Chapter 1

Integral Operators & Fredholm Theory

Boundary integral equation methods rest on a single observation: an elliptic boundary value problem posed on a domain can be recast as an integral equation posed only on its boundary, and the operator that comes out is a compact perturbation of the identity. This chapter is the theoretical groundwork for exploiting that structure, and it stands apart from any particular PDE so that the later chapters can lean on it without pausing. We follow [8] throughout. We first collect the properties of bounded and compact operators, and show that integral operators with continuous or weakly singular kernels are compact, which is exactly why the layer potentials of the coming chapters are. We then develop the Riesz theory of $I-A$, whose payoff is that injectivity alone gives us existence and continuous dependence on the data, so for a second-kind formulation it is enough to prove that the homogeneous problem has only the trivial solution. Lastly, we turn to Fredholm theory, which tells us what happens when injectivity fails. This is the case we meet at interior resonances, and it is what lets us pin down exactly the wave numbers at which the standard boundary integral formulations break down.

1.1 Bounded and Compact Operators

We will work almost exclusively with compact operators. The properties and results we will present will allow us to prove very nice existence and uniqueness statements for equations involving such operators.

1.1.1 Linear Bounded Operators

We start with linear bounded operators.

Definition 1.1.1 (Linear and Bounded Operator).

Let $X,Y$ be normed spaces, and define an operator $A : X \to Y$. We say a is linear if

A(\alpha \varphi + \beta \psi) = \alpha A \varphi + \beta \psi

for all $\alpha, \beta \in \C$ and $\varphi, \psi \in X$.

The operator $A$ is bounded if there exists a positive constant $C$ such that

||A \varphi|| \leq C || \varphi ||

We say that every $C$ this is true for is a bound on the operator.

We can interpret an operator as bounded if the image under the operation does not grow more than linearly. Alternatively, a linear operator $A: X \to Y$ is bounded if and only if $A$ maps bounded sets in $X$ to bounded sets in $Y$.

Theorem 1.1.2.

Let $X$ and $Y$ be normed spaces. A linear operator $A: X \to Y$ is bounded if and only if

||A|| := \sup_{||\varphi||=1} ||A \varphi|| < \infty

The number $||A||$ is the smallest bound for $A$ and is referred to as the norm of the operator.

Proof.

$\implies$) This is true by the definition of a bounded operator with $||\varphi|| < 1$ $\impliedby$) Take $\varphi \in X$

||A \varphi|| = \norm{A \frac{\norm{\varphi}}{\norm{\varphi}} \varphi} = \norm{\varphi} \norm{A \frac{\varphi}{\norm{\varphi}}} \leq \sup_{||\psi|| = 1} \norm{A\varphi} \norm{\psi}

Remark 1.1.3.

We note that any linear operator from a finite dimensional normed space into a normed space is bounded. This relies on the result that all norms are equivalent on a finite dimensional norm space.

It is true that linear combinations of bounded linear operators are bounded linear operators.

Theorem 1.1.4.

Let $X,Y,Z$ be normed spaces and define the bounded linear operators $A: X \to Y$ and $B: Y \to Z$. Then $BA: X \to Z$ is a bounded linear operator with norm $||(BA)\varphi || \leq ||A||\,||B||$

Proof.

This follows from linearity properties of $A$ and $B$ and Cauchy-Schwartz.

This let's us build new bounded linear operators from existing bounded linear operators. It also enables us to analyze sets of linear operators like sets of vectors using the tools of linear spaces.

Now we present the notion of convergence that we will need to analyze these spaces of operators.

Definition 1.1.5 (Cauchy Sequence).

A Cauchy sequence is a sequence $(\varphi_n)$ such that

\lim_{n,m \to \infty} ||\varphi_n - \varphi_m|| = 0

That is elements in the sequence keep getting closer together.

Definition 1.1.6 (Complete Set).

A subset $U \subseteq X$ is complete if every Cauchy sequence of elements in $U$ converges to an element in $U$.

Definition 1.1.7 (Banach Space).

A complete normed space is called a Banach Space.

Theorem 1.1.8.

Let $X,Y$ be normed spaces. The linear space $L(X,Y)$ of bounded linear operators $A: X \to Y$ is a normed space with norm

||A|| := \sup_{||\varphi||=1} ||A\varphi|| < \infty

Further, if $Y$ is a Banach space then $L(X,Y)$ is a Banach space.

Proof.

The spirit of the proof is to evoke the completeness and norm properties of $Y$ to show $L(X,Y)$ is a normed space since the operator norm is really looking at the norm of elements in $Y$ with the norm defined on $Y$.

With this result we have a way to measure how close operators are in a space. Naturally, we can consider sequences of operators.

Definition 1.1.9 (Operator Convergence).

A sequence of operators $(A_n)\subset L(X,Y)$ is norm convergent to an operator $A\in L(X,Y)$ if $A_n \to A$ as $n \to \infty$ in the $L(X,Y)$ norm.

A sequence $(A_n) \subset L(X,Y)$ is point-wise convergent to an operator $A$ if $A_n\varphi \to A\varphi$ as $n \to \infty$ for all $\varphi \in X$.

Note that norm convergence is the stronger convergence. Indeed, norm convergence implies point-wise convergence but the inverse is not true.

Definition 1.1.10 (Isomorphic Operator).

A bijective bounded linear operator $A: X \to Y$ is an isomorphism if the inverse operator $A^{-1}: Y \to X$ is bounded.

We say two normed spaces are isomorphic if there exists an isomorphism between them.

Note that isomorphic spaces have the same convergent sequences.

We are ready to introduce our next major player in our functional analysis lineup: the dual space.

Definition 1.1.11 (Dual space).

Let $X$ be a normed space. The dual space of $X$ is the set of bounded linear operators and is denoted $X^* := L(X, \C)$.

The dual space is a Banach space with norm

||F|| := \sup_{||\varphi|| =1} |F(\varphi)|, \quad F \in X^*, \varphi \in X

Notice that elements of $X^*$ are bounded linear functionals taking an element of $X$ to a complex scalar. It is also tempting to ask “what functionals are in the dual space”? The Hahn-Banach theorem ensures that any dual space has more than the trivial zero functional in it and is worth studying.

Theorem 1.1.12 (Hahn-Banach theorem).

Let $X$ be a normed space. Take $U \subset X$ and $F: U \to \C$. Then there exists bounded linear functional $G: X \to \C$ with the properties

$G(\varphi) = F(\varphi)$ for $\varphi \in U$
$||G|| = ||F||$

So the Hahn-Banach theorem allows us to extend any linear functional defined on a subset to the whole space such that the extension agrees on the subset, and the norm does not grow.

1.1.2 Compact Operators

Next we can move to the realm of compact operators.

Definition 1.1.13 (Relatively compact set).

A subset of a normed space is relatively compact (alternatively pre-compact) if its closure is compact. Or if every sequence in the subset has a convergent subsequence, but the limit need not be in the subset.

The Arzelà-Ascoli theorem gives an alternative way to determine if a subset is relatively compact or not.

Theorem 1.1.14 (Arzelà-Ascoli).

Define $C(G)$ as the set of continuous functions defined on a compact set $G \subset \C^m$. A set $U \subset C(G)$ is relatively compact if and only if it is bounded and equicontinuous. That is there exists a $C$ such that $|\varphi(x)| \leq C$ $\forall x \in G$ and $\forall \varphi \in U$ (this is bounded) and $\forall \epsilon > 0$, $\exists\delta > 0$ such that $|\varphi(x) - \varphi(y)| < \epsilon$ $\forall x,y \in G$ with $|x-y| < \delta$ (this is equicontinuous).

Equicontinuous is upgrading continuity to a whole set of functions such that the $\epsilon-\delta$ pair at a point holds for all of the functions in the set $C(G)$.

Definition 1.1.15 (Compact operator).

A linear operator $A: X \to Y$ is compact if it maps each bounded set in $X$ to a relatively compact set in $Y$.

Again, we have a theorem that gives us an alternative lens to view a compact operator

Theorem 1.1.16.

$A: X \to Y$ is compact if and only if for each bounded sequence $(\varphi_n) \subset X$ the sequence $(A\varphi_n) \in Y$ contains a convergent subsequence.

Proof.

$\implies)$ A set is relatively compact $\implies$ each sequence in the set contains a convergent subsequence. $\impliedby)$ Each sequence in the set contains a convergent subsequence $\implies$ the set is relatively compact $\implies$ A maps a compact set to a relatively compact set.

Now we list some important properties of compact operators.

Observation 1.1.17.

Compact linear operators are bounded.

Observation 1.1.18.

Linear combinations of compact linear operators are compact.

Observation 1.1.19.

Let $X,Y,Z$ be normed spaces, and $A:X \to Y$, $B: Y \to Z$ be bounded linear operators. The operator $BA: X \to Z$ is compact if $A$ or $B$ is compact.

Observation 1.1.20.

Bounded linear operators with finite dimensional range are compact operators.

Observation 1.1.21.

Let $(A_n)$ be a sequence of compact linear operators from normed space $X$ to a Banach space $Y$ that is norm convergent to a linear operator $A: X \to Y$. Then $A$ is compact.

This last property is useful when we have an operator $A$ that is not so nice to work with, but we can approximate it with more well behaved operators. Then we can say something about the compactness of $A$.

All in all, we should view compact operators a keeping elements close together from one space to another. These compact operators are very nice to work with, so it is good to know when an operator is or is not compact. The Riesz lemma gives us a tool to answer the negative question.

Lemma 1.1.22 (Riesz).

Let $X$ be a normed space and $U \subset X$ a closed subspace. Take $\alpha \in (0,1)$. Then there exists a $\psi \in X$ with $||\psi|| =1$ such that $|| \psi - \varphi || \geq \alpha$ for all $\varphi \in U$.

An immediate consequence of this lemma is that a subset being closed is not enough for it to be relatively compact. Thus an operator mapping to a closed set is not necessarily compact. It also allows us to prove the following theorem about the compactness of the identity operator.

Theorem 1.1.23.

The identity operator $I: X \to X$ is compact if and only if $X$ has finite dimension.

Proof.

$\impliedby)$ $I$ is a finite dimensional operator that is bounded and linear. Thus by Observation 1.1.20 $I$ is compact.

$\implies)$ We prove this direction with contradiction. Assume $I$ is compact and not finite dimensional. Choose $\varphi_1 \in X$ with $||\varphi_1||=1$. Denote $U_1 := \text{span}(\varphi_1)$. $U_1$ is finite dimensional and thus a closed subspace. Apply Lemma 1.1.22, so there exists $\varphi_2 \in X$ with $|| \varphi_2|| =1$ and $|| \varphi_2 - \varphi_1|| \geq 1/2$. Now define $U_2 := \text{span}(U_1, U_2)$. This is finite dimensional and closed, so apply Lemma 1.1.22 again. We can repeatedly apply the lemma to generate a sequence $(\varphi_n)$ with $|| \varphi_n || = 1$ and $|| \varphi_n - \varphi_m || \geq 1/2$ for $n \neq m$. So no subsequence converges! Thus a contradiction on the compact assumption of $I$.

This is an important result as we will see the identity operator plays a central role in the so called integral equations of the second kind we will encounter later.

1.2 Integral Operators

A mainstay in these notes are integral operators. These will allow us to define the namesake boundary integral equations which are well used in the solutions of elliptic PDEs. They are used in problems arising in acoustic and electromagnetic scattering, viscoelastic fluid flow, and fracture mechanics, to name a few.

In this section we will define an integral operator and show that they can be compact operators.

1.2.1 Introduction

First, we need to state some preliminaries.

Definition 1.2.1 (Characteristic Function).

The characteristic function $\chi_A$ is one such that for a set $A$

\chi_A(a) = \begin{cases} 1, &a \in A \\ 0, &a \notin A \end{cases}

Definition 1.2.2 (Jordan Measurable).

A set $G \subset \R^m$ is Jordan measurable if $\chi_{G}(x)$ is Riemann integrable.

We define the Jordan measure $|G|$ as the integral of $\chi_G$.

For any Jordan measurable $G$, $\overline{G}$ is Jordan measurable with measure $|G| = |\overline{G}|$. Additionally, $\partial G$ is Jordan measurable with measure $|\partial G| = 0$. Colloquially, we can say that the Jordan measure does not “see” the boundary.

Also, we have that if $G$ is compact and Jordan measurable, then every continuous function defined on $G$ is Riemann integrable.

Definition 1.2.3.

Take $G \in \R^m$ to be a nonempty, compact, Jordan measurable set that is the closure of its interior. Take $K: G \times G \to \C$ to be a continuous function.

The linear operator $A: C(G) \to C(G)$ defined by

[A\varphi](x) := \int_{G} K(x,y)\varphi(y) \, dy, \quad x \in G

is an integral operator with a continuous kernel $K$.

Theorem 1.2.4.

An integral operator with continuous kernel $K$ as defined in Definition 1.2.3 is a bounded linear operator.

It has norm

\norm[\infty]{A} = \max_{x \in G} \int|K(x,y)| \, dy

Integral operators can be used to form operator equations, which we can attempt to solve for the unknown. This is the idea of boundary integral equations we explore in these notes. A differential operator can be expressed as an integral operator with a kernel specific to the PDE. Then an equation is formed with respect to the known boundary data, and we can solve the equation this way.

With this goal in mind, we define operator and integral equations.

Definition 1.2.5 (Operator Equations).

Let $A: X \to Y$ be an operator. An operator equation of the first kind is $A\varphi = f$ where $\varphi \in X$ is unknown and $f \in Y$ is known.

An operator equation of the second kind is $(I-A)\varphi = f$.

An integral equation is an operator equation where $A$ is an integral operator. It turns out that equations of the second kind are much more well behaved mathematically and numerically. For this reason we will primarily concern ourselves with integral equations of the second kind.

We can study these integral operators with the tools we have previously developed

Theorem 1.2.6.

Integral operators with continuous kernels are compact linear operators on $C(G)$.

Proof.

Let $A$ be an integral operator with continuous kernels $K$, for all $\varphi \in C([a,b])$ with $\|\varphi\|_\infty\leq 1$ and all $x\in[a,b]$, we have that

|(A\varphi)(x)| \leq \int_a^b |K(x,y)\phi(y)|~\mathrm{d}y\leq \int_a^b \mathrm{d}y\max_{x,y\in[a,b]}|K(x,y)| \|\varphi\|_\infty \leq (b-a)\max_{x,y\in[a,b]}|K(x,y)|,

i.e., the set $U:=\{A\varphi~|~\varphi\in C([a,b]),~\|\varphi\|_\infty\leq 1\}\subset C([a,b])$ is bounded. Since $K$ is continuous, $K$ is uniformly continuous on the compact square $[a,b]\times[a,b]$, for every $\epsilon>0$, there exists $\delta >0$ s.t.

|K(x,z)-K(y,z)|< \frac{\epsilon}{b-a},

for all $x,y\in[a,b]$ with $|x-y|<\delta$. Then

|(A\phi)(x)-(A\phi)(y)| = \left|\int_a^b[K(x,z)-K(y,z)]\varphi(z)~\mathrm{d}z\right|\leq \epsilon,

for all $x,y\in[a,b]$ with $|x-y|<\delta$ and all $\varphi \in C([a,b])$ with $\|\varphi\|_\infty\leq 1$, i.e., $U$ is equicontinuous. Hence, by the Arzelà-Ascoli Theorem 1.1.14, $A$ is compact.

Note that we presented the proof for continuous functions on $G$, but the result can be upgraded to functions $\varphi \in L^2(G)$. We omit this proof. However, as we will see, many integral operators we will encounter in the wild will not have continuous kernels. We will prepare for these situations with the following definitions and theorems.

Definition 1.2.7 (Weakly Singular Kernel).

A kernel $K: G \times G \to \C$ is weakly singular if it is defined and continuous for all $x,y \in G \subseteq \R^m$ with $X \neq y$ and there exist $M > 0$ and $\alpha \in (0, m]$ such that

|K(x,y)| \leq \frac{M}{|x-y|^{m-\alpha}}

In other words, the singularity of the kernel is of order less than the dimension of the space. Thus, it blows up, but not too fast that it is not integrable. We would be remiss to not mention that many kernels of practical importance have a singularity with order equal to or greater than the dimension. These are called hypersingular kernels. We will not treat these kernels in these notes, but they are of great practical importance.

Theorem 1.2.8.

Integral operators with weakly singular kernels are compact linear operators on $C(G)$.

Proof.

The idea of the proof is to estimate the integral operator with weakly singular kernel with a sequence of integral operators with continuous kernels. Thus they are compact operators. Showing that the sequence is norm convergent to the operator with weakly singular kernel then tells us that it is compact by Observation 1.1.21.

We do so by defining the sequence of kernels

K_n(x,y) = \begin{cases} h(n|x-y)K(x,y), &x \neq y \\ 0, & x = y \end{cases}

where

h(t) = \begin{cases} 0, & 0 \leq t \leq 1/2 \\ 2t - 1, & 1/2 \leq t \leq 1 \\ 1, & t > 1 \end{cases}

These $K_n$ cut off the singularity and replace them with $0$. But as $n \to \infty$ we converge to the original, weakly-singular kernel $K$. The $K_n$ are continuous so if we consider the sequence of integral operators with kernel $K_n$ we have integral operators with continuous kernels. Thus they are compact by Theorem 1.2.6. Then as $n \to \infty$, Observation 1.1.21 tells us the integral operator with kernel $K$ is compact.

1.2.2 Volterra Integral Operators

We briefly present the case of Volterra integral equations, which can be thought of as Fredholm equations where the kernel is zero valued for $y > x$.

Definition 1.2.9.

Integral equations of the form

\int_{a}^{x} K(x,y)\varphi(y)\,dy = f(x), \quad x \in [a,b]

and

\varphi(x) - \int_{a}^{x} K(x,y) \varphi(y)\,dy = f(x), \quad x \in [a,b]

are Volterra integral equations of the first and second kind, respectively.

Note that for each right hand side of the integral equations, $f \in C[a,b]$ the Volterra integral equation of the second kind with a continuous kernel has a unique solution $\varphi \in C[a,b]$. Such a general statement cannot be made about the first kind equations, and in general working with a second kind, as said for Fredholm equations, is preferable. In many cases an equation of the first kind can be turned into a second kind equation by taking a derivative with respect to the non-integrating variable. We will see this in action later in the notes when we vist the Laplace and subsequently Helmholtz equations.

1.3 Riesz Theory

As mentioned a few times now, we want to solve operator equations of the form $(I-A)\varphi = f$ where $A$ is an integral operator. We have just shown that an integral operator with continuous or weakly singular kernels are compact. Now we develop the next tool to hang on our belt: Riesz Theory. In this section, define $L := I-A$, i.e. a perturbation of the identity. We present three Riesz theorems.

Theorem 1.3.1 (First Riesz Theorem).

The nullspace of the operator $L$, denoted $\bold{N}(L) := \{ \varphi \in X : L \varphi = 0 \}$ is a finite dimensional subspace.

Theorem 1.3.2 (Second Riesz Theorem).

The range of the operator $L$ denoted $L(X) := \{ L\varphi : \varphi \in X \}$ is a closed linear subspace.

Theorem 1.3.3 (Third Riesz Theorem).

There exists a unique $ r \geq 0$ called the Riesz number of the operator $A$ such that

\underbrace{\{ 0 \} = \bold{N}(L^0) \subsetneq \bold{N}(L^1) \subsetneq \dots \subsetneq \bold{N}(L^r) = \bold{N}(L^{r+1}) = \dots}_{\text{nullspace grows} ~\rightarrow}

and

\underbrace{X = L^0(X) \supsetneq L^1(X) \supsetneq \dots \supsetneq L^r(X) = L^{r+1}(X) = \dots}_{\text{range shrinks} ~ \rightarrow}

and we have the direct sum

X = \bold{N}(L^r) \oplus L^r(X)

This tells us that for this operator $L$, we keep taking powers of it, and we are guaranteed to at some point we split the whole space into the nullspace and range of $L$.

We say that an operator is of finite ascent if two successive iterated operators have the same nullspace. An operator is of finite descent if two successive iterated operators have the same range. In simpler words, an operator is of finite ascent if the nullspace levels out under iterations, and of finite descent if the range levels out. Note that if $A$ is a compact linear operator, then $I-A$ is of finite ascent and descent.

Theorem 1.3.4.

Let $X$ be a normed space and define a compact linear operator $A: X \to X$. Then $I-A$ is injective if and only if $I-A$ is surjective.

Proof.

Let $I-A$ be injective, then $\bold{N}(L^0) = \{ 0 \}$ which means $r=0$. Let $I-A$ be surjective, then $X = L^0(X)$ which means $r=0$. Thus injective and surjective are both equivalent to $r=0$.

Theorem 1.3.5.

Let $X$ be a normed space and define a compact linear operator $A: X \to X$. If $I-A$ is injective (therefore bijective) then $(I-A)^{-1}: X \to X$ is bounded and $I-A$ is an isomorphism.

Proof.

We will prove by contradiction. Assume $(I-A)^{-1}$ is not bounded. This means we can find a sequence $(f_n) \subset X$ with $\| f_n \| = 1$ such that $\| L^{-1}f_n \| \geq n$ for all $n \in \N$. Construct the following two sequences

g_n := \frac{f_n}{\| L^{-1} f_n \|}, \qquad \varphi_n := \frac{L^{-1}f_n}{\| L^{-1}f_n \|}, \quad n \in \N

Clearly, $g_n \to 0$ as $n \to \infty$ and $\| \varphi_n\| =1$ for all $n$. $A$ is compact so we get a subsequence of $(\varphi_n)$ that is convergent in $X$. Now consider

\varphi_n - A\varphi_n = (I-A)\varphi_n = (I-A) \frac{(I-A)^{-1}f_n}{\| L^{-1} f_n \|} = \frac{f_n}{\| L^{-1} f_n \|} = g_n

So we have

\varphi_n - A\varphi_n = g_n

Now take $n\to \infty$ and we see

(I-A)\varphi = 0 \implies \varphi \in \bold{N}(I-A)

This would mean $\| \varphi \| = 0$. Contradiction on $\| \varphi_n\| = 1$ for all $n$!

With this last theorem, we have a result on the inverse of the operator $I-A$. This puts us in the position to say something about the solution of equations with this operator!

Corollary 1.3.6.

Let $X$ be a normed space and $A: X \to X$ be a compact linear operator. If the homogeneous equation

(I-A)\varphi = 0

has only the trivial solution $\varphi = 0$, then for all $f \in X$, the inhomogeneous equation

(I-A)\varphi = f

has a unique solution that depends continuously on the data $f$.

Proof.

If we only have the trivial solution $\varphi = 0$ then $\bold{N}(I-A) = \{ 0 \}$ which means $I-A$ is injective thus bijective and thus an isomorphism.

Corollary 1.3.7.

Let $X$ be a normed space and $A: X \to X$ be a compact linear operator. If the homogeneous equation

(I-A)\varphi = 0

has a nontrivial solution, then it only has a finite number $m \in \N$ of linearly independent solutions $\varphi_1, \dots, \varphi_m$ and the inhomogeneous equation

(I-A)\varphi = f

is either unsolvable or has a general solution

\varphi = \tilde{\varphi} + \sum_{k=1}^{m}\alpha_k \varphi_k

where $\alpha_k \in \C$ and $\tilde{\varphi}$ is a particular solution of the homogeneous equation.

In the Riesz theory, we have looked at operators of the form $I-A$ where $I$ is the identity. We can generalize the results by replacing $I$ by any bounded operator with a bounded inverse.

Corollary 1.3.8.

The previous three results (Theorem 1.3.5 and Corollaries 1.3.6 and 1.3.7) hold when $I$ is replaced by $S$ where $S:X \to Y$ is a bounded linear operator with a bounded inverse $S^{-1}: Y \to X$.

Proof.

(S-A)\varphi = (I-S^{-1}A)\varphi

and $S^{-1}A$ is compact by Observation 1.1.19 since $A$ is compact and $S^{-1}$ is bounded.

Since we have seen that integral operators with continuous and weakly singular kernels are compact, we can apply the Riesz theory to $I-A$ where $A$ is an integral operator! The theory then allows us to replace the problem of establishing uniqueness of a solution to $(I-A)\varphi = f$ with the problem of showing $(I-A)\varphi = 0$ only has $\varphi=0$ as a solution. The latter is (generally) an easier problem.

1.3.1 Riesz Theory as Spectral Theory

We can frame the results of Riesz theory through the spectral analysis of compact operators. For this, we need more definitions, though hopefully some will be familiar to the reader.

Definition 1.3.9.

Let $X$ be a normed space and $A: X \to X$ a bounded linear operator.

$\lambda \in \C$ is an eigenvalue of $A$ if there exists $\varphi \in X$, $\varphi \neq 0$, such that $A\varphi = \lambda \varphi$.
$\varphi \in X$, $\varphi \neq 0$ such that $A \varphi = \lambda \varphi$ is called an eigenelement of $A$.
$\lambda \in \C$ is called a regular value of $A$ if $(\lambda I -A)^{-1}: X \to X$ exists and is bounded.
The set of all regular values is the resolvent set $\rho(A)$.
The resolvent of $A$ is the operator $R(\lambda;A) := (\lambda I - A)^{-1}$.
The complement of the resolvent set is the spectrum $\sigma(A)$.
The spectral radius of $A$ is $r(A) := \sup_{\lambda \in \sigma(A)} |\lambda|$

Theorem 1.3.10.

Let $X$ be an infinite dimensional norm space and $A:X \to X$ a compact linear operator. Then $\lambda = 0$ belongs to the spectrum $\sigma(A)$ and $\sigma(A)\setminus \{ 0 \}$ consists of at most a countable set of eigenvalues with no point accumulation except, possibly, at $\lambda = 0$.

1.4 Fredholm Theory

The compactness of integral operators is the backbone of boundary integral equations, in particular, the study of equation of the form $(I+K)\varphi = f$. In many ways, compact operators behave like finite-dimensional operators, despite the fact that they are acting on infinite-dimensional spaces. Fredholm theory characterizes the solvability of equations involving the identity plus a compact operator and establishes the backbone between invertibility and the presence of a finite-dimensional null-space or co-kernel. The treatment here follows [8].

1.4.1 Dual Systems

Riesz theory already tells us that $(I-A)$ is injective if and only if it is surjective, but it says nothing about what happens when injectivity fails. In finite dimensions we know the answer: $Mx=b$ is solvable precisely when $b$ is orthogonal to the null-space of $M^T$. To carry that statement into infinite dimensions we need a notion of transpose, and for that we need a pairing between two spaces. The pairing is supplied by a bilinear form.

Definition 1.4.1 (Bilinear form).

Let $X,Y$ be a linear spaces. A mapping

\langle\cdot,\cdot\rangle:X\times Y\to \C

is called a bilinear form if

\begin{align*} \langle\alpha_1\varphi_1+\alpha_2\varphi_2,\psi\rangle &= \alpha_1\langle \varphi_1,\psi\rangle + \alpha_2\langle \varphi_2,\psi\rangle,\\ \langle \varphi,\beta_1\psi_1+\beta_2\psi_2\rangle &= \beta_1\langle \varphi,\psi_1\rangle + \beta_2\langle \varphi,\psi_2\rangle \end{align*}

for all $\varphi_1,\varphi_2,\varphi\in X$, $\psi_1,\psi_2,\psi\in Y$ and $\alpha_1,\alpha_2,\beta_1,\beta_2\in\C$. The bilinear form is called non-degenerate if for every $\varphi\in X$ with $\varphi\not= 0$ there exists $\psi \in Y$ such that $\langle \varphi,\psi\rangle \not = 0$, and for every $\psi\in Y$ with $\psi \not = 0$ there exists $\varphi\in X$ such that $\langle\varphi,\psi\rangle\not = 0$.

Non-degeneracy is the condition that makes the pairing informative: no non-zero element of either space is invisible to the other. Without it, a statement such as “$f$ is orthogonal to the null-space of the adjoint” would carry no content, since a large subspace could be orthogonal to everything. A pair of spaces carrying such a form is exactly the structure we work in.

Definition 1.4.2 (Dual system).

Two normed spaces $X$ and $Y$ equipped with a non-degenerate bilinear form $\langle\cdot,\cdot\rangle:X\times Y\to \C$ are called a dual system and denoted by $\langle X,Y\rangle$.

Two dual systems will be used throughout. The first is available for free on any normed space, at the cost of carrying the dual space around in the second slot.

Theorem 1.4.3.

Each normed space $X$ together with its dual space $X^*$ forms the canonical dual system $\langle X,X^*\rangle$ with bilinear form

\langle \varphi, F\rangle := F(\varphi),\qquad \varphi\in X, F \in X^*.

The proof relies on the Hahn-Banach Theorem 1.1.12, which supplies, for every non-zero $\varphi$, a functional that does not annihilate it [8].
The second dual system is the one we actually want for boundary integral equations. Rather than pairing $C(G)$ with the space of measures that constitutes its dual, we pair $C(G)$ with itself through integration. This keeps both slots concrete and, as we will see, means the adjoint of an integral operator is again an integral operator on the same space.

Theorem 1.4.4.

Let $G\subset\R^m$ be a nonempty compact and Jordan measurable set that coincides with the closure of its interior. Then $\langle C(G),C(G)\rangle$ is a dual system with the bilinear form

\langle\varphi,\psi\rangle:=\int_G \varphi(x)\psi(x)\mathrm{d}x,\qquad \varphi,\psi\in C(G).

Proof.

Bilinearity is immediate from linearity of the integral, so only non-degeneracy needs an argument. Let $\varphi\in C(G)$ with $\varphi\not\equiv0$, say $\varphi(x_0)\neq0$. Since $G$ is the closure of its interior, $x_0$ can be taken in the interior of $G$, and by continuity there is a ball $B_\epsilon(x_0)\subset G$ on which $|\varphi|>\tfrac{1}{2}|\varphi(x_0)|$ and on which $\varphi$ does not change argument. Choose $\psi\in C(G)$ non-negative, supported in $B_\epsilon(x_0)$ and equal to $\overline{\varphi}/|\varphi|$ times a bump there. Then $\langle\varphi,\psi\rangle\neq0$. Jordan measurability guarantees the integral is defined in the Riemann sense for all of $C(G)$. The roles of the two slots are symmetric, so the same argument applies in reverse.

The hypotheses on $G$ are exactly what this proof consumes: without $G$ being the closure of its interior, a non-zero $\varphi$ could be supported on a lower-dimensional sliver of $G$ and be invisible to every pairing. With a pairing in hand we can transpose operators.

Definition 1.4.5 (Adjoint).

Let $\langle X_1,Y_1\rangle$ and $\langle X_2,Y_2\rangle$ be two dual systems. Then two operators $A:X_1\to X_2,~B:Y_2\to Y_1$ are called adjoint (w.r.t. these dual systems) if

\langle A\varphi, \psi\rangle = \langle\varphi,B\psi\rangle

for all $\varphi\in X_1,\psi\in Y_2$.

Note that the definition assumes neither linearity nor boundedness of either operator. Both come for free, and the adjoint is pinned down completely by the pairing.

Theorem 1.4.6.

Let $\langle X_1,Y_1\rangle$ and $\langle X_2,Y_2\rangle$ be two dual systems. If an operator $A:X_1\to X_2$ has an adjoint $B:Y_2\to Y_1$, then $B$ is uniquely determined and $A$ and $B$ are linear.

Proof.

Suppose $B_1$ and $B_2$ are both adjoint to $A$. Then $\langle \varphi, (B_1-B_2)\psi\rangle = 0$ for every $\varphi\in X_1$ and every $\psi\in Y_2$, and non-degeneracy in the second slot forces $(B_1-B_2)\psi=0$. For linearity of $A$, take $\varphi_1,\varphi_2\in X_1$ and scalars $\alpha_1,\alpha_2$. For every $\psi\in Y_2$,

\langle A(\alpha_1\varphi_1+\alpha_2\varphi_2),\psi\rangle = \langle \alpha_1\varphi_1+\alpha_2\varphi_2, B\psi\rangle = \langle \alpha_1 A\varphi_1 + \alpha_2 A\varphi_2,\psi\rangle,

using bilinearity of the form and the adjoint relation twice. Non-degeneracy in the first slot gives $A(\alpha_1\varphi_1+\alpha_2\varphi_2) = \alpha_1 A\varphi_1 + \alpha_2 A\varphi_2$. The argument for $B$ is identical.

Uniqueness is settled, but existence is a genuine question. In the canonical dual system it is always answered affirmatively.

Theorem 1.4.7.

Let $A:X\to Y$ be a bounded linear operator mapping a normed space $X$ into a normed space $Y$. Then the adjoint operator $A^*:Y^*\to X^*$ w.r.t. the canonical dual systems $\langle X,X^*\rangle$ and $\langle Y,Y^*\rangle$ exists. It is given by

A^*F:=FA,\qquad F\in Y^*,

and is called the dual operator of $A$. It is bounded with norm $\|A\| = \|A^*\|$.

In the dual system $\langle C(G),C(G)\rangle$ the adjoint is far more explicit, and this is the statement that makes Fredholm theory usable in practice. Transposing an integral operator amounts to swapping the arguments of its kernel.

Theorem 1.4.8.

Let $K$ be a continuous or weakly singular kernel. Then in the dual system $\langle C(G),C(G)\rangle$ the (compact) integral operators defined by

(A\varphi)(x):=\int_{G}K(x,y)\varphi(y)~\mathrm{d}y,\qquad x\in G,

and

(B\psi)(x) := \int_G K(y,x)\psi(y)~\mathrm{d}y,\qquad x\in G,

are adjoint.

Proof.

For $\varphi,\psi\in C(G)$,

\langle A\varphi,\psi\rangle = \int_G\left(\int_G K(x,y)\varphi(y)~\mathrm{d}y\right)\psi(x)~\mathrm{d}x = \int_G \varphi(y)\left(\int_G K(x,y)\psi(x)~\mathrm{d}x\right)\mathrm{d}y = \langle \varphi,B\psi\rangle,

where the interchange is Fubini's theorem. For a continuous kernel the double integral is over a compact set with a continuous integrand, so the hypothesis is trivially met. For a weakly singular kernel, $|K(x,y)|\leq C|x-y|^{\alpha-m}$ with $\alpha>0$, and the majorant $|\varphi(y)\psi(x)||x-y|^{\alpha-m}$ is integrable over $G\times G$, so Fubini again applies.

This is the exact analogue of $M\mapsto M^{T}$ for matrices, and it is worth keeping in mind that the adjoint here is a transpose and not a conjugate transpose. Note also that $B$ is the operator whose kernel is $K(y,x)$, which in the layer potential setting is precisely the relation between the double-layer operator $\mathcal{D}$ and its adjoint $\mathcal{D}'$. The conjugated version of all of this requires a sesquilinear pairing instead.

Definition 1.4.9 (Sesquilinear form).

Let $X,Y$ be linear spaces, A mapping

(\cdot,\cdot):X\times Y\to \mathbb{C}

is called a sesquilinear form if

\begin{align*} (\alpha_1\varphi_1+\alpha_2\varphi_2,\psi) & = \alpha(\varphi_1,\psi) + \alpha_2(\varphi_2,\psi)\\ (\varphi,\beta_1\psi_1 + \beta_2\psi_2) &= \overline{\beta_1}(\varphi,\psi_1) + \overline{\beta_2}(\varphi,\psi_2) \end{align*}

for all $\varphi_1,\varphi_2,\varphi\in X$, $\psi_1,\psi_2,\psi\in Y$ and $\alpha_1,\alpha_2,\beta_1,\beta_2\in\C$.

The sesquilinear form is related to the bilinear form

(\varphi,\psi) = \langle, \varphi,\psi^*\rangle,

except the sesquilinear form is nolonger holomorphic. The most common sesquilinear form on $G$ for $\varphi,\psi\in C(G)$ is given by

(\varphi,\psi) = \int_G\varphi(x)\overline{\psi}(x)\mathrm{d}x,

with an associated norm

\|\varphi\| =\sqrt{(\varphi,\varphi)}

The price of the conjugation is that antilinearity in the second slot breaks holomorphy in $\psi$, so the sesquilinear setting is the wrong one whenever we want to track dependence on a parameter analytically, such as the wave number $k$. Its advantage is that it comes with a norm, and with it the geometry of Hilbert space. The Riesz representation theorem then makes the canonical dual system of Theorem 1.4.3 collapse onto the space itself.

Theorem 1.4.10.

Let $X$ be a Hilbert space. Then for each bounded linear functional $F:X\to \mathbb{C}$ there exists a unique element $f\in X$ s.t.

F(\varphi) = (\varphi,f)

for all $\varphi\in X$. The norms of the element $f$ and the linear function $F$ coincide, i.e.,

\|f\| = \|F\|.

Identifying $X^*$ with $X$ in this way turns the dual operator into an operator on the original spaces, which is the familiar Hilbert space adjoint.

Theorem 1.4.11.

Let $X$ and $Y$ be Hilbert spaces, and let $A:X\to Y$ be a bounded linear operator. Then there exists a unquely determined lienar operator $A^*:Y\to X$ with the property

(A\varphi,\psi) = (\varphi,A^*\psi)

for all $\varphi\in X$ and $\psi\in Y$, i.e., $A$ and $A^*$ are adjoint w.r.t. the dual systems $(X,X)$ and $(Y,Y)$ generated by the scalar products on $X$ and $Y$. The operator $A^*$ is bounded and $\|A^*\|=\|A\|$.

Compactness also transfers across the adjoint [8], which we will need since both Fredholm theorems below require $A$ and $B$ to be compact, not just one of them.

Theorem 1.4.12.

Let $X$ and $Y$ be Hilbert spaces and let $A:X\to Y$ be a compact linear operator. Then the adjoint operator $A^*:Y\to X$ is also compact.

There is one loose end before we move on to solvability. Everything built so far, compactness of weakly singular integral operators and the Riesz theory, lives in $C(G)$ with the supremum norm. The scalar product above induces the $L^2$ norm instead, and on $C(G)$ the two are not equivalent. Passing between them is not a formality we can wave through, and the following theorem of Lax [8] is the tool that does it. It converts sup-norm boundedness into $L^2$ boundedness for free, provided an adjoint relation is available, which by Theorem 1.4.8 is always the case for our layer operators.

Theorem 1.4.13 (Lax).

Let $X$ and $Y$ be normed spaces, both of which have a scalar product $(\cdot,\cdot)$, and assume that there exists a positive constant $c$ such that

|(\varphi,\psi)|\leq c\|\varphi\|\|\psi\|

for all $\varphi,\psi\in X$. Let $U$ be a subspace of $X$ and let $A:X\to Y$ and $B:Y\to X$ be bounded linear operators satisfyin

(A\varphi,\psi) = (\varphi,B\psi)

for all $\varphi\in U$ and $\psi\in Y$. Then $A:U\to Y$ is bounded with respect to the norms $\|\cdot\|_s$ induces by the scalar products and

\|A\|_s^2\leq \|A\|\|B\|

The practical consequence is that an operator bounded on $C(G)$ with a transposed-kernel partner is bounded in the $L^2$ norm, and since $C(G)$ is dense in $L^2(G)$ it extends uniquely to a bounded operator on $L^2(G)$. This is how the single- and double-layer operators are moved off $C(\partial D)$ and onto $L^2(\partial D)$, which is where the hypersingular operator and the Dirichlet-to-Neumann map have to be analysed [8, 3].

1.4.2 Fredholm Alternative

We now have everything needed to state the solvability theory for $(I-A)\varphi=f$. Riesz theory already gives us that $\mathcal{N}(I-A)$ is finite-dimensional (Theorem 1.3.1) and that $(I-A)(X)$ is closed (Theorem 1.3.2). What Fredholm theory adds is that the null-space of the adjoint has the same dimension, and that it characterizes the range exactly. The bookkeeping in both proofs rests on the existence of a biorthogonal system, which is where non-degeneracy of the pairing is finally spent.

Lemma 1.4.14.

Let $\langle X,Y\rangle$ be a dual system. Then to every set of linearly independent elements $\varphi_1,\dots,\varphi_n\in X$ there exists a set $\psi_1,\dots,\psi_j\in Y$ such that

\langle\varphi_i,\psi_j \rangle = \delta_{ij},\qquad i,k = 1,\dots,n,

where $\delta_{ij} = 1$ for $i=j$ and $\delta_{ij} = 0$ otherwise. The same statement holds with the roles of $X$ and $Y$ interchanged.

The lemma is a dual basis statement, proved by induction on $n$ [8], and it is what allows the two null-spaces to be compared dimension by dimension.

Theorem 1.4.15 (First Fredholm Theorem).

Let $\langle X,Y\rangle$ be a dual system and $A:X\to X$, $B:Y\to Y$ be compact adjoint operators. Then the nullspaces of the operators $I-A$ and $I-B$ have the same finite dimension.

Finiteness of each dimension separately is the first Riesz Theorem 1.3.1; the content here is the equality of the two. The equality is not a soft statement and fails badly without compactness [8]. The unilateral shift on $\ell^2$ is injective while its adjoint has a one-dimensional null-space, so no such balance holds for general bounded operators.
Knowing that the two null-spaces have matching size, the remaining question is which right-hand sides are attainable. The second theorem answers this with the orthogonality condition we set out to reach.

Theorem 1.4.16 (Second Fredholm Theorem).

Let $\langle X, Y\rangle$ be a dual system and $A:X\to X$, $B:Y\to Y$ be compact adjoint operators. Then

(I-A)(X) = \{f\in X~|~\langle f,\psi\rangle = 0, \psi\in \mathcal{N}(I-B)\}

and

(I-B)(Y) = \{g\in Y~|~\langle\varphi,g\rangle = 0, \varphi\in \mathcal{N}(I-A)\}

Proof of the inclusion $\subseteq$.

Let $f\in(I-A)(X)$, so $f = (I-A)\varphi$ for some $\varphi\in X$, and let $\psi\in\mathcal{N}(I-B)$. Since $A$ and $B$ are adjoint, so are $I-A$ and $I-B$, and therefore

\langle f,\psi\rangle = \langle (I-A)\varphi,\psi\rangle = \langle \varphi,(I-B)\psi\rangle = 0.

The second identity follows by the same computation with the roles of the spaces exchanged.

The reverse inclusion is the substantial half, and it is where closedness of the range from the second Riesz Theorem 1.3.2 and the biorthogonal system of Lemma 1.4.14 are needed; the full argument is given in [8]. Combining the two Fredholm theorems with the Riesz dichotomy gives the statement that is actually used in practice.

Theorem 1.4.17 (Fredholm Alternative).

Let $A:X\to X$, $B:Y\to Y$ be compact adjoint operators in a dual system $\langle X,Y\rangle$. Then either $I-A$ and $I-B$ are bijective or $I-A$ and $I-B$ have nontrivial nullspaces with finite dimension

\dim{\mathcal{N}(I-A)}=\dim{\mathcal{N}(I-B)}\in\mathbb{N}

and the ranges are given by

(I-A)(X) = \{f\in X~|~\langle f,\psi\rangle = 0, \psi\in \mathcal{N}(I-B)\}

and

(I-B)(Y) = \{g\in Y~|~\langle\varphi,g\rangle = 0, \varphi\in \mathcal{N}(I-A)\}

The first case is the Riesz statement that injectivity implies surjectivity, now known to hold for $I-A$ and $I-B$ simultaneously by Theorem 1.4.15. The second case is Theorem 1.4.16. This is the infinite-dimensional analogue of the statement that a square matrix is either invertible or has a non-trivial kernel, and it is the reason second-kind formulations are so attractive: uniqueness implies existence, so for a boundary integral equation of the second kind it suffices to prove that the homogeneous problem has only the trivial solution. Specializing to the integral operators of Theorem 1.4.8 in the dual system of Theorem 1.4.4 gives the form in which we will actually invoke it.

Corollary 1.4.18.

Let $G\subset \R^m$ be as in Theorem 1.4.4 and let $K$ be a continuous or weakly singular kernel. Then either the homogeneous integrel equations

\varphi(x) - \int_G K(x,y)\varphi(y)~\mathrm{d}y = 0,\qquad x \in G,

and

\psi(x) - \int_G K(y,x)\psi(y)~\mathrm{d}y = 0,\qquad x \in G,

only have the trivial solutions $\varphi = 0$ and $\psi = 0$ and the inhomogeneous integral equations

\varphi(x) - \int_G K(x,y)\varphi(y)~\mathrm{d}y = f(x),\qquad x \in G,

and

\psi(x) - \int_G K(y,x)\psi(y)~\mathrm{d}y = g(x),\qquad x \in G,

have a unique solution $\varphi\in C(G)$ and $\psi\in C(G)$ for each $f\in C(G)$ and $g\in C(G)$, respectively, or the homogeneous integral equations have the same finite number $m\in \N$ of linearly independent solution and the inhomogeneous integral equations are solvable if and only if the right hand sides satisfy

\langle f,\psi\rangle = 0

for all solutions $\psi$ of the homogeneous adjoint equation and

\langle \varphi, g\rangle =0

for all solutions $\varphi$ of the homogeneous equation, respectively.

No proof is needed, the corollary is Theorem 1.4.17 read in the dual system $\langle C(G),C(G)\rangle$, with Theorem 1.4.8 supplying the adjoint pair and compactness coming from the weak singularity of the kernel. This is the tool we use for the Laplace and Helmholtz layer potential formulations, where the adjoint equation is the one obtained by swapping source and target in the kernel, and the solvability conditions above become concrete compatibility conditions on the boundary data.
It is worth closing the loop with the spectral reading of Riesz theory. Applying the Fredholm Alternative to $I-\lambda^{-1}A$ for $\lambda\neq0$ shows that $\lambda$ is an eigenvalue of $A$ if and only if it is an eigenvalue of the adjoint $B$, and by Theorem 1.4.15 the two eigenspaces have the same dimension. Combined with the fact that the non-zero spectrum of a compact operator consists of at most countably many eigenvalues accumulating only at the origin, this says $\sigma(A)\backslash\{0\}=\sigma(B)\backslash\{0\}$ with matching multiplicities. In the layer potential setting this is the statement that $\mathcal{D}$ and $\mathcal{D}'$ share a spectrum, which is why the interior Dirichlet and Neumann resonances appear symmetrically in the exterior sound-soft and sound-hard formulations [3].

References

[3]David Colton and Rainer Kress. Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol. 93, 4th edition, Springer, 2019.
[8]Rainer Kress, Vladimir Maz'ya, and Vladimir Kozlov. Linear integral equations. Springer, 1989.