Self-Adjoint Operators

Introduction

The self-adjoint operators of a Hilbert space are the fixed points of the involution $T\mapsto T^*$, and they form much more than a set: they are a real vector space, they carry the order of positivity, they are closed under the Jordan product $S\bullet T=\frac12(ST+TS)$, and they are an order-unit normed space whose order unit is the identity and whose norm is recovered from the order. In the $C^*$-algebraic language they are the self-adjoint part of $B(H)$, and the involution is the linear decomposition of every operator into a real and an imaginary part. The purpose of this article is to study that structure: the real vector space and its decomposition, the order with its order unit and its norm, the Jordan algebra and the triple product, the lattice that the commuting self-adjoint operators form, and the functional calculus as the embedding of the family generated by a single operator into the algebra of continuous functions on its spectrum. The spectral theorem itself, with its spectral measure and its representation as a multiplication, is Self-Adjoint Operators and the Spectral Theorem, and the present article is the structural side of the same family.

This article fixes the real vector space of self-adjoint operators and its decomposition, the order, the order unit and the order-unit norm, the Jordan and triple products, the lattice of commuting self-adjoint operators, and the functional calculus as the algebraic embedding. The spectral theorem is Self-Adjoint Operators and the Spectral Theorem; the positivity, the square root and the polar decomposition are Positive Operators and the Square Root; the numerical range and the positivity criterion are Hermitian Operators and the Numerical Range; the bounded operator theory is Bounded Operators on a Hilbert Space; the abstract $C^*$-algebraic background is Normed and Banach Algebras and The Gelfand Transform.

Throughout, $H$ is a complex Hilbert space with inner product linear in the first argument, $B(H)_{\mathrm{sa}}=\{T\in B(H):T^*=T\}$ is the set of self-adjoint operators, $T\ge0$ means that $T$ is self-adjoint and $\langle Tx,x\rangle\ge0$ for all $x$, and the order on $B(H)_{\mathrm{sa}}$ is $S\le T$ iff $T-S\ge0$.

The Real Vector Space of Self-Adjoint Operators

Proposition (the self-adjoint family is a real vector space). $B(H)_{\mathrm{sa}}$ is a real vector space closed under the Jordan product and under the norm, and it is closed but not an algebra under multiplication:

$$ S,T\in B(H)_{\mathrm{sa}}\ \Longrightarrow\ S+T\in B(H)_{\mathrm{sa}},\ \lambda S\in B(H)_{\mathrm{sa}}\ (\lambda\in\mathbb{R}),\ \tfrac12(ST+TS)\in B(H)_{\mathrm{sa}} . $$

Proof. The involution is linear over the reals and conjugate-linear over the complex scalars, so it fixes sums and real multiples; the adjoint of a product reverses the order, $(ST)^*=T^*S^*$, so a product of self-adjoints is self-adjoint exactly when they commute, and the symmetrised product is always self-adjoint; closedness in the norm is the continuity of the adjoint.

Theorem (unique decomposition). Every $T\in B(H)$ has a unique representation

$$ T=A+iB,\qquad A,B\in B(H)_{\mathrm{sa}},\qquad A=\tfrac12(T+T^*),\quad B=\tfrac1{2i}(T-T^*) , $$

and the map $T\mapsto(A,B)$ is a real-linear isomorphism $B(H)\to B(H)_{\mathrm{sa}}\oplus B(H)_{\mathrm{sa}}$; the operator $T$ is normal exactly when $A$ and $B$ commute, and it is self-adjoint exactly when $B=0$.

Proof. The formulas for $A$ and $B$ are real combinations of $T$ and $T^*$, hence self-adjoint, and they recover $T$; uniqueness is the real dimension count: a self-adjoint operator equal to an anti-self-adjoint one is both, hence zero. The normality criterion is the computation $TT^*-T^*T=2i(AB-BA)$, and the last statement is the definition.

Proposition (the involution as a real algebra structure). The involution $*$ is a real-linear map with $*^2=\mathrm{id}$, so $B(H)_{\mathrm{sa}}$ is the fixed-point set of a real-linear involution of the real algebra $B(H)$; the products satisfy

$$ (ST)^*=T^*S^*,\qquad (ST+S^*T^*)^*=ST+S^*T^* , $$

and the $\mathbb{C}$-linear structure of $B(H)$ is recovered from the real structure and the operator $i$.

Proof. The involution is conjugate-linear by definition, hence real-linear, and its square is the identity; the product rule is the definition of the adjoint, and the complex structure is $(\lambda T)=(\operatorname{Re}\lambda)T+(\operatorname{Im}\lambda)(iT)$.

The Order and the Order Unit

Definition. For $S,T\in B(H)_{\mathrm{sa}}$ the relation $S\le T$ means $T-S\ge0$; a linear functional $\varphi$ on a real vector space $V$ with a cone $V_+$ is positive if $\varphi(V_+)\subseteq[0,\infty)$, and an order unit is an element $e$ such that for every $v$ there is $\lambda>0$ with $-\lambda e\le v\le\lambda e$.

Theorem (the order is a partial order with order unit). The relation $\le$ is a partial order on $B(H)_{\mathrm{sa}}$, compatible with addition and with multiplication by positive scalars; it is antisymmetric, it is not total in dimension at least two, and the identity $I$ is an order unit, since

$$ -\|T\|I\le T\le\|T\|I\qquad(T\in B(H)_{\mathrm{sa}}). $$

Proof. Reflexivity is $T-T=0\ge0$; transitivity is the closure of the positive cone under addition; antisymmetry is the statement that $T\ge0$ and $-T\ge0$ force $T=0$, which follows from $\langle Tx,x\rangle=0$ for all $x$ and the polarisation identity; the order unit inequality is the norm formula $\|T\|=\sup_{\|x\|=1}|\langle Tx,x\rangle|$.

Proposition (the order and the norm). The norm of a self-adjoint operator is the order-unit norm,

$$ \|T\|=\inf\{\lambda>0:-\lambda I\le T\le\lambda I\} , $$

and the order unit is the smallest positive operator dominating every operator of norm at most one; the open unit ball is the set of $T$ with $-\lambda I

Proof. The inequality $-\|T\|I\le T\le\|T\|I$ gives the infimum no larger than $\|T\|$; conversely $-\lambda I\le T\le\lambda I$ gives $|\langle Tx,x\rangle|\le\lambda\|x\|^2$ and hence $\|T\|\le\lambda$ by the norm formula, so the infimum is exactly $\|T\|$.

Proposition (positivity of the form). A linear functional $\varphi$ on $B(H)_{\mathrm{sa}}$ is positive exactly when it is of the form $T\mapsto\sum_k\langle Tx_k,x_k\rangle$ for a sequence with $\sum\|x_k\|^2<\infty$ (the normal case) or is a weak limit of such; the positive functionals are the states, and the order is the dual order induced by the cone of positive operators.

Proof. The functionals $T\mapsto\langle Tx,x\rangle$ are positive, the sums of positives are positive, and the weak limits of positives are positive; conversely every positive functional on the self-adjoint part of a $C^*$-algebra extends to a positive functional on the algebra, which is of the stated form in the normal case by the standard representation theory for $B(H)$.

The Jordan Product and the Triple Product

Proposition (the Jordan algebra). $B(H)_{\mathrm{sa}}$ with the product $S\bullet T=\frac12(ST+TS)$ is a real Jordan algebra: the product is commutative, it is distributive over the real-linear structure, and it satisfies the Jordan identity

$$ (S\bullet T)\bullet(S\bullet S)=S\bullet\bigl(T\bullet(S\bullet S)\bigr), $$

and the involution is the identity on it, so $B(H)_{\mathrm{sa}}$ is a special Jordan algebra.

Proof. Commutativity and distributivity are immediate from the definition; the Jordan identity is verified by expanding both sides in the associative algebra $B(H)$; the last statement is the definition of a special Jordan algebra.

Proposition (the triple product). The ternary product

$$ \{S,T,U\}=\tfrac12(STU+UTS) $$

is real trilinear on $B(H)_{\mathrm{sa}}$, and the Jordan product is the ternary product $S\bullet T=\{S,I,T\}$; the norm and the order are compatible with the triple product in the sense

$$ \{T,T,T\}=T^3\ \text{on the commuting part},\qquad T\bullet T=T^2\ge0 . $$

Proof. The expansions are the definition; the specialisations are direct computations, and the positivity of the square of a self-adjoint operator follows from $A^*A\ge0$.

Remark (the abstract picture). These identities are the axioms of a JB-algebra, and the self-adjoint part of a $C^*$-algebra with the Jordan product is a JB-algebra; the spectral theorem can be developed from the Jordan axioms alone, and the functional calculus is then the representation of the Jordan algebra on the continuous functions of a compact Hausdorff space. The abstract theory is Jordan Algebras and the Jordan Decomposition (Part I).

The Lattice of Commuting Self-Adjoint Operators

Theorem (the real lattice). Let $\mathcal{C}$ be a commutative subalgebra of $B(H)_{\mathrm{sa}}$. Then $\mathcal{C}$ is a lattice under the order: every pair $S,T\in\mathcal{C}$ has a least upper bound $S\vee T$ and a greatest lower bound $S\wedge T$ with

$$ S\vee T=\tfrac12(S+T)+\tfrac12|S-T|,\qquad S\wedge T=\tfrac12(S+T)-\tfrac12|S-T| , $$

where $|S-T|$ is the modulus in the $C^*$-algebra generated by $S$ and $T$; the lattice is distributive, and the union and intersection of the spectral projections realise the lattice operations.

Proof. In the commutative $C^*$-algebra generated by $S$ and $T$, which is isomorphic to $C(X)$ for a compact Hausdorff $X$ by the Gelfand transform, the formulas are the pointwise ones $\max(s,t)$ and $\min(s,t)$; the modulus is the pointwise absolute value, and the spectral projections are the characteristic functions of the level sets, which realise the lattice operations.

Corollary (projections). The projections in a commutative subalgebra form a Boolean algebra under the operations $P\wedge Q=PQ$, $P\vee Q=P+Q-PQ$, and $P^{\perp}=I-P$, and the correspondence between the projections and the measurable sets of the spectrum is an isomorphism of Boolean algebras.

Proof. The lattice formulas for the projections are the special case of the theorem, and the Boolean laws are the algebraic translation of the set operations.

Theorem (the order is a lattice exactly in the commutative case). For a unital $C^*$-algebra $A$ the self-adjoint part $A_{\mathrm{sa}}$ is a lattice in the order if and only if $A$ is commutative. In particular $B(H)_{\mathrm{sa}}$ is a lattice if and only if $\dim H\le1$, so for a Hilbert space of dimension at least two there are pairs of self-adjoint operators with no least upper bound, and the obstruction is non-commutativity.

Proof. If $A$ is commutative the Gelfand transform identifies $A$ with $C(X)$ and the pointwise maximum and minimum give the lattice. Conversely, if $A_{\mathrm{sa}}$ is a lattice, then for any two self-adjoint $S,T$ the element $S\vee T$ is expressed in the algebra generated by $S$ and $T$, and the lattice operations force the pair to commute; the argument is tested on the two-dimensional matrix algebra $\mathbb{C}^{2\times2}$, where two non-commuting projections such as $\begin{pmatrix}1&0\\0&0\end{pmatrix}$ and $\begin{pmatrix}1/2&1/2\\1/2&1/2\end{pmatrix}$ have no least upper bound in the self-adjoint order, so a lattice forces commutativity.

The Functional Calculus as an Embedding

Theorem (the calculus embeds the generated algebra). For a self-adjoint $T$ the map $f\mapsto f(T)$ from $C(\sigma(T))$ to $B(H)_{\mathrm{sa}}$ is an isometric isomorphism onto the closed real subalgebra generated by $T$ and $I$; it carries the order of $C(\sigma(T))$ to the order of the operators, the pointwise product to the operator product, and the pointwise maximum to the lattice operation of the commutative algebra generated by $T$.

Proof. The calculus is an isometric $*$-isomorphism onto the closed algebra generated by $T$, by the spectral theorem and the Stone–Weierstrass theorem; the order preservation is the positivity criterion $f\ge0$ iff $f(T)\ge0$, and the product and lattice statements are the multiplicativity of the calculus and the identification of the lattice operations with pointwise max and min.

Corollary (the functional calculus and the order). For self-adjoint $S,T$ that commute, and for continuous functions on a common compact set containing their spectra,

$$ f(S)\le g(T)\ \Longleftrightarrow\ f\le g\ \text{pointwise on the joint spectrum}, $$

and the calculus is order-continuous: an increasing uniformly bounded net of continuous functions converges strongly to its pointwise limit.

Proof. The order criterion is the positivity of $g-f$ evaluated by the calculus; the order continuity is the spectral theorem applied to the joint algebra and the monotone convergence of the scalar integrals.

Example (the positive cone from the calculus). The cone of positive operators is the image of the cone of nonnegative continuous functions under the calculus, the square root is the image of $\sqrt{\cdot}$, the modulus is the image of $|\cdot|$, and the polar decomposition is the operator-valued form of $z=|z|e^{i\theta}$; these are the computations of Positive Operators and the Square Root, here read as statements about the order embedding.

Summary

The self-adjoint operators of a Hilbert space form the fixed-point set of the real-linear involution $T\mapsto T^*$, hence a real vector space closed under the Jordan product $S\bullet T=\frac12(ST+TS)$ and under the norm; every operator has a unique decomposition $T=A+iB$ into commuting real and imaginary self-adjoint parts, it is normal exactly when those parts commute, and it is self-adjoint exactly when the imaginary part vanishes. The order $S\le T$ iff $T-S\ge0$ is a partial order compatible with the real-linear structure, antisymmetric and not total, with the identity as order unit and with the norm recovered as the order-unit norm $\|T\|=\inf\{\lambda:-\lambda I\le T\le\lambda I\}$. With the Jordan product the self-adjoint operators form a special Jordan algebra, and the associated triple product and JB-algebra axioms underlie the abstract theory of the spectral theorem. Commuting self-adjoint operators form a distributive lattice with $S\vee T=\frac12(S+T)+\frac12|S-T|$ and its dual, and their projections form a Boolean algebra; the lattice can fail for non-commuting pairs. The functional calculus $f\mapsto f(T)$ is an isometric isomorphism of $C(\sigma(T))$ onto the commutative algebra generated by $T$, preserving the order, the product and the lattice, and the positive cone, the square root, the modulus and the polar decomposition are its images of the corresponding scalar functions.

Summary of Notation

Symbol	Meaning
$B(H)_{\mathrm{sa}}$	real vector space of self-adjoint operators
$T=A+iB$	unique decomposition into self-adjoint parts
$S\le T$	order, $T-S\ge0$
$\\|T\\|=\inf\{\lambda:-\lambda I\le T\le\lambda I\}$	order-unit norm
$S\bullet T=\frac12(ST+TS)$	Jordan product
$\{S,T,U\}=\frac12(STU+UTS)$	triple product
$S\vee T=\frac12(S+T)+\frac12\|S-T\|$	lattice join for commuting operators
$PQ$, $P+Q-PQ$, $I-P$	Boolean operations on projections
$f\mapsto f(T)$	functional calculus, order isomorphism
$T\ge0\Leftrightarrow f\ge0$	order preservation of the calculus