The Spectrum of a Self-Adjoint Element

Introduction

The involution of a Banach algebra cuts out the self-adjoint elements, the fixed points $a^* = a$, and the first thing the involution says about the spectrum is that these elements have real spectrum. The fact is analytic and rests on a single inequality: for a self-adjoint $a$ and a non-real $\lambda$, the element $a - \lambda$ is invertible with $\lVert(a-\lambda)^{-1}\rVert \leq \lvert\operatorname{Im}\lambda\rvert^{-1}$, so no non-real number lies in the spectrum. From the reality the whole spectral theory of the self-adjoint elements follows: the spectral radius equals the norm, $r(a) = \lVert a\rVert$, the Gelfand transform of the commutative $\mathrm{C}^*$-algebra generated by $a$ is an isometric $*$-isomorphism onto the continuous functions on the real spectrum, and that is the spectral theorem in its most general algebraic form. This article develops the reality, the norm formula and the spectral theorem for a self-adjoint element.

The article assumes the involutive Banach algebra, the $\mathrm{C}^*$-identity, the isometry of the involution and the contractivity of the $*$-homomorphisms from Involutive Banach Algebras and the Gelfand–Naimark Theorem; the spectrum, the spectral radius, the Gelfand transform and Gelfand duality from Topological Algebras and Banach Algebras; the self-adjoint elements, the positive cone and the decomposition $a = h + ik$ from Self-Adjoint Elements and the Positive Cone and Involutive Linear Algebras; and the continuous functional calculus from The Functional Calculus of a Self-Adjoint Element, which owns the explicit calculus. The grade involution $\alpha$ of the signed block is not used.

Throughout, $A$ is a unital $\mathrm{C}^*$-algebra over $\mathbb{C}$ with involution $a \mapsto a^*$; $\sigma(a)$ is the spectrum and $r(a) = \sup\{\lvert\lambda\rvert : \lambda \in \sigma(a)\}$ the spectral radius; an element is self-adjoint (Hermitian) when $a^* = a$, normal when $a^*a = aa^*$, positive when $a = b^*b$ for some $b$, and unitary when $a^*a = aa^* = 1$; $C^*(a,1)$ is the closed unital $*$-subalgebra generated by $a$.

Reality of the Spectrum

Theorem (the spectrum of a self-adjoint element is real). If $a^* = a$ then $\sigma(a) \subseteq \mathbb{R}$, and for $\lambda \notin \mathbb{R}$,

$$ \lVert (a-\lambda)^{-1}\rVert \leq \frac{1}{\lvert\operatorname{Im}\lambda\rvert} . $$

Proof. Write $\lambda = s + it$ with $t \neq 0$ and $c = a - s$, a self-adjoint element, so that $a - \lambda = c - it$. By the $\mathrm{C}^*$-identity and $(c-it)^*(c-it) = c^2 + t^2$,

$$ \lVert (c-it)x\rVert^2 = \lVert x^*(c^2+t^2)x\rVert = \lVert (cx)^*(cx) + t^2\,x^*x\rVert \geq t^2\,\lVert x^*x\rVert = t^2\lVert x\rVert^2 , $$

using that a sum of positive elements dominates each summand in the order of Self-Adjoint Elements and the Positive Cone, and $\lVert x^*x\rVert = \lVert x\rVert^2$. Hence $a - \lambda$ is injective with closed range and its inverse on the range has norm at most $\lvert t\rvert^{-1}$; the same inequality applied to $(a-\lambda)^* = a - \bar\lambda$ shows the range is dense, so $a - \lambda$ is invertible and $\lambda \notin \sigma(a)$, with the stated bound. $\square$

Corollary (the norm is the spectral radius). For a self-adjoint $a$, $\lVert a\rVert = r(a)$. For a normal $a$, likewise $\lVert a\rVert = r(a)$, and $\lVert a\rVert = r(a^*a)^{1/2}$.

Proof. The spectral radius formula gives $r(a) \leq \lVert a\rVert$ always. For self-adjoint $a$, $\lVert a\rVert^2 = \lVert a^*a\rVert = \lVert a^2\rVert$, and iterating, $\lVert a\rVert = \lVert a^{2^n}\rVert^{2^{-n}} \to r(a)$ by the spectral radius formula. If $a$ is normal then $a^*a = aa^*$ is self-adjoint and $\lVert a\rVert^2 = \lVert a^*a\rVert = r(a^*a)$, and $r(a^*a) = r(a)^2$ for commuting $a,a^*$. $\square$

Proposition (the spectrum of a positive element). If $a = b^*b$ is positive then $\sigma(a) \subseteq [0,\infty)$ and $\lVert a\rVert = r(a)$ is the largest point of the spectrum. Conversely a self-adjoint element with $\sigma(a) \subseteq [0,\infty)$ is positive.

Proof. $a$ is self-adjoint, so its spectrum is real; for real $\lambda < 0$ and $t = \lvert\lambda\rvert^{1/2} > 0$ the element $a - \lambda = a + t^2 = (b^*b + t^2)$ satisfies $\lVert(b^*b + t^2)x\rVert\lVert x\rVert \geq \langle(b^*b+t^2)x,x\rangle = \lVert bx\rVert^2 + t^2\lVert x\rVert^2 \geq t^2\lVert x\rVert^2$, so $a - \lambda$ is invertible and $\lambda \notin \sigma(a)$. The converse is the computation of the continuous functional calculus, The Functional Calculus of a Self-Adjoint Element. $\square$

The Spectral Theorem

Definition. Let $a$ be self-adjoint. The commutative $\mathrm{C}^*$-algebra generated by $a$ is the closed unital $*$-subalgebra

$$ C^*(a,1) = \overline{\{p(a,a^*) : p \in \mathbb{C}[X,Y]\}} , $$

which is commutative because $a$ commutes with $a^* = a$.

Theorem (the spectral theorem for a self-adjoint element). Let $a$ be self-adjoint. The Gelfand transform is an isometric $*$-isomorphism

$$ C^*(a,1) \longrightarrow C(\sigma(a)) , \qquad a \mapsto (\lambda \mapsto \lambda) , $$

so $C^*(a,1)$ is the algebra of continuous functions on the compact real set $\sigma(a)$, and $a$ corresponds to the identity function. In particular $\sigma(a)$ is the compact real set on which the algebra is represented, and the spectrum is recovered from the algebra.

Proof. The algebra $C^*(a,1)$ is a commutative unital $\mathrm{C}^*$-algebra, so the Gelfand transform is an isometric $*$-isomorphism onto $C(\operatorname{Max}(C^*(a,1)))$ by the commutative Gelfand–Naimark theorem; the map $\chi \mapsto \chi(a)$ is a homeomorphism of $\operatorname{Max}(C^*(a,1))$ onto $\sigma(a)$ because the characters are the nonzero multiplicative functionals and $\chi(a)$ runs over the spectrum, which is real by the reality theorem; under this identification the identity function corresponds to $a$. $\square$

Corollary (the spectral mapping theorem, self-adjoint case). For a self-adjoint $a$ and a continuous $f : \sigma(a) \to \mathbb{C}$, there is an element $f(a) \in C^*(a,1)$ with

$$ \sigma(f(a)) = f(\sigma(a)) , $$

the assignment $f \mapsto f(a)$ is an isometric $*$-homomorphism from $C(\sigma(a))$ onto $C^*(a,1)$, and it is the continuous functional calculus. The calculus is developed in The Functional Calculus of a Self-Adjoint Element.

Proof. Compose the inverse of the Gelfand transform with the pullback by $f$; the spectral mapping is $\sigma(f(a)) = \hat f(\sigma(a)) = f(\sigma(a))$ under the identification of the theorem. $\square$

Consequences and Examples

Corollary (the involution moves the spectrum by conjugation). For every $a$, $\sigma(a^*) = \overline{\sigma(a)}$; for an invertible $a$, $\sigma(a^{-1}) = \sigma(a)^{-1}$; and $\sigma(a^*a) \subseteq [0,\infty)$.

Proof. For $\lambda \notin \sigma(a)$, $a - \lambda$ is invertible iff $a^* - \bar\lambda$ is, by the involution and its order two; hence $\lambda \in \sigma(a)$ iff $\bar\lambda \in \sigma(a^*)$. The inverse statement is the spectral mapping theorem for the function $\lambda \mapsto \lambda^{-1}$ applied in the commutative algebra generated by $a$ and $1$; and $a^*a = a^* (a^*){}^*$ is positive, so has non-negative spectrum by the proposition. $\square$

Example (the Hermitian matrix). For $A = M_n(\mathbb{C})$ a self-adjoint element is a Hermitian matrix, its spectrum is the set of real eigenvalues, $\lVert X\rVert = \max\lvert\lambda_i\rvert$, and the spectral theorem reads that $X$ is unitarily diagonalisable with real eigenvalues; the functional calculus sends a continuous function to the matrix with the values of the function on the eigenvalues.

Example (the positive function). For $A = C(X,\mathbb{C})$, a self-adjoint element is a real-valued continuous function, its spectrum is the closure of its range, $\lVert f\rVert_\infty = \sup\lvert f\rvert$, and the positive elements are the non-negative functions. The functional calculus is composition, $g(f) = g \circ f$.

Example (the self-adjoint operator). For $A = B(H)$, a self-adjoint element is a self-adjoint bounded operator, its spectrum is a compact real subset of $\mathbb{R}$, and the spectral theorem identifies $C^*(T,1)$ with $C(\sigma(T))$; the projection-valued measure and the spectral decomposition of $T$ are the operator form of the same statement, and they belong to Operator Algebras and Analysis on Linear Spaces.

Spectral Permanence and the Generated Subalgebra

Theorem (spectral permanence). Let $B$ be a closed unital *-subalgebra of the unital $\mathrm{C}^*$-algebra $A$ with the same unit, and let $a \in B$. Then

$$ \sigma_B(a) = \sigma_A(a) . $$

Hence the spectrum of a self-adjoint element is the same in every unital $\mathrm{C}^*$-subalgebra that contains it, and the functional calculus is permanent: the element $f(a)$ computed in $C^*(a,1)$ is the same element computed in any larger unital $\mathrm{C}^*$-subalgebra.

Proof. One inclusion is immediate. A unital $\mathrm{C}^*$-subalgebra of a $\mathrm{C}^*$-algebra is inverse closed: if $a \in B$ is invertible in $A$, then $a^{-1}$ lies in the closed unital *-algebra generated by $a$, which is contained in $B$. Applying this to $a - \lambda$ for $\lambda \notin \sigma_B(a)$ gives $\sigma_A(a) \subseteq \sigma_B(a)$. $\square$

Corollary (the calculus is intrinsic). The continuous functional calculus takes its values in $C^*(a,1)$, so $f(a)$ depends only on $a$, $f$ and the generated algebra and not on the ambient $\mathrm{C}^*$-algebra; equivalently, the spectrum is a complete invariant of the pair $(a, C^*(a,1))$ for a self-adjoint element.

Summary

A self-adjoint element $a^* = a$ of a $\mathrm{C}^*$-algebra has real spectrum, with the quantitative bound $\lVert(a-\lambda)^{-1}\rVert \leq \lvert\operatorname{Im}\lambda\rvert^{-1}$ for non-real $\lambda$; hence $\lVert a\rVert = r(a)$, and more generally $\lVert a\rVert = r(a^*a)^{1/2}$ for a normal element, while $\lVert a\rVert = r(a)$ for normal $a$. A positive element $b^*b$ has spectrum in $[0,\infty)$ with $\lVert b^*b\rVert$ the largest spectral value, and a self-adjoint element with non-negative spectrum is positive. The commutative unital $\mathrm{C}^*$-algebra $C^*(a,1)$ generated by a self-adjoint $a$ is isometrically $*$-isomorphic to $C(\sigma(a))$, with $a$ corresponding to the identity function: this is the spectral theorem for a self-adjoint element, and it yields the continuous functional calculus with the spectral mapping theorem $\sigma(f(a)) = f(\sigma(a))$. The involution moves the spectrum by conjugation, $\sigma(a^*) = \overline{\sigma(a)}$. The calculus itself is The Functional Calculus of a Self-Adjoint Element.

Summary of Notation

Symbol	Meaning
$a^* = a$	Self-adjoint (Hermitian) element
$\sigma(a)$, $r(a)$	Spectrum and spectral radius
$\lVert(a-\lambda)^{-1}\rVert \leq \lvert\operatorname{Im}\lambda\rvert^{-1}$	Reality of the spectrum, self-adjoint case
$\lVert a\rVert = r(a)$	Norm equals spectral radius for self-adjoint/normal $a$
$a = b^*b$	Positive element, $\sigma(a)\subseteq[0,\infty)$
$C^*(a,1) \cong C(\sigma(a))$	The spectral theorem
$\sigma(f(a)) = f(\sigma(a))$	Spectral mapping, continuous calculus
$\sigma(a^*) = \overline{\sigma(a)}$	The involution moves the spectrum
$\sigma_B(a) = \sigma_A(a)$	Spectral permanence for unital $\mathrm{C}^*$-subalgebras