This opening chapter fixes the viewpoint of the course. Spectral theory begins with a simple finite-dimensional question: what remains of eigenvalues, diagonalisation, and matrix functions when the [vector space](/page/Vector%20Space) is infinite-dimensional? The answer is not only a list of analogues, because infinite-dimensional operators can fail to have eigenvectors while still having spectral behaviour detected by invertibility, approximation, and holomorphic methods.

The course will work mainly with bounded [linear operators on Banach spaces](/page/Linear%20Operators%20on%20Banach%20Spaces) and Hilbert spaces. Banach spaces provide the natural setting for resolvents and complex analysis; Hilbert spaces add orthogonality, adjoints, and normality, which make the strongest diagonalisation theorems possible. This chapter is a map of the terrain, the prerequisites, and the recurring examples.

# 0. Introduction

## What Spectral Theory Tries to Replace

The finite-dimensional model asks us to study a matrix $A \in \mathbb C^{n \times n}$ by the values of $\lambda \in \mathbb C$ for which $A - \lambda I$ is not invertible. In finite dimensions, non-invertibility is equivalent to the existence of a non-zero vector $v$ with $Av = \lambda v$. Infinite-dimensional spaces separate these two ideas, and that separation is the main source of the subject.

[definition: Bounded Linear Operator] Let $X$ and $Y$ be normed spaces over $\mathbb C$. A map $T: X \to Y$ is a [bounded linear operator](/page/Bounded%20Linear%20Operator) if $T$ is linear and there exists $C \ge 0$ such that $\|Tx\|_Y \le C\|x\|_X$ for every $x \in X$. [/definition]

Boundedness is the continuity condition that makes analysis possible. To compare different operators, discuss convergence of operator sequences, and treat perturbations quantitatively, we need a numerical size attached to each bounded operator.

[definition: Operator Norm] Let $X$ and $Y$ be normed spaces and let $T: X \to Y$ be a bounded linear operator. The operator norm of $T$ is \begin{align*} \|T\|_{\mathcal L(X,Y)} = \sup\{\|Tx\|_Y : x \in X,\ \|x\|_X \le 1\}. \end{align*} [/definition]

The notation $\mathcal L(X,Y)$ denotes the normed space of bounded linear operators from $X$ to $Y$, and $\mathcal L(X)$ denotes $\mathcal L(X,X)$. The course often studies $\mathcal L(X)$ not only as a vector space but as an algebra under composition.

[example: Multiplication By A Bounded Function] Let $(E,\mathcal E,\mu)$ be a [measure space](/page/Measure%20Space), let $1 \le p \le \infty$, and let $m \in L^\infty(E)$. Define $M_m:L^p(E)\to L^p(E)$ by $(M_m f)(x)=m(x)f(x)$ for representatives. Linearity follows pointwise: for scalars $a,b$ and functions $f,g$, \begin{align*} M_m(af+bg)(x)=m(x)(af(x)+bg(x))=a(M_m f)(x)+b(M_m g)(x). \end{align*} For $1\le p<\infty$, the defining property of the essential supremum gives $|m(x)|\le \|m\|_{L^\infty}$ for almost every $x$, hence \begin{align*} \|M_m f\|_{L^p}^p=\int_E |m(x)f(x)|^p\,d\mu(x)\le \int_E \|m\|_{L^\infty}^p |f(x)|^p\,d\mu(x)=\|m\|_{L^\infty}^p\|f\|_{L^p}^p. \end{align*} Taking $p$th roots gives $\|M_m f\|_{L^p}\le \|m\|_{L^\infty}\|f\|_{L^p}$. For $p=\infty$, the same almost-everywhere bound gives \begin{align*} \|M_m f\|_{L^\infty}=\operatorname*{ess\,sup}_{x\in E}|m(x)f(x)|\le \|m\|_{L^\infty}\|f\|_{L^\infty}. \end{align*} Thus $M_m$ is bounded and $\|M_m\|_{\mathcal L(L^p(E))}\le \|m\|_{L^\infty}$. To see the reverse inequality in the non-degenerate case, fix $0<c<\|m\|_{L^\infty}$ and set $A_c=\{x\in E:|m(x)|>c\}$. By the definition of essential supremum, $\mu(A_c)>0$. For $p=\infty$, take $f=\mathbf 1_{A_c}$; then $\|f\|_{L^\infty}=1$ and \begin{align*} \|M_m f\|_{L^\infty}=\operatorname*{ess\,sup}_{x\in A_c}|m(x)|\ge c. \end{align*} For $1\le p<\infty$, whenever $A_c$ contains a measurable subset $B$ with $0<\mu(B)<\infty$, take $f=\mathbf 1_B$. Then $\|f\|_{L^p}=\mu(B)^{1/p}$ and \begin{align*} \|M_m f\|_{L^p}^p=\int_B |m(x)|^p\,d\mu(x)\ge \int_B c^p\,d\mu(x)=c^p\mu(B)=c^p\|f\|_{L^p}^p. \end{align*} Hence $\|M_m\|\ge c$. Since this holds for every $c<\|m\|_{L^\infty}$, we get $\|M_m\|_{\mathcal L(L^p(E))}=\|m\|_{L^\infty}$ in the usual non-zero $L^p$ setting. The zero space is the degenerate exception. This example already suggests that the spectrum of an operator should remember the essential range of a function, not only eigenvalues. [/example]

The previous example is diagonal in the pointwise sense, but many important operators are not given by multiplication. The next object records the obstruction to inverting $T-\lambda I$ and is the organising definition for the first part of the course.

[definition: Spectrum And Resolvent Set] Let $X$ be a complex [Banach space](/page/Banach%20Space) and let $T \in \mathcal L(X)$. The resolvent set of $T$ is \begin{align*} \rho(T)=\{\lambda \in \mathbb C : T-\lambda I \text{ is bijective and } (T-\lambda I)^{-1}\in \mathcal L(X)\}. \end{align*} The spectrum of $T$ is \begin{align*} \sigma(T)=\mathbb C\setminus \rho(T). \end{align*} [/definition]

For Banach spaces, the bounded inverse theorem means that bijectivity of $T-\lambda I$ already gives boundedness of the inverse. The inverse is included in the definition because the normed-space setting requires it, and because the resolvent operator itself is the [analytic function](/page/Analytic%20Function) we study.

[definition: Resolvent Operator] Let $X$ be a complex Banach space, let $T \in \mathcal L(X)$, and let $\lambda \in \rho(T)$. The resolvent operator of $T$ at $\lambda$ is \begin{align*} R(\lambda,T)=(T-\lambda I)^{-1}:X\to X. \end{align*} [/definition]

Since $\lambda\in\rho(T)$, this inverse belongs to $\mathcal L(X)$. The sign convention here uses $T-\lambda I$, so later identities must keep this order. Some books use $(\lambda I-T)^{-1}$ instead; the mathematical content is the same after changing signs consistently.

## Why Infinite Dimensions Change Eigenvalues

The first warning is that spectral points need not be eigenvalues. In finite dimensions, injectivity and surjectivity are equivalent for linear maps on the same space; in infinite dimensions, an operator may be injective with dense non-closed range, injective with non-dense range, or fail to be injective. Spectral theory refines these failures rather than hiding them. We write $X^*$ for the [dual space](/page/Dual%20Space) of a normed space $X$, meaning the space of bounded linear functionals $f:X\to\mathbb C$.

[definition: Point Spectrum] Let $X$ be a complex Banach space and let $T \in \mathcal L(X)$. The point spectrum of $T$ is \begin{align*} \sigma_p(T)=\{\lambda \in \mathbb C : \ker(T-\lambda I)\ne \{0\}\}. \end{align*} [/definition]

The point spectrum is the set of genuine eigenvalues. It is often too small to control the operator, so the course introduces approximate eigenvectors and range conditions as complementary spectral data.

[definition: Approximate Point Spectrum] Let $X$ be a complex Banach space and let $T \in \mathcal L(X)$. The approximate point spectrum of $T$ is the set of all $\lambda \in \mathbb C$ for which there exists a sequence $(x_n)$ in $X$ such that $\|x_n\|_X=1$ for every $n$ and \begin{align*} \|(T-\lambda I)x_n\|_X \to 0. \end{align*} [/definition]