Strong spectral commutation of self-adjoint operators

(a) Setting. Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the choice hypotheses of the spectral theorem. Let $H$ be a nonzero complex Hilbert space (Hilbert space) and let $A$ and $B$ be self-adjoint operators on $H$ with domains $D(A)$ and $D(B)$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators). By the spectral theorem each of them has a unique spectral projection-valued measure on the Borel sigma-algebra of $\mathbb{R}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, The Borel sigma-algebra of a topological space); write $E_A$ for the spectral projection-valued measure of $A$ and $E_B$ for that of $B$.

(b) Strong spectral commutation. The operators $A$ and $B$ strongly commute, or are spectrally compatible, when their spectral projection-valued measures commute on every pair of Borel sets: $E_A(\mathrm{\Lambda })E_B(\mathrm{\Sigma })=E_B(\mathrm{\Sigma })E_A(\mathrm{\Lambda })\text{for all Borel }\mathrm{\Lambda },\mathrm{\Sigma }\subseteq \mathbb{R}\mathrm{.}$ This is the relation that the term "compatible sharp observables" abbreviates on the physical pages of the category. It is a condition on the two families of projections, not on a single vector or on a formal difference of operator products.

(c) Not to be confused with a commutator identity on a dense invariant domain. Let $D$ be a dense linear subspace of $D(A)\cap D(B)$ such that $AD\subseteq D$ and $BD\subseteq D$. Then $ABu$ and $BAu$ are defined for every $u\in D$. Strong spectral commutation implies $ABu=BAu$ on $D$: Teschl's bounded Borel calculus for a strongly commuting pair gives, for each bounded Borel function $h$, $h(B)D(A)\subseteq D(A)$ and $Ah(B)x=h(B)Ax$ for $x\in D(A)$ (Section 4.2, Lemma 4.5). Apply this with $h_n(\lambda )=\lambda {\mathbf{1}}_{[-n,n]}(\lambda )$. For $u\in D$, both $u$ and $Au$ belong to $D(B)$, so the spectral theorem gives $h_n(B)u\to Bu$ and $h_n(B)Au\to BAu$. Since $A{h}_n(B)u=h_n(B)Au$ and $A$ is closed, it follows that $Bu\in D(A)$ and $ABu=BAu$. The converse can fail even for a dense common invariant domain: the examples page records a pair of self-adjoint extensions for which the commutator vanishes on such a domain but the spectral projections do not commute.

For bounded everywhere-defined self-adjoint $A$ and $B$, if the commutator vanishes on a dense $D$, boundedness extends $AB=BA$ to all of $H$; the bounded Borel calculus then makes this equivalent to strong spectral commutation (Teschl, Section 4.2, Corollary 4.7). No graph-norm core property is needed in the comparison above. Clause (b), rather than a domain commutator identity, is the definition governing joint sharp outcomes.

(d) Status. Clauses (a)-(c) fix a definition and a comparison of conditions. They assert no physical claim, no joint measurement, no simultaneous sharp value and no dynamical statement. All further content about strong spectral commutation used by the category -- the bounded-calculus consequences, the joint projection-valued measure criterion and the physical readings -- is established as separate results with their own hypotheses.

• Why projections and not commutators. For unbounded $A$ and $B$ the products $AB$ and $BA$ may be defined on different domains, so the expression $AB-BA$ need not be an operator at all; the spectral projections $E_A(\mathrm{\Lambda })$ and $E_B(\mathrm{\Sigma })$ are bounded everywhere-defined projections, so their products are always defined and clause (b) makes literal sense. This is why the definition is stated at the level of the projection-valued measures.
• Symmetric role of the two operators. Clause (b) is manifestly symmetric in $A$ and $B$; the notation $E_A(\mathrm{\Lambda })E_B(\mathrm{\Sigma })=E_B(\mathrm{\Sigma })E_A(\mathrm{\Lambda })$ is a single condition on each pair of Borel sets, not two conditions.