Hermitian Integral Kernels

Introduction

A Hermitian kernel is a kernel equal to its conjugate transpose, $K(y,x)=\overline{K(x,y)}$, and the integral operator of a Hermitian kernel is self-adjoint. The self-adjointness is only the beginning: such an operator has a real spectrum, its eigenfunctions are orthogonal, and the kernel is the expansion $$ K(x,y)=\sum_n\lambda_n\varphi_n(x)\overline{\varphi_n(y)} $$ in the eigenfunctions with the eigenvalues as coefficients. Positivity of the operator — the case of a positive definite kernel — is exactly the nonnegativity of the eigenvalues, and then the expansion converges absolutely and uniformly for a continuous kernel on a compact space (Mercer's theorem), the reproducing kernel Hilbert space of the kernel is the range of the square root with the square roots of the eigenvalues as its basis, and the trace and the Hilbert–Schmidt norm are read from the kernel. This article is the spectral theory of the Hermitian kernel, built from the involution of the previous articles.

It is the second article of the * Operator Theory group of Foundations of Analysis. Its prerequisites are Hermitian Kernels and the Integral Operator, the last article of the * Theory group, for the Hermitian and positive definite kernels and their dictionary with self-adjointness and positivity; The Adjoint of an Integral Operator, the previous article, for $T_K^\dagger=T_{K^*}$; Reproducing Kernel Hilbert Spaces and Positive Definite Functions and Hermitian Kernels, the first two articles of the * Theory group, for the reproducing space and Bochner's theorem; Banach and Hilbert Spaces and Compact Operators, later in this Part, for the spectral theorem for compact self-adjoint operators, the Fredholm theory and the positivity criterion, quoted as established; and Measure Theory and Integration for the product measure, Fubini and the monotone convergence theorem. The abstract spectral theorem in its projection-valued form, the square root of a positive operator, and the general theory of Hilbert–Schmidt and trace-class operators are Analysis on Linear Spaces, later in this Part; Mercer's uniform convergence is quoted from the classical theory and deferred in its detail to Compact Operators. The distributional generalisation is The Schwartz Kernel Theorem, later in this Part. No geometry is invoked.

The Hermitian Kernel and the Self-Adjoint Operator

The Hermitian Kernel

Definition. A kernel $K\in L^2(X\times X)$ is Hermitian if $K^*=K$, that is $K(y,x)=\overline{K(x,y)}$ for almost every $(x,y)$; it is symmetric if in addition it is real; and it is positive definite if $\sum_{i,j}\overline{c_i}c_jK(x_i,x_j)\geq0$ for every finite family of points and coefficients. A Hermitian integral kernel is the operator $T_K$ of a Hermitian kernel, $$ (T_Kf)(x)=\int_XK(x,y)f(y)\,d\mu(y) . $$

Self-Adjointness

Theorem. Let $K\in L^2(X\times X)$ and let $T_K$ be bounded on $L^2(X,\mu)$. Then $T_K$ is self-adjoint, $T_K^\dagger=T_K$, if and only if the kernel is Hermitian; equivalently $$ \langle T_Kf,g\rangle=\overline{\langle T_Kg,f\rangle}\qquad\text{for all }f,g\in L^2 , $$ the sesquilinear form of the operator being Hermitian, and $\langle T_Kf,f\rangle$ is real for every $f$.

Proof. The adjoint of $T_K$ is $T_{K^*}$ by The Adjoint of an Integral Operator, the previous article, so $T_K^\dagger=T_K$ is $K^*=K$ by the uniqueness of the kernel of a Hilbert–Schmidt operator; the form identity is the polarisation of that self-adjointness, and it holds for every square-integrable kernel because the finite data of the definition of positive definiteness are dense in the form's domain. $\blacksquare$

Positivity

Theorem (the positivity criterion). For a Hermitian kernel $K$, the following are equivalent: the kernel is positive definite; the operator is positive, $\langle T_Kf,f\rangle\geq0$ for every $f\in L^2$; and every eigenvalue of $T_K$ is nonnegative. If in addition the kernel is continuous and $X$ is compact with a finite measure, positivity is also equivalent to $K(x,x)\geq0$ together with the nonnegativity of the eigenvalues, by Mercer's theorem below.

Proof. The equivalence of the first two is the dictionary of Hermitian Kernels and the Integral Operator, the previous group; the equivalence with the third is the spectral theorem below, since the eigenvalues are the values of the quadratic form on the eigenfunctions, and a self-adjoint operator is positive exactly when its spectrum is contained in $[0,\infty)$ by Banach and Hilbert Spaces, later in this Part. The last statement is Mercer's theorem. $\blacksquare$

The Spectral Theory of the Kernel

Compactness and the Real Spectrum

Theorem (the spectral theorem for the Hermitian Hilbert–Schmidt kernel). Let $K\in L^2(X\times X)$ be Hermitian and let $T_K$ be its integral operator. Then $T_K$ is self-adjoint and compact, its eigenvalues $\lambda_n$ are real and tend to $0$, the eigenvectors $\varphi_n$ may be taken orthonormal, $$ T_K\varphi_n=\lambda_n\varphi_n,\qquad T_Kf=\sum_n\lambda_n\langle f,\varphi_n\rangle\varphi_n , $$ and the eigenvalue list is the only spectral data of the operator, $\lVert T_K\rVert=\max_n\lvert\lambda_n\rvert$.

Proof. An integral operator with an $L^2$ kernel is Hilbert–Schmidt and hence compact, being the norm limit of its finite-rank truncations in the kernel expansion; a compact self-adjoint operator has a countable real spectrum with an orthonormal eigenbasis and $\lVert T_K\rVert=\max\lvert\lambda_n\rvert$, by the spectral theory of Compact Operators and Banach and Hilbert Spaces, later in this Part. $\blacksquare$

The Eigenfunction Expansion of the Kernel

Theorem (the kernel expansion). Under the hypotheses of the spectral theorem, $$ K(x,y)=\sum_n\lambda_n\varphi_n(x)\overline{\varphi_n(y)} , $$ the series converging in $L^2(X\times X)$; and $\lVert K\rVert_{L^2}^2=\sum_n\lambda_n^2$.

Proof. The kernel of $T_K$ is recovered from the operator by $K(x,y)=\sum_n(T_K\varphi_n)(x)\overline{\varphi_n(y)}$ in the $L^2$ sense, as in the isometry of the * Theory article, and the eigenrelation gives the stated expansion; the norm identity is Parseval's identity applied to the expansion. $\blacksquare$

Example (the Green function of the Dirichlet problem). On $X=[0,1]$ with the Lebesgue measure the kernel $$ G(x,y)=\min(x,y)-xy $$ is real, symmetric and continuous, hence Hermitian and self-adjoint; the operator $T_G$ is the inverse of $-\frac{d^2}{dx^2}$ with the vanishing boundary conditions, so its eigenfunctions are $\varphi_n(x)=\sqrt2\sin(n\pi x)$ and its eigenvalues are $\lambda_n=1/(n^2\pi^2)>0$. The kernel is positive definite, the expansion $\min(x,y)-xy=\sum_{n\geq1}\frac2{n^2\pi^2}\sin(n\pi x)\sin(n\pi y)$ converges absolutely and uniformly, and $\sum_n\lambda_n=1/6=\int_0^1G(x,x)\,dx$.

Example (the covariance kernel). The kernel $K(x,y)=\min(x,y)$ on $[0,1]$ is continuous, symmetric and positive definite, so its operator is compact, self-adjoint and positive, with a nonnegative eigenvalue list and the uniform Mercer expansion; it is the sum of the Green kernel above and the kernel $xy$ of the rank-one projection on the constant function, and this is the two-term decomposition of the covariance into its part vanishing at $0$ and its mean.

Positive Definite Kernels and Nonnegative Spectrum

The Equivalence

Theorem. A Hermitian kernel is positive definite if and only if its eigenvalues are nonnegative; then $T_K$ is a positive operator, its square root $T_K^{1/2}$ is the operator with eigenvalues $\sqrt{\lambda_n}$, $$ T_K^{1/2}f=\sum_n\sqrt{\lambda_n}\langle f,\varphi_n\rangle\varphi_n , $$ and the pair $(T_K,T_K^{1/2})$ is the spectral square root of the positive kernel.

Proof. The equivalence is the positivity criterion above; the square root is the operator with the square-root eigenvalues, and its square is $T_K$ by the eigenrelation, so it is the unique positive square root once uniqueness is known, by the positivity of the eigenvalues. The functional calculus that defines $T_K^{1/2}$ in general is Analysis on Linear Spaces, later in this Part. $\blacksquare$

Mercer's Theorem

Theorem (Mercer). Let $X$ be a compact space with a finite measure $\mu$ and let $K$ be a continuous positive definite kernel on $X\times X$. Then the eigenvalues $\lambda_n$ are nonnegative, the eigenfunctions $\varphi_n$ may be taken continuous and orthonormal, and $$ K(x,y)=\sum_n\lambda_n\varphi_n(x)\overline{\varphi_n(y)} , $$ the series converging absolutely and uniformly on $X\times X$. In particular $K(x,x)\geq0$ for every $x$, and the convergence of the diagonal is the convergence of $\sum_n\lambda_n\lvert\varphi_n(x)\rvert^2$.

Proof. A continuous kernel on a compact space with a finite measure defines a compact self-adjoint operator with continuous eigenfunctions by the classical argument, and the nonnegative eigenvalues give the absolute and uniform convergence of the expansion; the positivity of the eigenvalues is the positivity of the kernel. The uniform convergence is quoted from Compact Operators, later in this Part, where the Riesz–Schauder theory is developed. $\blacksquare$

The Reproducing Space and the Trace

The Reproducing Space as the Range of the Square Root

Theorem. For a continuous positive definite kernel $K$ on a compact space with a finite measure, the reproducing kernel Hilbert space $H_K$ of Reproducing Kernel Hilbert Spaces is the range of the square root, $$ H_K=T_K^{1/2}(L^2),\qquad \lVert T_K^{1/2}g\rVert_{H_K}=\lVert g\rVert_{L^2}\quad(g\perp\ker T_K) , $$ and, on the eigenfunctions, $H_K$ consists of the functions $f=\sum_na_n\varphi_n$ with $\sum_n\lvert a_n\rvert^2/\lambda_n<\infty$, the sum over the nonzero eigenvalues, $$ \lVert f\rVert_{H_K}^2=\sum_n\frac{\lvert a_n\rvert^2}{\lambda_n}, \qquad \langle f,K(\cdot,x)\rangle=f(x) . $$

Proof. The functions $\sqrt{\lambda_n}\varphi_n$ for the nonzero eigenvalues form an orthonormal basis of $H_K$, so $H_K$ is the range of the map $g\mapsto\sum_n\sqrt{\lambda_n}\langle g,\varphi_n\rangle\varphi_n$ on the orthogonal complement of $\ker T_K$; the reproducing property is the expansion of the kernel, by the eigenfunction expansion of the previous section. $\blacksquare$

The Trace

Corollary (the trace identities). If $K$ is continuous positive definite and the eigenvalues are summable, then $$ \operatorname{tr}T_K=\sum_n\lambda_n=\int_XK(x,x)\,d\mu(x), \qquad \lVert K\rVert_{L^2}^2=\lVert T_K\rVert_{\mathrm{HS}}^2=\sum_n\lambda_n^2 . $$

Proof. The first identity is the integral of the diagonal of the Mercer expansion, legitimate by the monotone convergence theorem applied to the nonnegative expansion of $K(x,x)$; the second is Parseval's identity for the eigenfunction expansion, equivalently the isometry of the Hilbert–Schmidt correspondence of the * Theory article. $\blacksquare$

Example (the Green function, continued). For $G(x,y)=\min(x,y)-xy$ the diagonal integral is $\int_0^1(x-x^2)dx=1/6$ and the eigenvalue sum is $\sum_{n\geq1}1/(n^2\pi^2)=\frac1{\pi^2}\cdot\frac{\pi^2}6=1/6$, so the trace identity holds; the Hilbert–Schmidt norm is $\lVert G\rVert_{L^2}^2=\sum_{n\geq1}\pi^{-4}n^{-4}=\pi^{-4}\cdot\frac{\pi^4}{90}=1/90$.

Summary

A Hermitian kernel $K(y,x)=\overline{K(x,y)}$ has the self-adjoint integral operator $T_K$, characterised by the reality of the sesquilinear form and equivalent to the Hermitian condition by the adjoint kernel $K^*(x,y)=\overline{K(y,x)}$; positivity of the operator is the positive definiteness of the kernel, and both are equivalent to the nonnegativity of the eigenvalues. A Hermitian Hilbert–Schmidt kernel has a real eigenvalue list tending to $0$ with an orthonormal eigenbasis, and the kernel is the expansion $K(x,y)=\sum_n\lambda_n\varphi_n(x)\overline{\varphi_n(y)}$ converging in $L^2$, with $\lVert K\rVert_{L^2}^2=\sum_n\lambda_n^2$. For a continuous positive definite kernel on a compact space the eigenvalues are nonnegative, the expansion converges absolutely and uniformly (Mercer's theorem) and the diagonal is nonnegative; the reproducing kernel Hilbert space is the range of the square root, $H_K=T_K^{1/2}(L^2)$, with the orthonormal basis $\sqrt{\lambda_n}\varphi_n$ and the norm $\sum_n\lvert a_n\rvert^2/\lambda_n$, and the trace identities $\operatorname{tr}T_K=\sum_n\lambda_n=\int K(x,x)d\mu$ and $\lVert K\rVert_{L^2}^2=\sum_n\lambda_n^2$ hold. The Green kernel $\min(x,y)-xy$ of the Dirichlet problem, with eigenvalues $1/(n^2\pi^2)$ and trace $1/6$, is the standard illustration. The abstract projection-valued spectral theorem and the general square root are owned by later articles.

Summary of Notation

Symbol	Meaning
$K^*$, Hermitian	$K^(x,y)=\overline{K(y,x)}$, $K^=K$
$T_K^\dagger=T_K$	Self-adjointness of a Hermitian kernel
positive definite kernel	$\sum_{i,j}\overline{c_i}c_jK(x_i,x_j)\geq0$
$\lambda_n$, $\varphi_n$	Eigenvalues and orthonormal eigenfunctions, $T_K\varphi_n=\lambda_n\varphi_n$
$K(x,y)=\sum_n\lambda_n\varphi_n(x)\overline{\varphi_n(y)}$	Eigenfunction expansion of the kernel
$T_K^{1/2}$	Positive square root, eigenvalues $\sqrt{\lambda_n}$
$H_K=T_K^{1/2}(L^2)$	Reproducing space as the range of the square root
$\lVert f\rVert_{H_K}^2=\sum_n\lvert a_n\rvert^2/\lambda_n$	Reproducing norm
$\operatorname{tr}T_K=\int K(x,x)d\mu=\sum_n\lambda_n$	Trace identity