{\mathsf T}E=\delta I$ of The Hermitian Cauchy Integral and the Boundary Values, which exhibits $E$ as the kernel of the operation inverse to the adjoint matrix operator. The verification in $\mathbb{H}\otimes_{\mathbb{R}}\mathrm{Cl}_{0,4}$ is in Clifford Analysis. $\square$

Theorem (the matrix kernel is the adjoint partner of the matrix operator). With the circulant matrix $D$ of the four Hermitian Dirac operators, the circulant matrix $E$ of the four Hermitian kernels satisfies

$$ D^{\mathsf T}E = \delta I , \qquad E\,D^{\mathsf T} = \delta I \ \text{ on the appropriate domain} , $$

so the matrix kernel is the two-sided inverse of the adjoint matrix operator on the level of distributions, and it is the kernel of the adjoint operation that inverts $D^{\mathsf T}$.

Proof. The first identity is the fundamental-solution theorem of The Hermitian Cauchy Integral and the Boundary Values; the second follows by the Hermitian symmetry of the matrix kernel and the commutativity of the circulant matrices, or by the second Borel–Pompeiu formula. $\square$

The Adjoint Pairing of the Cauchy Theory

Remark (the pairing). The Hermitian Cauchy theory is a duality between two spaces: the $q$-Hermitian monogenic functions on a domain and the densities on the boundary, paired by the integral of the pointwise module form,

$$ \langle f,\mu\rangle = \int_{\partial\Omega}(f,\mu) , $$

and the Cauchy kernel is the object that realises the duality: the transform $\mathcal{C}$ with kernel $E\,N^{\mathsf T}$, and its adjoint $\mathcal{C}^{\dagger}$ with the conjugate-reverse kernel, are the two members of the pairing, and the Cauchy integral formula is the statement that the pairing of the boundary values reproduces the interior function. In the complex refinement this pairing is the Martinelli–Bochner pairing of forms in several complex variables, and the conjugate-reverse kernel is the conjugate form; the general theory of the several-variable pairing is Several Complex Variables'.

Theorem (the boundary projections are self-adjoint). The Cauchy transform $\mathcal{C}$ has adjoint $\mathcal{C}^{\dagger}=I-\mathcal{C}$ up to the parity of the kernel, so that the boundary operators $\mathcal{C}^{\pm}$ of the Plemelj–Sokhotski decomposition of The Cauchy Integral Operator are complementary projections and the singular operator is self-adjoint with respect to the module form:

$$ (\mathcal{C}^{+})^{\dagger}=\mathcal{C}^{+} , \qquad (\mathcal{C}^{-})^{\dagger}=\mathcal{C}^{-} , \qquad \mathcal{S}^{\dagger}=\mathcal{S} , \qquad \mathcal{S}^2=I . $$

Proof. The one-operator statement is that of The Cauchy Integral Operator; passing to the matrix of the Hermitian theory, the conjugate-reverse kernel of the matrix transform is the same matrix transform with the conormal on the other side, and the Hermitian symmetry of the kernel $E$ and the reality of the conormal element give the self-adjointness of the singular operator. $\square$

Corollary (the Szegő projection and the adjoint). The Szegő projection $P^+$ of the monogenic Hardy space is self-adjoint, and its kernel — the Szegő kernel of Positive Definite Kernels in Clifford Analysis — is the Cauchy kernel read on the boundary; the Bergman projection is likewise self-adjoint. Hence the two projections of the monogenic $L^2$ theory are orthogonal projections in the sense of the module form, and the structure they define is the one of Hermitian Hilbert Modules over a Clifford Algebra.

Proof. A projection is self-adjoint exactly when it is the orthogonal projection onto its range; the range of $P^+$ is the monogenic Hardy space, closed by Hermitian Hilbert Modules over a Clifford Algebra, and the reproducing property of its kernel makes $P^+$ the orthogonal projection. $\square$

The Jump Relations

Theorem (the Plemelj formulae of the Hermitian Cauchy kernel). For a density $f$ of Hölder class on the boundary the one-sided extensions $\mathcal{C}^{\pm}f$ exist as continuous limits and satisfy the Plemelj–Sokhotski formulae

$$ \mathcal{C}^{\pm}f = \pm\tfrac12 f + \mathcal{S}f , \qquad \mathcal{C}^{+}f-\mathcal{C}^{-}f = f , \qquad \mathcal{C}^{+}f+\mathcal{C}^{-}f = 2\mathcal{S}f , $$

with $\mathcal{S}=2\mathcal{C}^{+}-I$ the singular operator of The Cauchy Integral Operator; the jump $\mathcal{C}^{+}-\mathcal{C}^{-}$ is the identity, and the sum is twice the singular operator.

Proof. The formulae are the decompositions of the identity with the projections $\mathcal{C}^{\pm}=\tfrac12(I\pm\mathcal{S})$; the existence of the limits is the Plemelj calculus of The Cauchy Integral Operator and The Hermitian Cauchy Integral and the Boundary Values read in the Hermitian refinement. $\square$

Theorem (the jump of the adjoint). The adjoint of the jump is the jump of the adjoint: with the adjoint taken with respect to the module form,

$$ (\mathcal{C}^{+}-\mathcal{C}^{-})^{\dagger} = I , \qquad (\mathcal{C}^{+}+\mathcal{C}^{-})^{\dagger} = 2\mathcal{S} , $$

so the jump operator is self-adjoint and the singular operator is self-adjoint. Read through the conjugate-reverse rule, the statement is that the difference of the two boundary values of the kernel $K(y,x)^{*}$ is the kernel of the identity, which is the distributional form of the jump relation.

Proof. The jump is the identity by the theorem above, and the identity is self-adjoint; the sum is twice $\mathcal{S}$, whose self-adjointness is the preceding section. The kernel form of the jump is the conjugate-reverse of the Plemelj formula $K\,N^{\mathsf T}$ computed on the two sides, where the two sides differ by the conormal term of the Cauchy–Pompeiu formula. $\square$

Remark (the jump as the boundary of the adjoint pairing). The jump relation and the adjointness of the boundary operators are two readings of the same structure: the pairing $\langle f,\mu\rangle=\int_{\partial\Omega}(f,\mu)$ of the Cauchy theory is non-degenerate, the kernel is its reproducing object, and the difference of the two one-sided extensions of the transform is the identity exactly because the transform is a projection differing from its adjoint by the parity of the kernel. In the Hermitian refinement the jump is the identity on the boundary spinors, the singular operator is the reflection $\mathcal{S}=\mathcal{C}^{+}-\mathcal{C}^{-}$ of the splitting $\mathcal{H}=P^+L^2\oplus P^-L^2$ of the boundary space, and the two projections are the self-adjoint projections of the monogenic Hardy theory.

Summary

A kernel $K(x,y)$ is the matrix of an operator, and the kernel of the adjoint is the conjugate-reverse $K^{\dagger}(x,y)=K(y,x)^{*}$; the kernel is self-adjoint exactly when it is Hermitian in the sense of Positive Definite Kernels in Clifford Analysis. The Hermitian Cauchy kernel $E_r(Z)=Z_r^{*}/(a_{4n}|Z|^{4n})$ is built from the adjoint Hermitian variable, is odd, and is carried by the involution to the kernel of the conjugate operator; the four kernels pair with the four conjugate Hermitian Dirac operators, and the circulant matrix $E$ is the two-sided inverse of the adjoint matrix operator, $D^{\mathsf T}E=\delta I$. The Hermitian Cauchy theory is a duality: the kernel realises the adjoint pairing between the $q$-Hermitian monogenic functions and the boundary densities, whose complex refinement is the Martinelli–Bochner pairing of several complex variables, and the conjugate-reverse kernel is the conjugate form. The boundary operators of the Plemelj–Sokhotski decomposition are complementary self-adjoint projections with respect to the module form, $\mathcal{S}^{\dagger}=\mathcal{S}$ and $\mathcal{S}^2=I$, and the Szegő and Bergman projections of the monogenic spaces are likewise self-adjoint, their kernels being the reproducing kernels of Positive Definite Kernels in Clifford Analysis. The one-operator statements are The Cauchy Integral Operator's and The Hilbert Adjoint on a Hilbert Module's; the integral formulae and the matrix device are The Hermitian Cauchy Integral and the Boundary Values; the operator-theoretic companion is The Hermitian Dirac Operator; and the general kernel theory of adjoints is Analysis on Linear Spaces'.

Summary of Notation

Symbol	Meaning
$K(x,y)$, $K^{\dagger}(x,y)=K(y,x)^{*}$	Kernel and the kernel of the adjoint
$K(y,x)^{*}=K(x,y)$	Hermitian symmetry (self-adjoint kernel)
$E_r(Z)=Z_r^{*}/(a_{4n}\|Z\|^{4n})$	Hermitian Cauchy kernel built from the adjoint variable
$E_r(-Z)=-E_r(Z)$	Oddness of the Hermitian kernel
$D^{\mathsf T}E=\delta I$	Matrix kernel as the inverse of the adjoint matrix operator
$\langle f,\mu\rangle=\int_{\partial\Omega}(f,\mu)$	Adjoint pairing of functions and densities
$\mathcal{C}^{\pm}$, $\mathcal{S}=2\mathcal{C}^+-I$	Plemelj projections and singular operator; $\mathcal{S}^{\dagger}=\mathcal{S}$, $\mathcal{S}^2=I$
$P^+$, Szegő kernel	Self-adjoint Szegő projection and its Cauchy-kernel realisation