The Adjoint of an Integral Operator

Introduction

The adjoint of an operator is the operator that moves the operator to the other side of the pairing, and for an integral operator it is again an integral operator, whose kernel is the conjugate transpose of the original one. If the pairing is the Hilbert pairing $\langle f,g\rangle=\int f\bar g$, then $$ T_K^\dagger=T_{K^*},\qquad K^*(x,y)=\overline{K(y,x)} , $$ and the operator $T_K$ is self-adjoint exactly when its kernel is Hermitian. The passage $T\mapsto T^\dagger$ is the involution on the operators, the archetype of the * Operator Theory group: it is conjugate-linear, involutive, an isometric anti-automorphism of the operator algebra, and it carries the element involution $K\mapsto K^*$ of the kernel to the operator adjoint. This article derives that correspondence, distinguishes the two adjoints of an integral operator — the Hilbert adjoint, taken with respect to the sesquilinear pairing, and the transpose, taken with respect to the bilinear pairing — and identifies the Hermitian case.

The article is the first of the * Operator Theory group of Foundations of Analysis. Its prerequisites are The Integral Operator, the first article of this category, for the integral operator and the kernel; Banach and Hilbert Spaces, later in this Part, for the adjoint of a bounded operator, its uniqueness and its algebraic properties, quoted as established; Conventions in Mathematics for the marks, where the dagger is the adjoint of an operator and the star the involution of an element; and Hermitian Kernels and the Integral Operator, the last article of the * Theory group, for the kernel involution and the dictionary of the two conditions. The spectral theory that the Hermitian case supports is Hermitian Integral Kernels, the next article; the general adjoint of a bounded operator, with domain considerations for the unbounded case, is The Adjoint of a Bounded Operator and the later unbounded-operator material, both later in this Part; the distributional kernel is The Schwartz Kernel Theorem, later in this Part. No geometry is invoked.

The Pairing and the Adjoint

The Pairing

Throughout, $X$ is a measure space with a $\sigma$-finite measure $\mu$, and $L^2=L^2(X,\mu)$ carries the Hilbert pairing $$ \langle f,g\rangle=\int_Xf\bar g\,d\mu , $$ conjugate-linear in the second slot and linear in the first. The bilinear pairing is $$ \{f,g\}=\int_Xfg\,d\mu , $$ with no conjugation, defined for the pairs for which the integral converges. The two pairings differ by the conjugate on the second slot, and they produce two different adjoints of the same operator.

The Adjoint of a Bounded Operator

Definition. For a bounded operator $T:L^2\to L^2$, the adjoint $T^\dagger$ is the unique bounded operator with $$ \langle Tf,g\rangle=\langle f,T^\dagger g\rangle\qquad\text{for all }f,g\in L^2 ; $$ it exists by the Riesz representation theorem, and it is characterised by those identities.

Theorem (properties of the adjoint). The map $T\mapsto T^\dagger$ is conjugate-linear and involutive, $$ (T+S)^\dagger=T^\dagger+S^\dagger,\qquad (\lambda T)^\dagger=\bar\lambda T^\dagger,\qquad T^{\dagger\dagger}=T , $$ it reverses products, $$ (ST)^\dagger=S^\dagger T^\dagger , $$ and it is isometric, $\lVert T^\dagger\rVert=\lVert T\rVert$. Consequently $T$ is self-adjoint if $T^\dagger=T$, unitary if $T^\dagger T=TT^\dagger=\mathrm{id}$, and normal if $TT^\dagger=T^\dagger T$.

Proof. The properties are those of the adjoint of a bounded operator in Banach and Hilbert Spaces, later in this Part, and are quoted from there; the anti-multiplicativity is the computation $\langle STf,g\rangle=\langle Tf,S^\dagger g\rangle=\langle f,T^\dagger S^\dagger g\rangle$, and the isometry is $\lVert T^\dagger\rVert=\sup_{\lVert g\rVert=1}\sup_{\lVert f\rVert=1}\lvert\langle Tf,g\rangle\rvert=\lVert T\rVert$. $\blacksquare$

Remark (the marks). By Conventions in Mathematics the dagger names the adjoint of an operator and the star names the involution of an element, the layer of the * Theory group. The two are distinct structures, and the identity $T_K^\dagger=T_{K^*}$ below is the statement that, for integral operators, the element involution $K\mapsto K^*$ produces the operator adjoint $T\mapsto T^\dagger$; that agreement is proved, never assumed.

The Transpose and the Adjoint

Definition. With respect to the bilinear pairing, the transpose of a bounded operator $T$ is the operator $T^{\mathrm t}$ with $$ \{Tf,g\}=\{f,T^{\mathrm t}g\}\qquad\text{for all }f,g , $$ when it exists. For an integral operator the transpose has the transposed kernel, while the Hilbert adjoint has the conjugate transposed kernel; the two coincide exactly for real kernels.

Theorem. Let $T_K$ be bounded with kernel $K$. Then provided the interchanges below are legitimate, $$ T_K^{\mathrm t}=T_{K^{\mathrm t}},\quad K^{\mathrm t}(x,y)=K(y,x), \qquad T_K^\dagger=T_{K^*},\quad K^*(x,y)=\overline{K(y,x)} , $$ so that the transpose, the adjoint and the conjugate are related by $K^*=\overline{K^{\mathrm t}}$ and $T_K^\dagger=\overline{T_K^{\mathrm t}}$.

Proof. For the transpose, $$ \{T_Kf,g\}=\iint K(x,y)f(y)g(x)\,d\mu(y)d\mu(x) =\int f(y)\Bigl(\int K(x,y)g(x)\,d\mu(x)\Bigr)d\mu(y)=\{f,T_{K^{\mathrm t}}g\} , $$ the middle integral being $(T_{K^{\mathrm t}}g)(y)$ with the kernel $K(y,x)$; the Hilbert case replaces $g$ by $\bar g$ throughout and produces $\overline{K(y,x)}$. The two differ by the conjugation on the second slot, which is exactly the relation $K^*=\overline{K^{\mathrm t}}$. $\blacksquare$

The Adjoint of an Integral Operator

The Kernel of the Adjoint

Theorem (the adjoint kernel). Let $K\in L^2(X\times X)$ and let $T_K$ be the integral operator of $K$, bounded on $L^2$. Then $$ T_K^\dagger=T_{K^*},\qquad K^*(x,y)=\overline{K(y,x)} , $$ and $T_{K^*}$ is the integral operator of the conjugate transpose kernel, which lies in $L^2(X\times X)$ with the same norm, $\lVert K^*\rVert_{L^2}=\lVert K\rVert_{L^2}$.

Proof. By Fubini and the definition of the pairing, $$ \langle T_Kf,g\rangle=\iint K(x,y)f(y)\overline{g(x)}\,d\mu(y)d\mu(x) =\iint f(y)\overline{\Bigl(\int\overline{K(x,y)}g(x)\,d\mu(x)\Bigr)}\,d\mu(y) =\langle f,T_{K^*}g\rangle , $$ and the inner integral is $(T_{K^*}g)(y)$ with $K^*(y,x)=\overline{K(x,y)}$, that is $K^*(x,y)=\overline{K(y,x)}$. The uniqueness of the adjoint identifies $T_K^\dagger$ with $T_{K^*}$, and the norm identity is the substitution $x\leftrightarrow y$. $\blacksquare$

Corollary (the Hermitian case). The following are equivalent: $T_K$ is self-adjoint; $K^*=K$, that is $K(y,x)=\overline{K(x,y)}$ for almost every $(x,y)$; and the sesquilinear form $(f,g)\mapsto\langle T_Kf,g\rangle$ is Hermitian, $\langle T_Kf,g\rangle=\overline{\langle T_Kg,f\rangle}$. In particular a real kernel is self-adjoint exactly when it is symmetric, $K(y,x)=K(x,y)$.

Proof. The first two are equivalent by the theorem and the uniqueness of the kernel of a Hilbert–Schmidt operator; the Hermitian property of the form is the polarisation of self-adjointness and is the kernel form of the Hermitian condition, as in the * Theory article. For a real kernel, $\overline{K(x,y)}=K(x,y)$, so $K^*=K$ is $K(y,x)=K(x,y)$. $\blacksquare$

The Involution on the Operators

Theorem. The map $T\mapsto T^\dagger$ restricted to the integral operators is the conjugate transport of the kernel involution: on the class of integral operators with $L^2$ kernels, $$ T_K^\dagger=T_{K^*},\qquad (T_KT_L)^\dagger=T_L^\dagger T_K^\dagger,\qquad (T_K^\dagger)^\dagger=T_K , $$ and it is an isometry of the Hilbert–Schmidt class onto itself.

Proof. The first identity is the theorem above; the anti-multiplicativity is the property of the adjoint of a bounded operator, read on the kernels through the correspondence $\langle T_K,T_L\rangle_{\mathrm{HS}}=\iint K\bar L$ of the * Theory article; the involution property $T^{\dagger\dagger}=T$ and the isometry are the same properties transported. $\blacksquare$

Example (the Volterra operator). The Volterra operator $Vf(x)=\int_0^xf(y)\,dy$ has the kernel $K(x,y)=\mathbf 1_{0\leq y\leq x}$, and its adjoint has the kernel $K^*(x,y)=\mathbf 1_{0\leq x\leq y}$, so that $$ (V^\dagger f)(x)=\int_x^1f(y)\,dy . $$ The operator is not self-adjoint, and $V+V^\dagger$ and $V-V^\dagger$ are its self-adjoint and skew-adjoint parts; the example shows that the kernel involution is the sharpening of the operator adjoint for integral operators.

Example (rank one). For $K(x,y)=\varphi(x)\overline{\psi(y)}$ the operator is $T_Kf=\varphi\int\bar\psi f$, and the adjoint kernel is $\overline{K(y,x)}=\overline{\varphi(y)}\psi(x)$, so that $T_K^\dagger g=\psi\int\bar\varphi g$; the two coincide with $K$ when $\varphi=\psi$ and $\lVert\varphi\rVert=1$, which recovers the orthogonal projection of rank one as the self-adjoint case.

Example (the Fourier and Hilbert operators). The Fourier operator $\mathcal F$ of The Fourier Operator is symmetric, $\mathcal F^{\mathrm t}=\mathcal F$, and unitary, so that its adjoint is its inverse, $\mathcal F^\dagger=\mathcal F^{-1}=\overline{\mathcal F}$; the adjoint is worked out in full in The Adjoint of the Fourier Operator, the last article of this group. The Hilbert transform $H$ of The Fourier Transform and Conjugate Symmetry has the purely imaginary multiplier $-i\operatorname{sgn}$, so it is skew-adjoint, $H^\dagger=-H$, and $iH$ is self-adjoint; this is the multiplier form of the kernel involution.

The Distributional Case

Theorem (distributional adjoint). Let $T$ be a bounded operator $C_c^\infty(Y)\to\mathcal D'(X)$ with distribution kernel $K\in\mathcal D'(X\times Y)$, by the Schwartz kernel theorem. Then the adjoint $T^\dagger$ has the conjugate transpose kernel, $$ K^\dagger(x,y)=\overline{K(y,x)} , $$ the conjugate being taken in the distributional sense, $\langle K^\dagger,\Phi\rangle=\overline{\langle K,\Phi^{\dagger}\rangle}$ for a test function $\Phi$, and $T$ is self-adjoint exactly when $K$ is Hermitian as a distribution.

Proof. The Schwartz kernel theorem of The Schwartz Kernel Theorem, later in this Part, provides the kernel; the computation of the previous sections carries over with the pairings of test functions against distributions, and the Hermitian condition is the distributional form of $K^*=K$, as in Positive Definite Distributions, the * Theory article. $\blacksquare$

Remark (unbounded operators). An unbounded operator has an adjoint only after its domain is fixed, and the adjoint may be densely defined and closed without the original being so; the distinction between a symmetric operator and a self-adjoint one, which coincide for bounded operators, is exactly the failure of the domain to match under the passage to the adjoint. This is The Adjoint of a Bounded Operator and the unbounded-operator material of Analysis on Linear Spaces, later in this Part; here the integral operators are bounded, and the subtlety does not arise.

Summary

The adjoint of a bounded operator on $L^2(X,\mu)$ is characterised by $\langle Tf,g\rangle=\langle f,T^\dagger g\rangle$ for the Hilbert pairing $\langle f,g\rangle=\int f\bar g$; it is conjugate-linear, involutive, isometric and reverses products, and it is the involution of the * Operator Theory group, written with the dagger by Conventions in Mathematics, while the star is reserved for the involution of the elements. For an integral operator the adjoint is the integral operator with the conjugate transpose kernel, $T_K^\dagger=T_{K^*}$ with $K^*(x,y)=\overline{K(y,x)}$, and the transpose with respect to the bilinear pairing has the transposed kernel $K(y,x)$; the two differ by the conjugation, $K^*=\overline{K^{\mathrm t}}$, and agree exactly for real kernels. The operator $T_K$ is self-adjoint exactly when its kernel is Hermitian, $K(y,x)=\overline{K(x,y)}$, which for a real kernel is symmetry; examples are the Volterra operator, whose adjoint integrates from $x$ to $1$, the rank-one projections, the symmetric unitary Fourier operator with $\mathcal F^\dagger=\mathcal F^{-1}$ and the skew-adjoint Hilbert transform $H^\dagger=-H$. The distributional case follows from the Schwartz kernel theorem, with the same conjugate transpose kernel. The spectral theory of the self-adjoint case, and the distinction of symmetric from self-adjoint for unbounded operators, are owned by later articles.

Summary of Notation

Symbol	Meaning
$\langle f,g\rangle=\int f\bar g$	Hilbert pairing of $L^2(X,\mu)$
$\{f,g\}=\int fg$	Bilinear pairing
$T^\dagger$	Adjoint of the operator $T$
$T^{\mathrm t}$	Transpose of $T$ with respect to the bilinear pairing
$K^*(x,y)=\overline{K(y,x)}$	Conjugate transpose kernel
$K^{\mathrm t}(x,y)=K(y,x)$	Transposed kernel
$T_K^\dagger=T_{K^*}$	The adjoint of the integral operator
self-adjoint / unitary / normal	$T^\dagger=T$ / $T^\dagger T=TT^\dagger=\mathrm{id}$ / $TT^\dagger=T^\dagger T$

The Adjoint of an Integral Operator

Introduction

The Pairing and the Adjoint

The Pairing

The Adjoint of a Bounded Operator

The Transpose and the Adjoint

The Adjoint of an Integral Operator

The Kernel of the Adjoint

The Involution on the Operators

The Distributional Case

Summary

Summary of Notation

Further Reading