Lecture 14: Spectral Theorem for Bounded Operators

$$

tags: 224a

The goal of this lecture is to prove the following important theorem of von Neumann.
Theorem.(Spectral Theorem for Bounded Operators. If

$A=A^{\ast }\in L(H)$, there is a finite measure space

$(X,\mu )$, a function

$g\in L^{\mathrm{\infty }}(X,\mu )$ and a unitary

$U:H\to L^2(X,\mu )$ such that

$$(UA{U}^{-1})(f)=M_{g}f.$$

Continuous Functional Calculus

Let

$C(X)$ denote the Banach space of continuous functions on a compact set

$X$ with the sup norm.

Theorem. (Continuous Functional Calculus) If

$A=A^{\ast }$ there is a linear map

$\varphi ={\varphi }_A:C(\sigma (A))\to L(H)$ with the following properties:

1. $\ast$-homomorphism:
   $\varphi (fg)=\varphi (f)\varphi (g),\varphi (1)=I,\varphi (\bar{f})=\varphi (f{)}^{\ast }$.
2. Spectral Mapping:
   $\sigma ({\varphi }_A(f))=f(\sigma (A))$
3. Isometry:
   $\Vert \varphi (f)\Vert =\Vert f{\Vert }_{\mathrm{\infty }}$.
4. Eigenvector Mapping: If
   $A\psi =\lambda \psi$ then
   ${\varphi }_A(f)\psi =f(\lambda )\psi$.
5. Positivity: If
   $f\ge 0$ then
   $\varphi (f)\ge 0$.

With the normalization that

${\varphi }_A(x)=A$, the above is unique (in fact, assuming only (1) and continuity). We denote it by

${\varphi }_A(f)=f(A)$.

To prove the existence of such a mapping, we first show that it exists for the dense subspace

$P\subset C(\sigma (A))$ of polynomial functions.

Lemma 1.(Spectral Mapping for Polynomials) If

$p\in \mathbb{C}[x]$,

$p(\sigma (A))=\sigma (p(A))$.
Proof. Reed and Simon VII.1

$◻$

Lemma 2.(Isometry for Polynomials) If

$p\in \mathbb{C}[x]\subset C(\sigma (A))$,

$$\Vert p(A)\Vert =\Vert p{\Vert }_{\mathrm{\infty }}.$$

The BLT theorem now implies that

${\varphi }_A(\cdot )$ has a unique extension from

$P$ to

$C(\sigma (A))$. Note that self-adjointness was used in two places: (1) the polynomial functions are dense in

$C(\sigma (A))$ (2)

$r(A)=\Vert A\Vert$ for self-adjoint matrices. The theorem is simply not true without self-adjointness; consider the

$2\times 2$ Jordan block with

$\sigma (J)=\{0\}$ but

$\Vert J\Vert =1$.

Consequences and Spectral Measures

Theorem. If

$A$ is self-adjoint

$\lambda$ is an isolated point in

$\sigma (A)$, then

$\lambda \in {\sigma }_p(A)$.
Proof. Since

$\lambda$ is isolated, there is an

$f\in C(\sigma (A))$ such that

$f(\lambda )=1$ and

$f(x)=0$ on

$\sigma \setminus \{\lambda \}$. We also have

$f^2=f$ and

$\bar{f}=f$, so by the functional calculus,

$f(A)$ must be a nonzero orthogonal projection

$P$. Since

$(x-\lambda )f(x)\equiv 0$, we have

$(A-\lambda )P=0$, whence

$\lambda \in {\sigma }_p(A)$.

$◻$

A more substantial punch line is that the calculus yields a canonical way to associate a probability measure with a state. 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

$\ell :C(X)\to \mathbb{C}$ 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.

Cyclic Vectors

Defn. A vector

$\psi \in H$ is called cyclic if the closure of the span of

$\{p(A)\psi :p\in \mathbb{C}[x]\}$ is all of

$H$.

Thm. If

$A=A^{\ast }\in L(H)$ 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.

$◻$

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

$A:H_n\to H_n$ has a cyclic vector for every

$n$.