# Lecture 12: Spectral Theorem for Bounded Operators, Multiplicative Form
$\newcommand{\C}{\mathbb{C}}$
###### tags: `224a 2020`

## Measure Spaces and Multiplication Operators

**Defn. (Measure Space)** The Borel sigma algebra of a topological space $X$ is the collection of all sets obtainable from the open sets by countable union, countable intersection, and complements. A *Borel measure* is a countable additive measure defined on the Borel sets satisfying certain regularity properties (see RS I.4 for details) which will always be satisfied in this course. We will refer to the pair $(X,\mu)$ as a *measure space*, and call it *finite* if $\mu(X)<\infty$.

**Defn.(Lebesgue-Stieltjes Integral)** Given a nonnegative measure $\mu$ on $X$, a function $f:X\rightarrow \C$ is measurable if $\mu(\{f(x)>t\})$ is measurable for every $t$. We then define its *Lebesgue-Stieltjes* integral as:
$$ \int_X f(x)d\mu(x):=\int_0^\infty \mu(\{f>t\})dt.$$

Lebesgue measure is a special case of this, but there are other very different examples, e.g., the discrete measure $\delta_{x}$ supported on one point, and mixtures of such measures.

### Basic Examples

1. Shift Operators. $\sigma(L)=\{|z|\le 1\}$, with point spectrum in the interior. Adjoint $R=L^*$ has no point spectrum. Residual and continuous spectrum on the homework.

2. Multiplication Operators. $g\in L^\infty(X,\mu)$ for a finite measure space $\mu$ on $X\subset \R$. If $g$ is real-valued, $\sigma(M_g)=\{y: \mu(g^{-1}(y-\epsilon,y+\epsilon))>0\forall \epsilon>0\}$, the essential range of $g$.

3. Adjacency matrix of a path: $(Ax)_n = x_{n-1} + x_{n+1}$ on $H=\ell^2(\mathbb{Z})$. Consider the isometry $U:H\rightarrow L^2(S^1)$ via Fourier series:
$$ Ux = \sum_{n\in \mathbb{Z}} e^{inx}x_n.$$
In this basis, the operator is
$$(UAU^{-1})f = e^{ix}f+e^{-ix}f = 2\cos(x)f(x).$$
Since conjugating by an isometry does not change the resolvent set or spectrum, $$\sigma(A)=\sigma(UAU^{-1})=\mathrm{ess-range}(2\cos(x)) = [-2,2].$$

## Spectral Measures

The functional calculus yields a canonical way to associate a probability measure with a unit vector (state in quantum mechanics). Given self-adjoint $A$ and any unit vector $\psi\in H$, consider the bounded linear map
$$ \ell_\psi(f)= (\psi, f(A) \psi).$$ Notice that $\ell_\psi$ is *positive* because $(\psi, f(A)\psi)\ge 0$ whenever $f\ge 0$ by the functional calculus.

We now appeal to the fact that positive linear functionals on continuous functions always come from positive measures.
**Riesz-Markov Theorem.** If $\ell:C(X)\rightarrow \C$ is a positive, continuous linear functional, there is a unique finite positive Borel measure $(X,\mu)$ such that
$$ \ell(f)=\int_X f(x)d\mu(x).$$

Applying this to $\ell_\psi,$ we obtain a canonical mapping $\psi\mapsto \mu_\psi$. By pluggin in $f=1$ we see that $\mu(X)=(\psi,\psi)=1$, so it is a probability measure.

## Spectral Theorem: Multiplicative Form
The content of the spectral theorem is that we can *always* find a unitary like the one in the last example above.

**Theorem.(Spectral Theorem for Bounded Operators.** If $A=A^*\in L(H)$, there is a finite measure space $(X,\mu)$, a function $g\in L^\infty(X,\mu)$ and a unitary $U:H\rightarrow L^2(X,\mu)$ such that
$$ (UAU^{-1})(f) = M_g f.$$

We begin by recording some consequences of the continuous functional calculus developed in the last lecture.

**Defn.** A vector $\psi\in H$ is called *cyclic* if the closure of the span of $\{p(A)\psi:p\in \C[x]\}$ is all of $H$.

**Thm.** If $A=A^*\in L(H)$ has a cyclic vector $\psi$, then there is an isometry $U:L^2(\sigma(A),\mu_\psi)$ such that
$$ (U^{-1}AU)(f) = M_\lambda f.$$
*Proof.* Reed and Simon VII.2 Lemma 1.$\square$

The full spectral theorem follows because for any $A\in L(H)$, $H$ can be decomposed into (possibly infinitely many) orthogonal invariant subspaces $H_n$ such that $A:H_n\rightarrow H_n$ has a cyclic vector for every $n$. See Reed and Simon VII.2 or the class notes for details.