Proofs and complements: the basic idea

Observables \(\cdot\) Companion I. October 23, 2026.

Companion to the introductory post.

Each operator includes its domain. Inner products are conjugate-linear in the first argument and linear in the second. Section and theorem numbers are local to this companion; references to another companion name its part explicitly.

The background is that of a first graduate course: completeness and orthogonal projection in Hilbert spaces, the closed graph and uniform boundedness theorems, Hahn–Banach, Banach–Alaoglu, the bicommutant theorem, elementary Banach-algebra spectral theory, the Radon–Nikodym and Carathéodory extension theorems, Fubini and dominated convergence, and Fourier analysis and Sobolev spaces on Euclidean domains. We spell out the spectral, extension and perturbation arguments instead of treating the unbounded spectral theorem as an unexplained prerequisite. When an analytic lemma is needed for the more advanced examples, it is stated where it is used. Historical remarks and descriptions of research programmes in the post are not additional mathematical hypotheses.

Contents.

1. Domains, adjoints and positive functionals

2. Bounded commutative algebras and their spectral measures

3. The unbounded spectral theorem, with domains

4. Cores and resolvent coordinates

5. Cauchy inversion and cyclic observables

6. Unitary groups and Fourier reconstruction

7. The Cayley correspondence

8. Random variables and pullbacks

9. Convergence of observables and resolvent tests

1Domains, adjoints and positive functionals

Proof. The definition of \(S^{\ast} v = w\) is \(\langle Su, v \rangle = \langle u, w \rangle\) for every \(u \in D (S)\). Passing to limits in this identity proves closedness of \(S^{\ast}\); applying it with \(w = \bar{z} v\) gives the orthogonal-complement identity. If \(S\) is symmetric, \(D (S) \subset D (S^{\ast})\), so \(D (S^{\ast})\) is dense. A sequence \(u_n \to 0\) with \(Su_n \to v\) satisfies \(\langle v, w \rangle = 0\) for every \(w \in D (S^{\ast})\), hence \(v = 0\). Its graph closure is therefore an operator graph.

For symmetric \(S\) and \(z = x + iy\), expanding the norm gives

If \(S\) is closed and \((S - z) u_n\) converges with \(y \ne 0\), this identity makes \(u_n\) Cauchy; \(Su_n = (S - z) u_n + zu_n\) also converges. Closedness gives a limit in the range. If \(S = S^{\ast}\), the orthogonal-complement identity and the same estimate show that this closed range is dense, hence all of \(H\).

Conversely suppose both ranges at \(\pm i\) are all of \(H\). Given \(v \in D (S^{\ast})\), solve \((S - i) u = (S^{\ast} - i) v\). Then \(v - u \in \ker (S^{\ast} - i) = \text{Ran} (S + i)^{\perp} = 0\). Thus \(D (S^{\ast}) \subset D (S)\), and equality of operators follows. This argument does not presume that \(S\) is closed; self-adjointness proves it.\(\Box\)

Proof. Here are the measure-construction details. For open \(G\) set

All these numbers lie in \([0, \| \ell \|]\). Monotonicity is immediate. If \(G \subset \bigcup_j G_j\) and \(f\) is an admissible test for \(G\), its compact support has a finite subcover. A continuous partition of unity for that finite cover decomposes \(f = \sum_j f_j\) with \(0 \le f_j \le 1\) and support in \(G_j\). Hence \(\ell (f) \le \sum_j m (G_j)\), proving countable subadditivity for open sets and then for \(\mu^{\ast}\) by outer approximation. If \(G_1, G_2\) are disjoint open sets, admissible tests add without exceeding one; taking suprema gives \(m (G_1 \cup G_2) \ge m (G_1) + m (G_2)\). For sets at positive distance, disjoint open neighbourhoods inside any prescribed open cover give the same inequality for \(\mu^{\ast}\). Thus \(\mu^{\ast}\) is a metric outer measure. Carathéodory's theorem makes all Borel sets measurable; write \(\mu\) for the restriction. Since \(m\) is monotone and \(G\) itself is an allowed outer approximation, \(\mu (G) = m (G)\).

We verify the integral identity, rather than assuming it in the construction. Let \(f \ge 0\) be continuous with compact support, and partition its range into intervals of length \(h\). For each positive level \(jh\), the open set \(G_j = \{f > jh\}\) has mass \(\sup \ell (g)\) over compactly supported \(0 \le g \le 1_{G_j}\). Choose finitely many such \(g_j\) with \(\ell (g_j)\) arbitrarily close to \(\mu (G_j)\). Since \(h \sum_{j \ge 1} g_j \le f\), positivity yields

For the reverse inequality choose a compactly supported cutoff \(\chi\) equal to one on \(\text{supp} f\) and continuous functions \(h_j\) equal to one on \(\{f \ge jh\}\), supported in \(\{f > (j - 1) h\}\), for \(j \ge 2\). Then \(f \le 2 h \chi + h \sum_{j \ge 2} h_j\) and

The upper and lower sums converge, as \(h \downarrow 0\), to \(\int_0^{\infty} \mu (f > t) \hspace{0.17em} dt = \int f \hspace{0.17em} d \mu\). This proves the identity on nonnegative \(C_c\), hence on all of \(C_c\) by linearity. Uniform approximation by compactly supported cutoffs extends it to \(C_0\).

Every open set is an increasing union of compact subsets, for example \(\{x \in [- n, n] : \text{dist} (x, G^c) \ge 1 / n\}\). Countable continuity gives inner regularity on open sets. Outer regularity is built into \(\mu^{\ast}\); finiteness and passage to complements give inner regularity on every Borel set. The measure is Radon. Continuous cutoffs increasing to one give \(\mu (\mathbb{R}) \le \| \ell \|\), while \(| \ell (f) | \le \mu (\mathbb{R}) \|f\|_{\infty}\) gives the reverse inequality. Finally any Radon measure representing \(\ell\) has the displayed formula for its mass on open sets, by compact approximation and continuous cutoffs, so is unique.\(\Box\)

The same proof works on a second-countable locally compact Hausdorff space: use a compact exhaustion and finite partitions of unity. We will use that version for manifolds. We also use the elementary consequence that finite Borel measures on such a space are Radon. One proof is to check inner and outer approximation first on open sets using a countable relatively compact base; the sets with both approximation properties form a sigma-algebra, since the measure is finite.

2Bounded commutative algebras and their spectral measures

Proof. Characters have norm one: \(\chi (1) = 1\) and \(\chi (a)\) belongs to the spectrum of \(a\), so \(| \chi (a) | \le \|a\|\). The character equations define a weak-star closed subset of the unit ball of \(\mathcal{A}^{\ast}\); weak-star compactness therefore makes \(K\) compact. Maximal ideals in a unital commutative Banach algebra are closed: a proper ideal cannot contain an element within distance less than one of the identity, since that element is invertible. The quotient by a maximal ideal is a complex Banach division algebra and is \(\mathbb{C}\) by the Gelfand–Mazur theorem. Thus every spectral value of \(a\) equals \(\chi (a)\) for a character: apply this to a maximal ideal containing \(a - \lambda 1\).

For self-adjoint \(a\) the spectrum is real, and hence \(\chi (a^{\ast}) = \overline{\chi (a)}\) by splitting an arbitrary \(a\) into its real and imaginary self-adjoint parts. The \(C^{\ast}\)-identity and the spectral-radius formula give \(r (a) = \|a\|\) for normal \(a\): repeated squaring gives \(\|a^{2^n} \| = \|a\|^{2^n}\). All elements of a commutative \(C^{\ast}\)-algebra are normal, so the Gelfand map \(a \mapsto \hat{a}\), \(\hat{a} (\chi) = \chi (a)\), is isometric. Its image is closed, self-adjoint, contains the constants and separates characters. The complex Stone–Weierstrass theorem makes it all of \(C (K)\).

For completeness, the approximation step in that theorem follows from its real version. The closure of a real separating algebra containing constants is a lattice: uniform polynomial approximation to \(|t|\) on bounded intervals gives \(|f|\), and hence maxima and minima. Given \(f \in C (K, \mathbb{R})\) and \(\varepsilon > 0\), separation and affine rescaling give, for each pair \(x, y\), an algebra element agreeing with \(f\) at \(x, y\). For fixed \(x\), finitely many neighbourhoods of \(y\) cover \(K\); the minimum of their chosen functions is below \(f + \varepsilon\) everywhere and agrees with \(f\) at \(x\). Finitely many neighbourhoods of \(x\) then cover \(K\); their maximum lies between \(f - \varepsilon\) and \(f + \varepsilon\). This proves density. Complexification proves the version just used.

Apply the representation to \(C^{\ast} (I, T)\). Its characters are determined by \(\chi (T)\), because polynomials in \(T, T^{\ast}\) are dense, and their range is the spectrum of \(T\). Spectral permanence for a unital \(C^{\ast}\)-subalgebra identifies this with the operator spectrum; it follows, for instance, by constructing inverses through continuous approximation to \(1 / t\) for positive elements bounded away from zero. This yields the asserted calculus and its uniqueness.\(\Box\)

The locally compact Stone–Weierstrass theorem follows by adjoining constants on the one-point compactification: a self-adjoint subalgebra of \(C_0 (X)\) which separates points and vanishes at no point is dense. A \(\ast\)-homomorphism of \(C^{\ast}\)-algebras is contractive, since it preserves positivity in unitizations and \(a^{\ast} a \le \|a\|^2 I\); taking norms of the image proves the assertion.

Proof. For \(u \in H\), positivity and contractivity make \(f \mapsto \langle u, \Phi (f) u \rangle\) a bounded positive functional. Proposition 2 gives \(\mu_u\). The parallelogram and complex polarization identities pass to measures by uniqueness. Thus

is conjugate-linear in \(u\), linear in \(v\), and represents \(\langle u, \Phi (f) v \rangle\). For every Borel set \(B\), positivity of \(\mu_{u + tv} (B)\) for complex \(t\) gives \(| \mu_{u, v} (B) |^2 \le \mu_u (B) \mu_v (B)\).

If \(0 \le \chi_n \le 1\) are compactly supported cutoffs increasing to one, then \(\Phi (\chi_n) \Phi (g) v \to \Phi (g) v\) by uniform convergence of \(\chi_n g\) to \(g\). Nondegeneracy and the uniform norm bound extend this to every vector. Consequently \(\mu_u (\mathbb{R}) = \|u\|^2\) and the sesquilinear form \(\mu_{u, v} (B)\) defines a positive contraction \(E (B)\). More generally integration of a bounded Borel \(f\) defines an operator \(\Psi (f)\) of norm at most \(\|f\|_{\infty}\): for simple \(f = \sum c_j 1_{B_j}\) on disjoint sets, Cauchy–Schwarz over the partition bounds \(| \sum c_j \mu_{u, v} (B_j) |\) by \(\|f\|_{\infty} \|u\| \|v\|\); uniform simple approximation proves the general assertion.

We justify multiplication before using any projection property. For \(g \in C_0\) and any \(u, v\), uniqueness of the representing measures for \(f \mapsto \langle u, \Phi (f) \Phi (g) v \rangle\) gives

Hence \(\Psi (f) \Phi (g) = \Psi (fg)\) for bounded Borel \(f\). The class of bounded Borel \(g\) for which this identity holds for every bounded Borel \(f\) contains \(C_0\) and constants and is closed under uniformly bounded pointwise limits: both sides converge weakly by dominated convergence of scalar measures. The functional monotone-class theorem, applied to the algebra \(C_0 +\mathbb{C}1\) generating the Borel sets, gives all bounded Borel \(g\). Thus \(\Psi (f) \Psi (g) = \Psi (fg)\); the adjoint identity follows directly from polarization. In particular \(E (B)^2 = E (B) = E (B)^{\ast}\).

Disjoint projections are orthogonal. If \(B = \bigcup B_j\) disjointly, scalar countable additivity and orthogonality show \(E (B) u = \sum_j E (B_j) u\) in norm. Finally

under bounded pointwise convergence. Uniqueness follows from scalar Riesz–Markov on \(C_0\) and the same monotone-class argument. This proves every assertion without presupposing that the initially constructed positive operators \(E (B)\) were projections.\(\Box\)

The proof for a compact spectrum, or for a second-countable locally compact space, is identical. A unital Borel homomorphism with bounded pointwise weak continuity produces countably additive scalar measures by testing increasing unions of sets; multiplicativity upgrades its weak continuity to strong continuity by the last displayed identity. Without that continuity, evaluation of \(f (n)\) along a free ultrafilter is a multiplicative state annihilating \(C_0 (\mathbb{R})\): continuous cutoffs then all have value zero although they increase to one. This proves the regularity warning in the post.

3The unbounded spectral theorem, with domains

Proof. The truncations are Cauchy precisely when the indicated integral is finite, because the squared norm of their difference is the integral of the squared difference of their multipliers. The projections \(1_{\{|g| \le n\}} (E)\) increase strongly to \(I\), so the domain is dense. Truncation and passage to limits give symmetry. For \(z \not\in \mathbb{R}\), the bounded multiplier \((g - z)^{- 1}\) maps \(H\) into \(D (g (E))\), since \(|g / (g - z) |\) is bounded. Multiplication of truncated integrals followed by norm convergence gives

Both nonreal ranges are therefore onto. Lemma 1 proves self-adjointness, including closedness.

Conversely let \(A = A^{\ast}\). Its bounded resolvent exists by Lemma 1. Define

For \(h = (A + i) u\) the norm identity in that lemma gives \(\|Vh\| = \|h\|\); surjectivity of \(A - i\) makes \(V\) unitary. Also \(\ker (I - V) = 0\). Apply the bounded normal calculus of Theorem 3 and Theorem 4 on the circle to obtain its spectral measure \(F\). Since \(F (\{1\}) H = \ker (I - V) = 0\), the inverse Cayley function

defines a real spectral integral \(B = a (F)\). Its resolvent at \(- i\) is \((I - V) / (2 i) = (A + i)^{- 1}\). Equality of these injective operators gives equality of ranges, hence \(D (B) = D (A)\), and inversion on that range gives \(B = A\). Push \(F\) forward under \(a\) to obtain \(E\) on the line. Its continuous calculus is nondegenerate by dominated convergence of cutoffs.

These constructions are inverse. Indeed the Cayley transform of \(\int x \hspace{0.17em} dE\) is \(\int (x - i) / (x + i) \hspace{0.17em} dE\), whose bounded normal calculus is unique. Thus its circle spectral measure, and then \(E\), are unique. Theorem 4 identifies the equivalent representation of \(C_0\). This also proves uniqueness of the bounded Borel extension, rather than adding it as an assumption.\(\Box\)

Proof. The upper norm estimate follows from \(\|f (A) u\|^2 = \int |f|^2 d \mu_u\). If \(E (\{|f| > c\}) \ne 0\), choose a unit vector in that range; its image has norm at least \(c\). This proves the norm formula. A real \(\lambda\) outside the support has a neighbourhood with zero projection, so \((x - \lambda)^{- 1}\) is bounded on the support and gives a bounded inverse of \(A - \lambda\). If every neighbourhood of \(\lambda\) has nonzero projection, unit vectors \(u_n \in E ((\lambda - 1 / n, \lambda + 1 / n)) H\) belong to \(D (A)\) and satisfy \(\|(A - \lambda) u_n \| \le 1 / n\). No bounded inverse can exist. The support is therefore the spectrum. The remaining assertions follow by applying the norm formula to continuous functions and to coordinate truncations.\(\Box\)

4Cores and resolvent coordinates

Proof. The graph norm is \(\|u\|_A = (\|u\|^2 +\|Au\|^2)^{1 / 2}\). We have \(\|(A - z) u\| \le \|Au\| + |z| \|u\|\). Conversely \(\|u\| \le | \text{Im} z|^{- 1} \|(A - z) u\|\) and \(\|Au\| \le \|(A - z) u\| + |z| \|u\|\). These estimates and surjectivity prove the isomorphism and the density criterion. For \(u \in D (A)\),

Replace \(x^2\) by \(x^{2 k}\) for \(A^k\). A restriction to a core has closure \(A\). Conversely an essentially self-adjoint restriction of \(A\) has self-adjoint closure contained in \(A\), and self-adjoint operators have no proper symmetric extensions; therefore that closure equals \(A\).\(\Box\)

Proof. The norm bound and range statements were proved above. The adjoint and difference formulas follow by applying the bounded calculus to the corresponding scalar identities; thus no unbounded product is being manipulated. The function \(r_z\) never vanishes and separates points. Its conjugate is included in the generated algebra, so the locally compact Stone–Weierstrass theorem applies. Equality of one resolvent identifies the domain by its range and the action by its inverse, even without using the density assertion.\(\Box\)

5Cauchy inversion and cyclic observables

Proof. On a compact subset off the real line, the integrand and all its complex derivatives are uniformly bounded by constants integrable against the finite measure. Differentiation under the integral proves holomorphy. Its imaginary part is \(\int \varepsilon / ((x - s)^2 + \varepsilon^2) \hspace{0.17em} d \mu (x)\). The kernel \(P_{\varepsilon} (t) = \varepsilon / (\pi (t^2 + \varepsilon^2))\) has integral one. For each fixed \(x\) and bounded continuous \(f\), split \(\int P_{\varepsilon} (t) (f (x + t) - f (x)) dt\) at \(|t| = \delta\); continuity controls the inner part and \(\int_{|t| > \delta} P_{\varepsilon} \to 0\) the outer part. Thus \(P_{\varepsilon} \ast f (x) \to f (x)\), bounded by \(\|f\|_{\infty}\). Fubini and dominated convergence give weak convergence and hence uniqueness by Riesz–Markov.

Integrating the kernel over \((a, b)\) gives the difference of two arctangents divided by \(\pi\). Its limit is the indicator of the interval with value \(1 / 2\) at either endpoint. It is bounded between zero and one, so dominated convergence proves the first formula. For the atom formula the integrand is \(\varepsilon^2 / ((x - a)^2 + \varepsilon^2)\), bounded by one and tending to \(1_{\{a\}}\). Finally \(- iy / (x - iy) \to 1\) with modulus at most one, proving the mass formula.\(\Box\)

The boundary density assertion in the post is stronger than weak inversion but follows from measure differentiation. Write \(\mu = f \hspace{0.17em} dx + \mu_s\). At almost every \(s\), \(s\) is a Lebesgue point of \(f\) and \(\mu_s ((s - r, s + r)) / (2 r) \to 0\). For a positive measure \(\nu\) with this last density zero, split its Poisson integral into dyadic annuli \(2^k \varepsilon \le |x - s| < 2^{k + 1} \varepsilon\). Inside a fixed radius \(\delta\), the bound \(\nu ((s - r, s + r)) \le \eta r\) makes the sum at most a constant times \(\eta \sum_{k \ge 0} 2^{- k}\). The part outside \(\delta\) is at most \(\varepsilon \nu (\mathbb{R}) / (\pi \delta^2)\). First let \(\varepsilon \downarrow 0\), then \(\eta \downarrow 0\). The same estimate applied to \(|f (x) - f (s) | dx\) near a Lebesgue point shows \(P_{\varepsilon} \ast f (s) \to f (s)\). Thus the boundary density is \(f\) almost everywhere, and loses \(\mu_s\) unless weak inversion is retained.

For \(\mu = (\delta_{- 1} + \delta_1) / 2\), direct addition gives \(m (z) = z / (1 - z^2)\) and \(m (2 i) = m (i / 2) = 2 i / 5\). This shows that injectivity of the transform on measures is not injectivity of the function \(z \mapsto m (z)\). If \(u\) is cyclic for \(A\), the map \(f \mapsto f (A) u\) is an isometry from \(C_0\) with its \(L^2 (\mu_u)\) norm onto a dense subspace of \(H\). Since \(C_c\) is dense in \(L^2 (\mu_u)\), it extends to a unitary intertwining multiplication by \(x\) and \(A\), including their domains by truncation. This proves the cyclic classification claimed in the post.

6Unitary groups and Fourier reconstruction

Proof. The identity \(\|(U (t) - I) u\|^2 = 2 \| u \|^2 - 2 \text{Re} \langle u, U (t) u \rangle\) upgrades weak continuity at zero to strong continuity there, and the group law extends it to all times. Define \(B\) by the strong derivative at zero. For \(\varphi \in C_c^{\infty} (\mathbb{R})\), the Bochner integral \(u_{\varphi} = \int \varphi (t) U (t) u \hspace{0.17em} dt\) belongs to \(D (B)\), because translation of the integrand gives \(Bu_{\varphi} = - \int \varphi' (t) U (t) u \hspace{0.17em} dt\). Approximate identities give \(u_{\varphi} \to u\), so \(D (B)\) is dense.

The group leaves \(D (B)\) invariant and \(BU (t) u = U (t) Bu\). Consequently

If \(u_n \to u\) and \(Bu_n \to v\), passage to the limit in this identity gives \(U (t) u - u = \int_0^t U (s) vds\). Dividing by \(t\) and letting \(t \to 0\) shows \(u \in D (B)\) and \(Bu = v\). Thus \(B\) is closed. Differentiating \(\langle U (t) u, U (t) v \rangle\) gives \(\langle Bu, v \rangle + \langle u, Bv \rangle = 0\); \(A = - iB\) is symmetric.

For \(\lambda > 0\), put \(R_+ u = \int_0^{\infty} e^{- \lambda t} U (t) u \hspace{0.17em} dt\). A change of variable in \(U (h) R_+ u\) shows that \(R_+ u \in D (B)\) and \(BR_+ u = \lambda R_+ u - u\). For example the difference quotient of the exponential factor tends to \(\lambda R_+ u\), and the missing interval at the origin contributes \(- u\). Replacing \(U (t)\) by \(U (- t)\) gives a right inverse for \(\lambda + B\). Thus \(A \pm i \lambda\) are onto. The proof of Lemma 1, with \(1\) replaced by \(\lambda\), makes \(A\) self-adjoint.

The spectral calculus constructs \(e^{itA}\) with derivative \(iA\) on \(D (A)\). For \(u \in D (A)\), both groups preserve the domain and commute with \(A\); differentiating \(U (- t) e^{itA} u\) gives zero. Its value is therefore \(u\). Density extends equality to every vector. Conversely, for a spectral group and \(u \in D (A)\), dominated convergence with \(| (e^{itx} - 1) / t| \le |x|\) proves the derivative assertion. If the difference quotients converge, their norms remain bounded; Fatou applied to their squared spectral multipliers gives \(\int x^2 d \mu_u < \infty\). This proves the converse domain inclusion and uniqueness.\(\Box\)

Proof. On finite linear combinations of symbols \([t]\), set \(\langle [s], [t] \rangle = F (t - s)\). Positive definiteness makes this a positive semidefinite form. Quotient its null space and complete. Translation \([s] \mapsto [s + t]\) preserves the form, defines a unitary group, and is strongly continuous: first check \(\|[s + t] - [s]\|^2 = 2 F (0) - 2 \text{Re} F (t)\), then finite sums and density. Apply Stone's theorem and its spectral measure to the vector \([0]\). This gives \(F (t) = \int e^{itx} d \mu (x)\) and the mass assertion.

For uniqueness, if a finite complex measure has zero Fourier transform, Fourier inversion and Fubini give zero integral against every \(C_c^{\infty}\) function. Uniform density in \(C_0\) and uniqueness in Riesz–Markov give the zero measure. This also justifies the Fourier reconstruction of the calculus described in the post: \(\int g (t) U (t) dt\) has multiplier \(\int g (t) e^{itx} dt\) for \(g \in L^1\), and inverse Fourier transforms of smooth compactly supported functions give a uniformly dense class in \(C_0\).\(\Box\)

The Laplace representation \(R_A (i) u = i \int_0^{\infty} e^{- t} e^{- itA} u \hspace{0.17em} dt\) follows by integrating the scalar exponential and then using Fubini against the spectral measure. The integral is norm-convergent on each vector. Finite sums of \(e^{itx}\) close in supremum norm to the almost periodic functions, by definition; their closure is not \(C_b\). A single value \(e^{iA}\) does not identify \(A\), since replacing a scalar eigenvalue by that eigenvalue plus \(2 \pi\) preserves it.

7The Cayley correspondence

Proof. The forward direction was established in Theorem 5. Conversely a unitary \(V\) with no fixed vector has \(\overline{\text{Ran} (I - V)} = \ker (I - V^{\ast})^{\perp} = H\). Injectivity of \(I - V\) makes the proposed action well-defined. Expanding inner products and using \(V^{\ast} V = I\) proves symmetry. Directly

Both ranges are onto, so the two-range criterion proves self-adjointness. These identities also recover \(V\) and prove that the two assignments are inverse. The scalar map sends \(0\) to \(- 1\), \(1\) to \(- i\), \(- 1\) to \(i\), and the two ends of the real line to the missing point \(1\); its inverse is \(i (1 + \zeta) / (1 - \zeta)\). If \(1\) lies in the spectrum of \(V\), it need not be an eigenvalue; the inverse can therefore be unbounded while still densely defined. This is exactly the geometry illustrated in the post.\(\Box\)

8Random variables and pullbacks

Proof. Pointwise addition, multiplication and conjugation establish the homomorphism laws; \(\|j_X (f)\| \leqslant \|f\|_{\infty}\). Choose \(0 \leqslant \chi_n \leqslant 1\) in \(C_c (\mathbb{R})\), equal to one on \([- n, n]\). Since \(X\) is finite almost everywhere, dominated convergence gives \(\chi_n (X) u \to u\) for every \(u \in H\). The indicator formula gives orthogonal projections, \(E_X (\mathbb{R}) = I\), and countable additivity in the strong operator topology: for disjoint sets, the remainder of a finite union acts as multiplication by an indicator decreasing pointwise to zero, so dominated convergence applies again. Direct integration gives \(\langle u, j_X (f) u \rangle = \int f \mathrm{d} \mu_u\) with \(\mu_u (B) = \int_{X^{- 1} (B)} |u|^2 \mathrm{d} \mathbb{P}\).

The stated domain makes \(A_X\) well defined and symmetric. It is dense because \(1_{|X| \leqslant n} u \to u\) for every \(u \in H\). To prove self-adjointness, let \(v \in D (A_X^{\ast})\) and write \(w = A_X^{\ast} v\). Testing the adjoint identity on arbitrary functions supported where \(|X| \leqslant n\) yields \(1_{|X| \leqslant n} w = X 1_{|X| \leqslant n} v\). As \(n \to \infty\), these equalities imply \(w =\operatorname{Xv}\) almost everywhere. Since \(w \in H\), the maximal domain condition holds for \(v\), so \(D (A_X^{\ast}) \subseteq D (A_X)\). Symmetry gives the reverse inclusion.\(\Box\)

In the example \(X (t) = 1 / t\) on \((0, 1)\), the domain condition is \(\int_0^1 |u (t) |^2 t^{- 2} \mathrm{d} t < \infty\). It excludes \(u = 1\). An eigenvector with eigenvalue \(\lambda\) would vanish outside \(\{t : 1 / t = \lambda\}\), a null set, so it is zero. The substitution \(x = 1 / t\) gives \(X_{\ast} \mathbb{P}= 1_{(1, \infty)} x^{- 2} \mathrm{d} x\).

Proof. All algebra identities are pointwise and remain true modulo null sets. Compactly supported cutoffs \(\chi_n \uparrow 1\) give \(\chi_n (X) u \to u\) by dominated convergence in \(L^2\), proving nondegeneracy. The pushforward identity follows first for indicators, then simple functions and bounded pointwise approximation. Finiteness and second countability make the Borel probability law Radon by the regularity argument after Proposition 2. A countable base and compact cutoffs give a countable family of continuous functions separating points. If two pullbacks agree, remove the countable union of null exceptional sets for this family. Outside it, all separating values agree, so the maps agree. This proves determination of the random variable, as distinct from its law.\(\Box\)

9Convergence of observables and resolvent tests

Proof. Write \(R_n = R_{A_n} (z)\) and \(R = R_A (z)\). Strong convergence and the common bound imply \(R_n^{\ast} \to R^{\ast}\) weakly. These resolvents are normal, so \(\|R_n^{\ast} u\| = \|R_n u\| \to \|Ru\| = \|R^{\ast} u\|\). Weak convergence together with convergence of norms in a Hilbert space is strong convergence. Products of uniformly bounded strongly convergent operators converge strongly, by adding and subtracting one mixed product. Thus every polynomial without constant term in \(R_n, R_n^{\ast}\) converges. Density of the resolvent algebra in \(C_0\) and contractivity extend this to all \(C_0\) tests, in particular all other resolvents.

For \(f \in C_b\), fix \(u\) and choose \(0 \le \chi \le 1\) in \(C_c\) with \(\|(I - \chi (A)) u\| < \varepsilon\). Strong convergence of \(\chi (A_n)\) makes the corresponding tail for \(A_n\) less than \(2 \varepsilon\) eventually. Since the factors from the same calculus commute,

The first term tends to zero. This proves the bounded continuous assertion, hence pointwise unitary convergence. For \(|t| \le T\), the functions \(x \mapsto e^{itx} \chi (x)\) form a compact subset of \(C_0\) in supremum norm. A finite uniform net and contractivity make their convergence uniform in \(t\); the same tails bound the remaining error. With common nonnegative spectrum, use the compact family \(e^{- tx} \chi (x)\) on \([0, \infty)\), \(0 \le t \le T\), to prove the heat assertion.

Conversely pointwise strong convergence of the unitary groups permits dominated convergence in \(R_{A_n} (i) u = i \int_0^{\infty} e^{- t} e^{- itA_n} u \hspace{0.17em} dt\). This proves equivalence of the resolvent and unitary conditions.

Strong convergence for every \(f \in C_b\) gives weak convergence of each vector's spectral law by taking the diagonal matrix coefficient. Conversely suppose the probability laws converge weakly for every unit vector. By rescaling, the corresponding finite spectral measures converge against bounded continuous functions for every vector. Fix \(f \in C_b\). Polarization of \(\langle u, f (A_n) u \rangle\) shows convergence of \(\langle v, f (A_n) u \rangle\) for every \(u, v\). Thus \(f (A_n) u\) converges weakly to \(f (A) u\). Multiplicativity and preservation of adjoints give

because \(| f |^2 \in C_b\). The Hilbert-space identity for the squared distance now proves strong convergence. This establishes the equivalence with convergence of laws in all vector states.\(\Box\)

The Cayley description gives the same topology. Put \(V_n = I - 2 iR_{A_n} (- i)\) and \(V = I - 2 iR_A (- i)\). Strong convergence of these unitaries is equivalent to convergence of the resolvents at \(- i\), and hence to the preceding conditions. This equivalence is on the space of Cayley unitaries with \(\ker (I - V) = \{ 0 \}\); a strong limit of such unitaries may leave that space, as the escaping-mass example below shows.

Convergence is stable under finite ordered products of bounded continuous tests, even when the observables within a product do not commute. Indeed, if \(T_n \rightarrow T\) and \(S_n \rightarrow S\) strongly and \(\sup_n \| T_n \| < \infty\), write

Each term tends to zero; induction handles any finite product. The spectral bound \(\| f (A_n) \| \leqslant \| f \|_{\infty}\) supplies the needed uniform bounds.

Bounded approximations and the role of a core

For the clipping functions \(b_n (x) = \max (- n, \min (x, n))\), the bounded Borel calculus gives a bounded self-adjoint \(A_n = b_n (A)\) of norm at most \(n\). The substitution rule makes \(f (A_n) = (f \circ b_n) (A)\) for bounded continuous \(f\). Its spectral norm identity is

The integrand tends pointwise to zero and is bounded by \(4 \| f \|_{\infty}^2\), integrable against the finite measure. This proves strong resolvent convergence to \(A\). If \(u \in D (A)\), apply the unbounded spectral norm identity to \(b_n (x) - x\). Its squared modulus is bounded by \(x^2\), integrable for precisely such \(u\), and tends to zero. Hence \(A_n u \rightarrow Au\) on that domain.

Proof. For \(f = (A - z) u\) with \(u \in D_0\), the resolvent equation gives

The core criterion makes these \(f\) dense, and the uniform resolvent bound extends convergence to all \(H\).\(\Box\)

The same algebra approximation proves norm convergence of every bounded test from \(C_0\) under norm resolvent convergence. The extension to \(C_b\) need not hold in norm: on \(L^2 (\mathbb{R})\) take \(A = Q\), \(A_n = Q + 1 / n\). Their resolvents converge in norm by the resolvent identity, whereas \(f (x) = e^{ix^2}\) gives \(\|f (A_n) - f (A)\| = 2\), because \(2 x / n + 1 / n^2\) takes every real value. If an interval has zero limiting spectral projection at each finite endpoint, approximate its indicator from above and below by bounded continuous functions with transitions near those endpoints. Tightness of each vector's spectral laws, furnished by the cutoff argument, deals with unbounded ends. The scalar integrals converge; polarization gives weak convergence of the projections, and convergence of \(\|E_n (I) u\|^2 = \langle u, E_n (I) u \rangle\) gives strong convergence. The example \(A_n = n^{- 1} I\), \(A = 0\), and \(I = (- \infty, 0]\) shows the need for the endpoint condition.

For \(A_n = nI\), all nonreal resolvents converge in norm to zero, which is not an injective operator with dense range and hence is not a resolvent on nonzero \(H\). A proposed self-adjoint limit cannot be omitted from the theorem. For \(A_n = n^2 P_n\) on \(\ell^2\), bounded continuous tests satisfy \((f (A_n) - f (0)) u = (f (n^2) - f (0)) u_n e_n \to 0\) for every \(u\). Nevertheless the unit vector \(u_n = c / n\), with \(c^{- 2} = \sum n^{- 2}\), has \(\langle u, A_n u \rangle = c^2\). This proves the moment warning. For \(A \ge 0\), the bounded approximations \(A_N = AE ([0, N])\) agree with \(A\) eventually on each bounded spectral subspace. The core criterion applies, proving the assertion about bounded spectral approximations.

10References and further reading

• M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, revised edition, Academic Press, 1980; and II: Fourier Analysis, Self-Adjointness, 1975. Volume I develops spectral theory and convergence; Volume II, Theorems X.37–X.38, treats Nelson and Faris–Lavine. Volume I; Volume II bibliographic record.

• G. Teschl, Mathematical Methods in Quantum Mechanics, second edition, AMS, 2014. A complementary textbook treatment of domains, spectral measures, extensions and perturbations. Section 6.6 treats strong and norm resolvent convergence, bounded continuous tests, the core criterion and spectral projections. Author's text and book page.

• E. A. Carlen, Functional Analysis, 2017, Section 7.2. For Riesz–Markov, measure regularity and the distinction between compact and locally compact formulations. Lecture notes.

• J. van Neerven, Functional Analysis, Cambridge University Press, 2022. Chapters 9–10 treat bounded and unbounded normal operators; Chapter 15 connects states and observables. Chapter 10.

• C. Bordenave, Lecture notes on random matrix theory, 11 January 2019, Lecture 3, especially Lemma 3.2. Author's notes page. Cauchy–Stieltjes inversion, recovery of atoms, and weak convergence.

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