Self-Adjoint Operators and the Spectral Theorem

Introduction

A bounded self-adjoint operator is one equal to its own adjoint. The equality is a symmetry that forces the whole spectral theory: the spectrum is real, the operator has a spectral measure, and every bounded Borel function of the operator is defined and is again an operator of the same algebra. The spectral theorem is the precise form of this statement, $T=\int\lambda\,dE(\lambda)$, and the functional calculus $f\mapsto f(T)$ is its most useful corollary, since it converts a statement about a single operator into a statement about a commutative algebra of operators. This article develops the self-adjoint operator, the spectral measure, the theorem and the calculus, and it presents the two structural consequences that classify the operator: the Cayley transform, which converts self-adjointness into unitarity, and the multiplication model, in which the operator acts as a multiplication by the independent variable.

The bounded operator theory is Bounded Operators on a Hilbert Space; the spectral theorem in its measure-theoretic form and the compact case are stated and proved in Banach and Hilbert Spaces, and the multiplication model of the theorem is developed in The Spectral Operator. The algebraic and order-theoretic study of the self-adjoint family is Self-Adjoint Operators; the unitary case with its own measure on the circle is Unitary Operators and the Spectral Measure; the positivity and the square root are Positive Operators and the Square Root; the unbounded case is Unbounded Operators and Spectral Measures. The measure and integration used are Measure Theory and Integration, and the $L^2$ theory is Banach and Hilbert Spaces.

Throughout, $H$ is a complex (or real) Hilbert space with inner product $\langle\cdot,\cdot\rangle$ linear in the first argument, $T\in B(H)$ is bounded, $T^*$ is its adjoint, $\sigma(T)$ is its spectrum and $E$ is a spectral measure on the Borel subsets of $\sigma(T)$.

Self-Adjoint Operators

Definition. $T$ is self-adjoint if $T^*=T$; it is positive if $T=T^*$ and $\langle Tx,x\rangle\ge0$ for all $x$; the self-adjoint operators form a real vector space and are ordered by $S\le T$ iff $T-S$ is positive.

Theorem (reality of the spectrum and the norm formula). For a self-adjoint $T$ the spectrum is real, $\sigma(T)\subseteq\mathbb{R}$, and

$$ \|T\|=\sup_{\|x\|=1}|\langle Tx,x\rangle|=\max_{\lambda\in\sigma(T)}|\lambda| . $$

Proof. The numerical range of a self-adjoint operator lies in $\mathbb{R}$ and its closure contains the spectrum, so the spectrum is real. For the norm formula, the bound $\|T\|\ge\sup|\langle Tx,x\rangle|$ is Cauchy–Schwarz; the reverse inequality follows by polarising $|\langle T(x+y),x+y\rangle|$ and using that the diagonal is real, which gives $\|Tx\|\le\sup_{\|x\|=1}|\langle Tx,x\rangle|\|x\|$; the identification with $\max|\lambda|$ is the spectral theorem below.

Proposition (real and imaginary parts; the Cayley transform). Every bounded operator is $T=A+iB$ with $A,B$ self-adjoint and this decomposition is unique; the Cayley transform

$$ C(T)=(T-iI)(T+iI)^{-1} $$

is a unitary of $H$ whose spectrum does not contain $1$, and the assignment $T\mapsto C(T)$ is a bijection from the bounded self-adjoint operators onto the unitaries $U$ with $1\notin\sigma(U)$, with inverse $U\mapsto i(I+U)(I-U)^{-1}$.

Proof. Since $T\pm iI$ is invertible for self-adjoint $T$ by the reality of the spectrum, the transform is defined; $C(T)^*C(T)=(T+iI)^{-1}(T-iI)(T+iI)(T-iI)^{-1}=I$ by the commutativity of the resolvent factors, so $C(T)$ is unitary, and the inverse formula inverts the Möbius transformation; the bijection is the two-sided verification of the compositions.

Proposition (the order). The positive operators are exactly the self-adjoint operators with $\sigma(T)\subseteq[0,\infty)$; the order is compatible with addition and with the unit and is an order on the self-adjoint family: every bounded set has a least upper bound, $0\le T\le I$ exactly when $T$ is a contraction that is positive, and $T^*T\ge0$ with $\|T^*T\|=\|T\|^2$ for every $T$.

Proof. The spectral criterion for positivity is the spectral theorem; the lattice statement is the spectral theorem applied to the spectral families of two commuting self-adjoint operators; the remaining identities are the $C^*$-identity and the reality of the diagonal.

The Spectral Measure and the Spectral Theorem

Definition. A spectral measure on $H$ is a map $E$ from the Borel subsets of a compact set $\Sigma\subseteq\mathbb{C}$ to projections with

$$ E(\varnothing)=0,\quad E(\Sigma)=I,\quad E(S\cap S')=E(S)E(S'),\quad E\Bigl(\bigcup_nS_n\Bigr)=\sum_nE(S_n) $$

for the disjoint countable unions, the sum converging in the strong operator topology.

Theorem (the spectral theorem, self-adjoint case). Let $T$ be a bounded self-adjoint operator. Then there is a spectral measure $E$ on the Borel subsets of the compact set $\sigma(T)\subseteq\mathbb{R}$ with

$$ T=\int_{\sigma(T)}\lambda\,dE(\lambda), $$

and for every bounded Borel function $f$ on $\sigma(T)$ the operator

$$ f(T)=\int_{\sigma(T)}f(\lambda)\,dE(\lambda) $$

is bounded with $\|f(T)\|=\sup_{\lambda\in\sigma(T)}|f(\lambda)|$; the map $f\mapsto f(T)$ is a $\mathbb{K}$-algebra homomorphism sending $1$ to $I$, the identity function to $T$, and $\bar f$ to $f(T)^*$.

Proof. The spectral measure is constructed from the continuous functional calculus: polynomials in $T$ are computed on the spectrum, the Stone–Weierstrass theorem extends the calculus to the continuous functions, and the Riesz representation theorem converts the positive linear functionals $f\mapsto\langle f(T)x,x\rangle$ into measures; the strong-operator additivity and the multiplicativity are then the countable additivity and the multiplicativity of the integral. The norm identity is the isometry of the $*$-homomorphism of the bounded Borel functions.

Theorem (the spectral theorem, normal case). A bounded operator $T$ on a complex Hilbert space is normal, $T^*T=TT^*$, if and only if there is a spectral measure $E$ on the Borel subsets of $\sigma(T)\subseteq\mathbb{C}$ with $T=\int\lambda\,dE(\lambda)$; equivalently, $T$ is unitarily equivalent to a multiplication by the independent variable on a direct sum of spaces $L^2(\sigma(T),\mu_j)$.

Proof. The self-adjoint case applied to the commuting operators $\operatorname{Re}T$ and $\operatorname{Im}T$ produces their joint spectral measure, which is a measure in the plane supported on the joint spectrum; normality is the commutativity needed for the two real parts to be simultaneously diagonalised, and the multiplication model is the spectral theorem of The Spectral Operator.

The Functional Calculus

Proposition (the calculus). The map $f\mapsto f(T)$ from the bounded Borel functions on $\sigma(T)$ is a $*$-homomorphism onto the von Neumann algebra generated by $T$, it is isometric for the uniform norm, it is weak-operator continuous on bounded sets, and its kernel is the ideal of functions vanishing on the support of $E$.

Proof. Each statement is the corresponding statement for the integral against a spectral measure: multiplicativity is the multiplicativity of the integral, the $*$-property is the reality of $E$, the isometry is the norm formula, and the kernel is the vanishing of the measure on the support.

Corollary (the calculus acts by spectral projection). For a Borel set $S$ the operator $E(S)$ is the characteristic function of $T$, $E(S)=\mathbf{1}_S(T)$; conversely the spectral measure is recovered from the calculus, and the projections in the von Neumann algebra generated by $T$ are exactly the $E(S)$.

Proof. The characteristic function of $S$ integrated against $E$ is $E(S)$ by the definition of the integral, and the converse is the inversion formula for the measure.

Corollary (the spectral mapping theorem). For a continuous function $f$ one has $\sigma(f(T))=f(\sigma(T))$, and the eigenvalues of $T$ are the atoms of $E$, with the eigenspace at $\lambda$ the range of the projection $E(\{\lambda\})$.

Proof. $f(T)-\mu I=(f-\mu)(T)$ is invertible exactly when $f-\mu$ does not vanish on the spectrum; the eigenvalue statement is the atomic part of the spectral measure.

The Multiplication Model and Multiplicity

Theorem (the model). A bounded self-adjoint operator $T$ on a separable Hilbert space is unitarily equivalent to the multiplication by the independent variable on $L^2(\sigma(T),\mu)$ when it has a cyclic vector; in general it is unitarily equivalent to $M_\lambda$ on a direct sum $\bigoplus_nL^2(\sigma(T),\mu_n)$, and the measure class together with the multiplicity function is a complete unitary invariant.

Proof. This is the spectral operator of The Spectral Operator; the cyclic vector produces the transform that sends $T^nx_0$ to $\lambda^n$, and the general case decomposes the space into cyclic subspaces.

Proposition (support and point spectrum). The spectral measure of $T$ is supported on $\sigma(T)$, and its atoms are exactly the eigenvalues: $E(\{\lambda\})\neq0$ iff $\lambda$ is an eigenvalue, in which case the range of $E(\{\lambda\})$ is the eigenspace. The continuous part of $E$ corresponds to the continuous spectrum, and $T$ has no eigenvectors exactly when $E$ is non-atomic.

Proof. The complement of the support is the set on which $E$ vanishes, and there $T-\lambda I$ is invertible; an atom at $\lambda$ gives a nonzero projection whose range consists of eigenvectors, and conversely an eigenvector lies in the range of the corresponding atom.

Example (compact and finite-rank). For a compact self-adjoint operator the spectral measure is purely atomic, supported on $\{0\}\cup\{\lambda_n\}$ with $\lambda_n\to0$, and the spectral theorem is the orthonormal eigenbasis expansion of Compact Operators. For a finite-rank self-adjoint operator the measure is a finite atomic measure and the theorem is the orthogonal diagonalisation of a Hermitian matrix.

Example (multiplication operators). For $H=L^2(X,\mu)$ and a real $g\in L^\infty(X,\mu)$ the operator $M_g f=gf$ is self-adjoint, its spectral measure is $E(S)=M_{\mathbf{1}_{g^{-1}(S)}}$, and the spectral theorem is the identity $M_g=\int\lambda\,dE(\lambda)$; the model theorem in the other direction is exactly this computation.

Summary

A bounded self-adjoint operator has real spectrum, satisfies $\|T\|=\sup_{\|x\|=1}|\langle Tx,x\rangle|=\max|\sigma(T)|$, is positive exactly when its spectrum lies in $[0,\infty)$, and is related to the unitaries by the Cayley transform $C(T)=(T-iI)(T+iI)^{-1}$. The spectral theorem represents it as $T=\int\lambda\,dE(\lambda)$ against a spectral measure on its real spectrum, and the functional calculus $f(T)=\int f\,dE$ is an isometric $*$-homomorphism from the bounded Borel functions onto the von Neumann algebra generated by $T$, sending $\mathbf{1}_S$ to $E(S)$, satisfying the spectral mapping theorem, and having as its kernel the functions vanishing on the support of $E$. A bounded operator on a complex Hilbert space is normal exactly when it has such a representation with a measure in the plane, equivalently when it is a multiplication by the independent variable on a direct sum of $L^2$ spaces; the measure class and the multiplicity form a complete unitary invariant, and the atoms of the measure are precisely the eigenvalues. The compact and finite-rank cases are the purely atomic specialisations, and the order structure and the algebraic properties of the self-adjoint family are Self-Adjoint Operators.

Summary of Notation

Symbol	Meaning
$T=T^*$	self-adjointness, spectrum real
$\\|T\\|=\sup_{\\|x\\|=1}\|\langle Tx,x\rangle\|$	the norm formula
$C(T)=(T-iI)(T+iI)^{-1}$	the Cayley transform, a unitary avoiding $1$
$E(S)$	spectral measure, projection-valued
$T=\int\lambda\,dE(\lambda)$	the spectral theorem
$f(T)=\int f\,dE$	the functional calculus
$\\|f(T)\\|=\sup\|f\|$	isometry of the calculus
$\sigma(f(T))=f(\sigma(T))$	spectral mapping
$E(\{\lambda\})$	eigenspace projection at an eigenvalue
$\bigoplus_nL^2(\sigma(T),\mu_n)$	multiplication model with multiplicity