# spectral theorem

## 1 Introduction

Roughly speaking, the spectral theorems state that normal operators (or self-adjoint operators) are diagonalizable and can be expressed as a sum or, more generally, as an integral of projections  . More specifically, a normal (or self-adjoint  ) operator $T$ is unitarily equivalent  to a multiplication operator in some $L^{2}$-space (http://planetmath.org/L2SpacesAreHilbertSpaces) and we can to it a spectral measure (or a resolution of identity) whose integration gives is $T$.

There is a wide of spectral theorems, each one with its own , applicable to many classes of normal and self-adjoint operators.

## 2 Motivation

We explore here two to motivate the spectral theorem. The first is by recalling the finite-dimensional case, corresponding to the well known result in linear algebra, the spectral theorem for Hermitian matrices (http://planetmath.org/SpectralTheoremForHermitianMatrices) (or the self-adjoint analog). The second motivation comes from the $C^{*}$-algebra  (http://planetmath.org/CAlgebra) theory, by regarding a normal (or self-adjoint) operator as a continuous function  .

### 2.0.1 Finite-dimensional case

Suppose $T$ is a self-adjoint operator in a finite-dimensional Hilbert space $H$. An important fact about self-adjoint operators (not just in finite-dimensional spaces) is the following:

Fact 1 - If $V\subseteq H$ is an invariant subspace by $T$, then so it is $V^{\perp}$, the orthogonal complement   of $V$.

Proof: Let $x\in V$ and $y\in V^{\perp}$. Then $\langle x,Ty\rangle=\langle Tx,y\rangle=0$, where the last term is zero because $V$ is invariant  by $T$, i.e. $Tx\in V$. But this proves that $Ty\in V^{\perp}$. $\square$

Let $v_{1}$ be an eigenvector of $T$ and $V_{1}$ the subspace generated by it. For self-adjoint transformations, Fact 1 above says that $V_{1}^{\perp}$ is also invariant by $T$. Thus, by restriction   (http://planetmath.org/RestrictionOfAFunction), we have a self-adjoint operator $T:V_{1}^{\perp}\longrightarrow V_{1}^{\perp}$ and we could again find an eigenvector and repeat the same . Thus, we are decomposing $H$ as a direct sum of orthogonal  one-dimensional subspaces $H=V_{1}\oplus\dots\oplus V_{n}$, and the operator $T$ can be expressed as a sum

 $T=\sum_{i=1}^{n}\lambda_{i}P_{i}$

where each $\lambda_{i}$ is the eigenvalue     associated with the eigenvector $v_{i}$ and each $P_{i}$ is the orthogonal projection onto the subspace $V_{i}$.

This is exactly the process of diagonalization of a self-adjoint matrix.

For normal operators it is more subtle as Fact 1 is no longer true. The idea to overpass this is that eigenvectors of normal operators are always orthogonal to each other (see this entry (http://planetmath.org/EigenvaluesOfNormalOperators)).

### 2.0.2 $C^{*}$-algebras

Suppose $T$ is a self-adjoint operator in some Hilbert space $H$. The closed *-algebra generated by $T$ and the identity operator  is a commutative $C^{*}$-algebra (http://planetmath.org/CAlgebra), which we denote by $C^{*}(T)$. Hence, the Gelfand-Naimark theorem   (http://planetmath.org/GelfandTransform) and the continuous functional calculus provide an isomorphism      $\displaystyle C^{*}(T)\cong C(\sigma(T))$

where $\sigma(T)$ stands for the spectrum of $T$ and $C(\sigma(T))$ is the $C^{*}$-algebra of continuous functions $\sigma(T)\to\mathbb{C}$.

Recall that the spectrum of a self-adjoint operator is a always a compact subset of $\mathbb{R}$. Thus, we can think of $T$ as a continuous function defined in a subset of $\mathbb{R}$.

It is a well known fact from measure theory that every continuous function $f:X\longrightarrow\mathbb{C}$ can be approximated by linear combinations  of characteristic functions    . With some additional effort it can be seen that each continuous function $f$ is in fact a (vector valued) integral of characteristic functions

 $\displaystyle\displaystyle f=\int_{X}f\,d\chi$

where $\chi$ is the vector measure of characteristic functions $\chi(A):=\chi_{A}$.

We now notice that characteristic functions in $\sigma(T)$ are not continuous in general. Hence, they may not have a correspondent in the $C^{*}(T)$, the $C^{*}$-algebra generated by $T$ and the identity    . The fact is that they do have a correspondent in the von Neumann algebra    generated by $T$. Informally, this is the same as saying that characteristic functions belong to $L^{\infty}(\sigma(T))$ rather then $C(\sigma(T))$.

The correspondent operators in the von Neumann algebra generated by $T$ must be projections (since characteristic functions are projections in $L^{\infty}$), and similarly, $T$ can be approximated by linear combinations of projections and can, in fact, be expressed as an integral of projections:

 $\displaystyle T=\int_{\sigma(T)}\lambda\;dP(\lambda)$

where $P(\lambda)$ is a resolution of identity for $T$ (or a projection valued measure, when $T$ is a normal operator).

## 3 Spectral Theorems

Here we list a series of spectral theorems, applicable to different classes of normal or self-adjoint operators.

### 3.0.1 Normal Operators

• spectral theorem for Hermitean matrices (http://planetmath.org/SpectralTheoremForHermitianMatrices)

This is the finite-dimensional case. It is often referred to as ”the spectral theorem”, especially in linear algebra.

• spectral theorem for bounded (http://planetmath.org/OperatorNorm) normal operators in separable Hilbert spaces

• spectral theorem for bounded normal operators in inseparable Hilbert spaces

• spectral theorem for compact (http://planetmath.org/CompactOperator) normal operators

• spectral theorem for unbounded (http://planetmath.org/OperatorNorm) normal operators