The Nuclear Spectral Theorem of Gelfand and Maurin

Contents

Hypotheses and statement
The direct integral
What nuclearity buys
Generalized eigenvectors, and completeness

This appendix proves Theorem 16.107 of Hilbert Spaces — that a self-adjoint operator acting continuously on the test space of a Gelfand triple possesses a complete set of generalized eigenvectors — and says exactly what the proof assumes. It is the theorem that licenses the completeness relation $\int\ketbra{\lambda}{\lambda}\,\dd\mu(\lambda)=\identity$ of Remark 16.108, and with it Dirac's formalism [Dirac:1930b] in the form physics actually uses it. The result is due to Gelfand and his collaborators [Gelfand:1964], where it appears in volume 4 of Generalized Functions; Maurin obtained it independently at about the same time, and the theorem carries both names.

The architecture has three stages, and only the second imports anything. First, the spectral theorem of The Spectral Theorem for a Bounded Self-Adjoint Operator and Stone's Theorem on One-Parameter Unitary Groups is read as a direct-integral decomposition: $\mathcal{H}$ is realised as a space of square-integrable fields $\lambda\longmapsto\hat x(\lambda)$ over the spectrum, in which $A$ acts by multiplication by $\lambda$. At that stage the assignment $x\longmapsto\hat x(\lambda)$ is defined only up to a $\mu$-null set, and only as a map of equivalence classes: for a fixed $\lambda$ it is not a map on $\mathcal{H}$ at all, so there are no eigenvectors yet, generalized or otherwise. Second, nuclearity of the test space is used — and this is the only import — to produce, for $\mu$-almost every $\lambda$, a genuine continuous map $\Phi\longrightarrow\mathcal{H}_{\lambda}$ that is a version of the fibre map. Third, each vector in the fibre is paired with that map, giving a continuous linear functional on $\Phi$, and the multiplication form of $A$ makes it a generalized eigenvector of eigenvalue $\lambda$; the isometry of the first stage, read fibrewise, is the completeness relation.

Hypotheses and statement

The standing hypotheses of Definition 16.103 are made explicit here, because the proof uses each of them.

Definition A27.1 (Countably Hilbert nuclear space).

A countably Hilbert space is a vector space $\Phi$ whose topology is given by an increasing sequence of norms $\norm{\cdot}_{1}\leq\norm{\cdot}_{2}\leq\cdots$, each coming from an inner product, and which is complete for the resulting metric; $\Phi_{p}$ denotes the Hilbert space completion of $\Phi$ in $\norm{\cdot}_{p}$. It is nuclear if for every $p$ there is $q\geq p$ such that the canonical map $\Phi_{q}\longrightarrow\Phi_{p}$ is nuclear, that is, of the form $u\longmapsto\sum_{k}s_{k}\braket{a_{k}}{u}\,b_{k}$ with $\sum_{k}\abs{s_{k}}<\infty$ and $\set{a_{k}}$, $\set{b_{k}}$ orthonormal. Rests on Definitions 16.2 and 16.103.

Theorem A27.2 (Gelfand–Maurin).

Let $\Phi\subseteq\mathcal{H}\subseteq\Phi'$ be a Gelfand triple (Definition 16.103) in which $\mathcal{H}$ is separable and $\Phi$ is a separable countably Hilbert nuclear space, densely and continuously embedded in $\mathcal{H}$. Let $A$ be self-adjoint on $\mathcal{H}$ and map $\Phi$ continuously into itself. Then there exist a finite Borel measure $\mu$ on $\sigma(A)$, separable Hilbert spaces $\mathcal{H}_{\lambda}$ and, for $\mu$-almost every $\lambda$, a continuous linear map $\Lambda_{\lambda}\!:\Phi\longrightarrow\mathcal{H}_{\lambda}$ such that

$\Lambda_{\lambda}(A\varphi)=\lambda\,\Lambda_{\lambda}(\varphi)$ for every $\varphi\in\Phi$ and $\mu$-almost every $\lambda$;
for every $\xi\in\mathcal{H}_{\lambda}$ the functional $F_{\lambda,\xi}(\varphi) =\braket{\xi}{\Lambda_{\lambda}\varphi}_{\mathcal{H}_{\lambda}}$ belongs to $\Phi'$ and is a generalized eigenvector of $A$ with eigenvalue $\lambda$ (Definition 16.105);
writing $\set{\xi_{n}}$ for an orthonormal basis of $\mathcal{H}_{\lambda}$ and $F_{\lambda,n}=F_{\lambda,\xi_{n}}$,

\begin{equation}\tag{A27.1} \braket{\varphi}{\psi}=\int_{\sigma(A)}\sum_{n} F_{\lambda,n}(\varphi)^{\ast}\,F_{\lambda,n}(\psi)\, \dd\mu(\lambda)\ec\qquad \forall\,\varphi,\psi\in\Phi\ep \end{equation}

When the spectrum of $A$ is simple — when $\mathcal{H}$ has a single cyclic vector — every $\mathcal{H}_{\lambda}$ is one dimensional, the sum in Equation (A27.1) has one term, and the identity is Equation (16.69) exactly as stated in Hilbert Spaces. Rests on Definition 16.105, Definition 16.103 and Theorem A24.1.

The direct integral

Proposition A27.3 (Direct-integral form of the spectral theorem).

Let $A$ be self-adjoint on a separable $\mathcal{H}$, with projection-valued measure $E$ (Theorem A24.1 and Proposition A25.10). There are a finite Borel measure $\mu$ on $\R$, supported on $\sigma(A)$, and an isometry

\begin{equation}\tag{A27.2} J\!:\mathcal{H}\longrightarrow \set{\text{fields }\lambda\longmapsto\hat x(\lambda) \in\ell^{2}(\N)}\ec\qquad \norm{x}^{2}=\int_{\R}\norm{\hat x(\lambda)}^{2}_{\ell^{2}} \,\dd\mu(\lambda)\ec \end{equation}

such that for $x\in D(A)$ one has $\widehat{Ax}(\lambda)=\lambda\,\hat x(\lambda)$ for $\mu$-almost every $\lambda$. Rests on Theorem A24.1, Lemma A24.13 and Proposition A25.10.

Proof.

Derives Proposition A27.3. Cyclic decomposition. The construction of Lemma A24.13 applies verbatim with the bounded Borel calculus of Proposition A24.9 — or, for unbounded $A$, of Proposition A25.10 — in place of the continuous one: it used only that the cyclic subspaces $\mathcal{H}_{n}=\overline{\set{g(A)x_{n}\mid g\ \text{bounded Borel}}}$ are $A$-invariant with $A$-invariant orthogonal complements, and that $\mathcal{H}$ is separable. So $\mathcal{H}=\bigoplus_{n}\mathcal{H}_{n}$ with each $x_{n}$ cyclic in $\mathcal{H}_{n}$.

Each block. As in Lemma A24.12, $\norm{g(A)x_{n}}^{2}=\int\abs{g}^{2}\dd\mu_{n}$ with $\mu_{n}(\Omega)=\braket{x_{n}}{E(\Omega)x_{n}}$, so $g(A)x_{n}\longmapsto g$ extends to a unitary $U_{n}\!:\mathcal{H}_{n}\longrightarrow L^{2}(\R,\mu_{n})$ — the bounded Borel functions are dense in $L^{2}(\mu_{n})$, since the simple functions already are — and $U_{n}AU_{n}^{-1}$ is multiplication by $\lambda$.

One measure. Put $\mu=\sum_{n}2^{-n}\norm{x_{n}}^{-2}\mu_{n}$, a Borel measure with $\mu(\R)\leq1$, supported on $\sigma(A)$ because every $\mu_{n}$ is. Each $\mu_{n}$ is absolutely continuous with respect to $\mu$, so the Radon–Nikodym theorem (Remark A27.7) provides densities $\rho_{n}=\dd\mu_{n}/\dd\mu\geq0$.

The isometry. For $x=\sum_{n}v_{n}$ with $v_{n}\in\mathcal{H}_{n}$ write $h_{n}=U_{n}v_{n}\in L^{2}(\mu_{n})$ and define

\begin{equation}\tag{A27.3} \hat x(\lambda) =\Bigl(\sqrt{\rho_{n}(\lambda)}\;h_{n}(\lambda)\Bigr)_{n\in\N} \in\ell^{2}(\N)\ep \end{equation}

Then, exchanging sum and integral by monotone convergence (Remark 16.1),

\begin{equation*} \int_{\R}\norm{\hat x(\lambda)}^{2}_{\ell^{2}}\dd\mu =\sum_{n}\int_{\R}\abs{h_{n}}^{2}\rho_{n}\,\dd\mu =\sum_{n}\int_{\R}\abs{h_{n}}^{2}\,\dd\mu_{n} =\sum_{n}\norm{v_{n}}^{2}=\norm{x}^{2}\ec \end{equation*}

which is Equation (A27.2); in particular $\hat x(\lambda)$ lies in $\ell^{2}(\N)$ for $\mu$-almost every $\lambda$. Since $A$ acts on each block as multiplication by $\lambda$, and multiplication commutes with the fibrewise rescaling by $\sqrt{\rho_{n}}$, $\widehat{Ax}(\lambda)=\lambda\hat x(\lambda)$ almost everywhere for $x\in D(A)$.

∎

The map $J$ is the whole content of the spectral theorem written fibrewise, and it is also the whole of the difficulty: $\hat x$ is an equivalence class of fields modulo $\mu$-null sets, so the expression $\hat x(\lambda)$ for one fixed $\lambda$ has no meaning, and there is nothing yet that could be evaluated on a test function.

What nuclearity buys

Theorem A27.4 (Nuclear spaces embed by Hilbert–Schmidt maps; quoted).

Let $\Phi$ be a countably Hilbert nuclear space (Definition A27.1) continuously embedded in a Hilbert space $\mathcal{H}$. Then there is an index $q$ such that the embedding extends to a Hilbert–Schmidt map $J_{q}\!:\Phi_{q}\longrightarrow\mathcal{H}$, that is, one for which

\begin{equation}\tag{A27.4} \sum_{k}\norm{e_{k}}_{\mathcal{H}}^{2}<\infty \end{equation}

for one, hence for every, orthonormal basis $\set{e_{k}}$ of $\Phi_{q}$. Rests on Definition A27.1.

The Schwartz triple Equation (16.66) is the case to keep in mind: $\mathcal{S}(\R)$ is countably Hilbert for the norms built from the harmonic-oscillator Hermite expansion, and Equation (A27.4) holds there because the Hermite coefficients of a Schwartz function decay faster than any power.

Proposition A27.5 (The fibre maps are continuous on $\Phi$).

Under the hypotheses of Theorem A27.2 there are, for $\mu$-almost every $\lambda$, a constant $C(\lambda)<\infty$ and a linear map $\Lambda_{\lambda}\!:\Phi_{q}\longrightarrow\ell^{2}(\N)$ with

\begin{equation}\tag{A27.5} \norm{\Lambda_{\lambda}\varphi}_{\ell^{2}} \leq C(\lambda)\,\norm{\varphi}_{q}\ec\qquad \forall\,\varphi\in\Phi_{q}\ec \end{equation}

such that for each fixed $\varphi\in\Phi$ one has $\Lambda_{\lambda}\varphi=\hat\varphi(\lambda)$ for $\mu$-almost every $\lambda$. In particular $\Lambda_{\lambda}$ is continuous for the topology of $\Phi$. Rests on Theorem A27.4 and Proposition A27.3.

Proof.

Derives Proposition A27.5. Let $q$ and $\set{e_{k}}$ be as in Theorem A27.4. Applying Equation (A27.2) to each $e_{k}$ and summing,

\begin{equation}\tag{A27.6} \int_{\R}\left(\sum_{k}\norm{\hat e_{k}(\lambda)}^{2}_{\ell^{2}} \right)\dd\mu(\lambda) =\sum_{k}\norm{e_{k}}^{2}_{\mathcal{H}}<\infty\ec \end{equation}

the exchange of sum and integral being monotone convergence for a series of non-negative terms (Remark 16.1). A non-negative function with finite integral is finite almost everywhere, so

\begin{equation}\tag{A27.7} C(\lambda)^{2}=\sum_{k}\norm{\hat e_{k}(\lambda)}^{2}_{\ell^{2}} <\infty\qquad\text{for }\mu\text{-almost every }\lambda\ep \end{equation}

This is the entire role of nuclearity: without Equation (A27.4) the left-hand side of Equation (A27.6) is a divergent series and Equation (A27.7) says nothing.

Fix such a $\lambda$. For $\varphi\in\Phi_{q}$ expand $\varphi=\sum_{k}c_{k}e_{k}$ with $c_{k}=\braket{e_{k}}{\varphi}_{q}$ and $\sum_{k}\abs{c_{k}}^{2}=\norm{\varphi}_{q}^{2}$ (Theorem 16.30), and put

\begin{equation}\tag{A27.8} \Lambda_{\lambda}\varphi=\sum_{k}c_{k}\,\hat e_{k}(\lambda)\ep \end{equation}

The series converges absolutely in $\ell^{2}(\N)$, since by Cauchy–Schwarz

\begin{equation*} \sum_{k}\abs{c_{k}}\,\norm{\hat e_{k}(\lambda)} \leq\Bigl(\sum_{k}\abs{c_{k}}^{2}\Bigr)^{1/2} \Bigl(\sum_{k}\norm{\hat e_{k}(\lambda)}^{2}\Bigr)^{1/2} =C(\lambda)\norm{\varphi}_{q}\ec \end{equation*}

which is also Equation (A27.5). Linearity is clear.

It remains to identify $\Lambda_{\lambda}\varphi$ with the fibre of $\varphi$. Fix $\varphi\in\Phi$ and let $S_{N}=\sum_{k\leq N}c_{k}e_{k}$. Then $S_{N}\longrightarrow\varphi$ in $\Phi_{q}$, hence in $\mathcal{H}$ because the embedding is bounded, and $J$ is isometric, so

\begin{equation*} \int_{\R}\norm{\hat S_{N}(\lambda)-\hat\varphi(\lambda)}^{2} \dd\mu(\lambda)=\norm{S_{N}-\varphi}^{2}_{\mathcal{H}} \longrightarrow0\ep \end{equation*}

A sequence converging in $L^{2}(\mu)$ has a subsequence converging $\mu$-almost everywhere (Remark A27.7), so along it $\hat S_{N}(\lambda)\longrightarrow\hat\varphi(\lambda)$ for almost every $\lambda$. But $\hat S_{N}(\lambda)$ is the $N$th partial sum of Equation (A27.8), which converges to $\Lambda_{\lambda}\varphi$ for every $\lambda$ satisfying Equation (A27.7). Limits are unique, so $\Lambda_{\lambda}\varphi=\hat\varphi(\lambda)$ almost everywhere. Finally $\norm{\cdot}_{q}$ is one of the norms defining the topology of $\Phi$, so Equation (A27.5) is continuity on $\Phi$.

∎

Generalized eigenvectors, and completeness

Proof of Theorem A27.2. Derives Theorem A27.2. Take $\mu$, $\mathcal{H}_{\lambda}=\ell^{2}(\N)$ and $\Lambda_{\lambda}$ from Propositions A27.3 and A27.5.

(1) The eigenvalue relation. Since $A$ maps $\Phi$ into $\Phi$, in particular $\Phi\subseteq D(A)$, and Proposition A27.3 gives, for each fixed $\varphi\in\Phi$,

\begin{equation}\tag{A27.9} \Lambda_{\lambda}(A\varphi)=\widehat{A\varphi}(\lambda) =\lambda\,\hat\varphi(\lambda) =\lambda\,\Lambda_{\lambda}\varphi \qquad\text{for }\mu\text{-almost every }\lambda\ec \end{equation}

the exceptional null set depending on $\varphi$. Let $\set{\varphi_{j}}$ be dense in $\Phi$, which exists because $\Phi$ is separable, and let $Z$ be the union of the countably many exceptional sets attached to the $\varphi_{j}$ together with the null set outside which Equation (A27.7) holds; $Z$ is $\mu$-null. For $\lambda\notin Z$ the two maps $\varphi\longmapsto\Lambda_{\lambda}(A\varphi)$ and $\varphi\longmapsto\lambda\Lambda_{\lambda}\varphi$ are continuous on $\Phi$ — the first because $A$ is continuous from $\Phi$ to $\Phi$ and $\Lambda_{\lambda}$ is continuous on $\Phi$, the second by Equation (A27.5) — and they agree on the dense set $\set{\varphi_{j}}$, hence everywhere. This is (1), now with one null set for all $\varphi$ at once.

(2) The functionals. For $\lambda\notin Z$ and $\xi\in\mathcal{H}_{\lambda}$ put $F_{\lambda,\xi}(\varphi) =\braket{\xi}{\Lambda_{\lambda}\varphi}$. It is linear in $\varphi$, because $\Lambda_{\lambda}$ is linear and the inner product is linear in its second slot (Equation (9.34)), and continuous, since $\abs{F_{\lambda,\xi}(\varphi)}\leq\norm{\xi}C(\lambda) \norm{\varphi}_{q}$ by Equation (16.1) and Equation (A27.5); so $F_{\lambda,\xi}\in\Phi'$ (Definition 16.103). By (1) and the reality of $\lambda$,

\begin{equation*} F_{\lambda,\xi}(A\varphi) =\braket{\xi}{\Lambda_{\lambda}(A\varphi)} =\braket{\xi}{\lambda\Lambda_{\lambda}\varphi} =\lambda\,F_{\lambda,\xi}(\varphi)\ec \end{equation*}

which is Equation (16.68): $F_{\lambda,\xi}$ is a generalized eigenvector of $A$ with eigenvalue $\lambda$.

(3) Completeness. Polarising the isometry Equation (A27.2) — by Equation (16.3), an isometry of Hilbert spaces preserves inner products — gives, for $\varphi,\psi\in\Phi$,

\begin{equation}\tag{A27.10} \braket{\varphi}{\psi}_{\mathcal{H}} =\int_{\R}\braket{\hat\varphi(\lambda)}{\hat\psi(\lambda)} _{\ell^{2}}\,\dd\mu(\lambda)\ep \end{equation}

Let $\set{\xi_{n}}$ be the standard orthonormal basis of $\ell^{2}(\N)$, the same for every $\lambda$, and $F_{\lambda,n}=F_{\lambda,\xi_{n}}$. Parseval's identity Equation (16.20) in the fibre reads

\begin{equation*} \braket{\hat\varphi(\lambda)}{\hat\psi(\lambda)} =\sum_{n}\braket{\hat\varphi(\lambda)}{\xi_{n}} \braket{\xi_{n}}{\hat\psi(\lambda)} =\sum_{n}F_{\lambda,n}(\varphi)^{\ast}\,F_{\lambda,n}(\psi)\ec \end{equation*}

using $\Lambda_{\lambda}\varphi=\hat\varphi(\lambda)$ almost everywhere and $\braket{\hat\varphi}{\xi_{n}} =\braket{\xi_{n}}{\hat\varphi}^{\ast}=F_{\lambda,n}(\varphi)^{\ast}$. Substituting into Equation (A27.10) gives Equation (A27.1). Since $\mu$ is supported on $\sigma(A)$, the integral may be written over $\sigma(A)$.

Simple spectrum. If a single cyclic vector generates $\mathcal{H}$ the decomposition of Proposition A27.3 has one block, $\ell^{2}(\N)$ is replaced by $\C$, and Equation (A27.1) collapses to $\braket{\varphi}{\psi}=\int F_{\lambda}(\varphi)^{\ast} F_{\lambda}(\psi)\,\dd\mu$, which is Equation (16.69).

∎

Example A27.6 (Momentum on the line).

For the Schwartz triple Equation (16.66) and $A=P=-\ii\hbar\,\dd/\dd x$, the spectrum is simple, $\mu$ may be taken to be Lebesgue measure on $\R$ in the variable $p$, and $\Lambda_{p}\varphi$ is the Fourier transform Equation (16.67), so that $F_{p}=F_{p}$ of Proposition 16.106. Statement (2) of Theorem A27.2 is then the computation already carried out there, and Equation (A27.1) is the Plancherel identity of Fourier Analysis and Integral Transforms. The plane waves are a complete set of generalized eigenvectors of the momentum, and no one of them is a state. Rests on Theorem A27.2, Proposition 16.106 and Example 16.104.

Remark A27.7 (What is quoted here).

This is the one long proof of Hilbert Spaces that rests on a theory the treatise does not build, and the reader is owed a precise account of where.

Theorem A27.4 — that a countably Hilbert nuclear space embeds in $\mathcal{H}$ through some Hilbert–Schmidt map — is assumed. It is the definition of nuclearity (Definition A27.1) plus the elementary fact that the composition of a nuclear map with a bounded one is Hilbert–Schmidt, but the theory of nuclear spaces itself, including the proof that $\mathcal{S}(\R)$ is nuclear, is not developed anywhere in this treatise. Reference of record: [Gelfand:1964], volume 4, chapter I. Everything that nuclearity is used for is Equations (A27.6) and (A27.7) and nothing else — which is why the architecture above is worth having even though the input is quoted: the reader can see exactly how much is being bought, and how little.
Three standard theorems of measure theory are used: the Radon–Nikodym theorem, in Proposition A27.3; the monotone convergence theorem, twice; and the fact that a sequence convergent in $L^{2}$ has an almost-everywhere convergent subsequence. The treatise develops no measure theory and says so in Remark 16.1, where the convergence theorems are already declared.
The spectral theorem is not quoted: the direct integral of Proposition A27.3 is built from The Spectral Theorem for a Bounded Self-Adjoint Operator for bounded $A$ and from Proposition A25.10 for unbounded $A$ — the momentum operator of Example A27.6 is the unbounded case.
Two hypotheses of Theorem A27.2 are suppressed in the chapter's statement Theorem 16.107 and are used here: separability of $\mathcal{H}$, without which the cyclic decomposition need not be countable, and separability of $\Phi$, without which the single null set of Equation (A27.9) cannot be assembled. Both hold for the Schwartz triple. A third point of honesty: for a spectrum of multiplicity greater than one the completeness relation carries the sum over $n$ displayed in Equation (A27.1), which Equation (16.69) suppresses; the two agree exactly when the spectrum is simple.

Nothing above claims more. In particular the theorem does not say that the $F_{\lambda,n}$ are unique, nor that every generalized eigenvector arises this way, nor that $\mu$ is canonical — only its measure class is — and no use made of the theorem in this treatise requires any of those.

Remark A27.8.

The Nuclear Spectral Theorem of Gelfand and Maurin discharges the proof obligation of Theorem 16.107 (Section 16.6.3). What it licenses is the working rule of Remark 16.108: a ket belonging to a point of continuous spectrum is an element of $\Phi'$, the pairing $\braket{\lambda}{\varphi}$ is the number $F_{\lambda}(\varphi)$ for a test function $\varphi$, and the “normalisation” $\braket{\lambda}{\lambda'} =\delta(\lambda-\lambda')$ is shorthand for Equation (A27.1) — an identity between functionals, never between numbers. Proposition 16.102 is the proof that no other reading is available, and Example A27.6 is the case every scattering calculation of Scattering Theory silently uses.