Functional and Spectral Analysis

Functional and spectral analysis is needed whenever a QFT statement depends on completion, continuity, an operator domain, a spectrum, or a topology that finite-dimensional notation hides. This chapter has two independent entrances. The finite-dimensional route studies convex cones, separation, conic duality, and semidefinite certificates. The infinite-dimensional route starts with Banach and Hilbert spaces and then branches toward compact operators, unbounded operators, spectral theory, trace ideals, nuclear spaces, or a short operator-algebra bridge.

Choose the smallest route that exposes the hypotheses behind the intended claim. A reader checking a finite semidefinite exclusion does not first need the whole operator sequence. A reader deciding whether a Hamiltonian generates unitary evolution does need Hilbert spaces, domains, adjoints, and self-adjointness, but need not first study trace ideals or nuclear spaces. This page lets a reader choose a route from a stated QFT need and diagnose missing preparation without treating the whole chapter as a prerequisite. The sidebar order is therefore a reference order, not a compulsory linear course.

The chapter makes finite-dimensional convex certificates and infinite-dimensional state, operator, spectrum, and topology claims precise enough to expose their hypotheses and limits. Developed physical applications remain in Foundations, Renormalization and EFT, Conformal Bootstrap, and Mathematical QFT. Infinite-dimensional convex duality, production solvers, theorem-first specialist operator theory, and the primary theory of operator algebras and algebraic QFT remain with their dedicated destinations.

Enter this chapter

Use these checks to find the smallest useful entry point.

Readiness check	Ready	If unsure	Repair and return
Can you distinguish a positive-semidefinite matrix from one whose eigenvalues are merely numerically close to nonnegative, and can you use the trace pairing?	Enter Convex Cones, Separation, Conic Duality, and Semidefinite Programs.	Check whether Hermitian adjoints and finite-dimensional spectral projectors are already familiar.	Review Bilinear and Hermitian Forms, Adjoints, and Isometries and Normal Forms, Spectra, and Projectors. Both are hard preparation for the convex route.
Can you explain why completing a normed space can add limits not present in the original vector space?	Enter Banach and Hilbert Spaces, Completion, and Riesz Representation.	Test whether “Cauchy,” “convergent,” and “complete” are being treated as different statements.	The Hilbert-space page has no hard prerequisite. Lp Spaces, Inequalities, and Weak Convergence is recommended preparation.
Given a linear operator, can you say whether it is bounded, compact, or defined only on a proper dense domain?	Choose the bounded/compact or unbounded branch after Banach and Hilbert Spaces.	Check whether the operator norm, relatively compact image of the unit ball, and graph of an operator answer three different questions.	Start at Banach and Hilbert Spaces, then continue to Bounded, Compact, and Integral Operators or Unbounded Operators, Domains, Closure, and Adjoints.
Can you distinguish a formally symmetric expression, a symmetric operator, an essentially self-adjoint operator, and a chosen self-adjoint extension?	Take Unbounded Operators, Domains, Closure, and Adjoints before self-adjointness.	Ask whether the domain has been stated and whether it equals the adjoint domain.	Review Banach and Hilbert Spaces and Unbounded Operators in dependency order before Self-Adjointness, Extensions, and Unitary Evolution.
Can you classify a spectral point by injectivity, density of the range, and surjectivity rather than by searching only for eigenvectors?	Enter spectral theory after unbounded operators; self-adjointness is recommended.	Test whether continuous and residual spectrum have been silently merged.	Read Banach and Hilbert Spaces → Unbounded Operators → Spectra, Resolvents, Spectral Measures, and Functional Calculus; add Self-Adjointness for the recommended self-adjoint branch.
Can you name the topology on a test-function space and distinguish its continuous dual from a Hilbert-space antidual?	Combine Banach and Hilbert Spaces with Test-Function Spaces, Distributions, Support, and Convergence, then take the locally convex route.	Check whether one norm is being assumed where a family of seminorms is required.	Repair both hard inputs in the preceding column before Locally Convex, Nuclear, and Rigged Hilbert Spaces.
Can you state positivity of a linear functional without assuming that every state is a density matrix?	Take Banach and Hilbert Spaces, then the operator-algebra bridge.	Check the distinction among an algebra element, a representation, a vector state, and an abstract positive functional.	Use Quantum states and operators repair for orientation, then read Banach and Hilbert Spaces → Operator Algebras and Positive Functionals: a Bridge.

For a broader self-check, use Diagnose mathematical readiness or Diagnose quantum-mechanics readiness. If a gap appears, continue to Linear and tensor methods repair or Quantum states and operators repair.

Choose a route

An arrow means a hard dependency in that route. A plus sign means that both inputs are required. “Recommended” material improves fluency but is not promoted to a hard prerequisite.

Reader goal	Minimum coherent route	Useful addition	Capability at the end
Interpret a finite conic or semidefinite certificate	Forms and Adjoints + Finite-Dimensional Spectra → Convex Cones and Conic Duality	From Crossing Equations to Convex Optimization supplies the physical problem	Put a finite problem into declared primal and dual form, derive weak duality, name any qualification used for strong duality, and state exactly what a certificate proves
Build an infinite-dimensional state space	Banach and Hilbert Spaces	Lp Spaces is recommended	Distinguish normed, Banach, and Hilbert spaces; form completions; and use Riesz representation with the site’s conjugate-linearity convention
Decide whether a kernel operator is continuous or compact	Banach and Hilbert Spaces → Bounded, Compact, and Integral Operators	For resolvents, separately add Unbounded Operators → Spectra and Resolvents	Test boundedness and compactness without inferring an orthonormal eigenbasis for a non-normal operator
Treat a Hamiltonian or differential operator	Banach and Hilbert Spaces → Unbounded Operators	None required before the domain analysis	Include the domain in the operator, determine density and closability, and compute the adjoint domain
Justify unitary evolution or boundary realizations	Banach and Hilbert Spaces → Unbounded Operators → Self-Adjointness	Compare exactly solvable boundary conditions and their spectra	Distinguish symmetry, essential self-adjointness, and extensions, and apply the correct self-adjoint generator theorem
Analyze poles, continua, or functional calculus	Banach and Hilbert Spaces → Unbounded Operators → Spectra and Resolvents	Self-Adjointness is recommended	Separate point, continuous, and residual spectrum in the general closed case and specialize spectral measures and Borel calculus to self-adjoint operators
Decide whether an infinite-dimensional trace or determinant exists	Banach and Hilbert Spaces → Bounded, Compact, and Integral Operators → Trace Ideals and Fredholm Determinants	Use Heat Kernels, Zeta Functions, and Spectral Determinants only when a separate regularization problem is intended	Check Schatten-class hypotheses and distinguish an ordinary Fredholm determinant from a regularized QFT determinant
Use test-function topology, kernels, or generalized eigenvectors	Banach and Hilbert Spaces + Test Functions and Distributions → Locally Convex and Nuclear Spaces	Distributional Kernels and Distributions on Manifolds	Name the locally convex topology, dual, and rigging, and state separately which nuclear or spectral theorem licenses the desired kernel or generalized eigenvector
Prepare for algebraic QFT	Banach and Hilbert Spaces → Operator-Algebra Bridge	Bounded, Compact, and Integral Operators is recommended	Recognize normed star-algebras, positivity, states, and the statement of GNS without importing nets, folia, or AQFT theorems into this chapter

The hard-dependency graph is branched:

• the finite-dimensional forms and spectral pages feed the convex/conic page, which is independent of the Hilbert-space branch;
• Banach and Hilbert spaces feed both bounded/compact operators and unbounded operators;
• unbounded operators feed self-adjointness and, separately, general spectra and resolvents; self-adjointness is recommended rather than hard preparation for the latter;
• bounded/compact operators feed trace ideals and Fredholm determinants;
• Hilbert spaces and test-function distributions are both required for the locally convex, nuclear, and rigged-space route; and
• Hilbert spaces feed the operator-algebra bridge, while bounded/compact operators are recommended there.

Consequently, neither the finite convex route nor the two advanced topology and algebra routes should be inserted as a prerequisite for every operator-theory question.

The chapter’s objects are related but not interchangeable

Object	Defining data or test	Licensed conclusion	Common overreach
Closed convex cone $K\subset V$	A finite-dimensional real vector space, its dual pairing, closure, convexity, and $\lambda K\subset K$ for $\lambda\geq0$	Its dual cone $K^*=\{s\in V^*:\langle s,x\rangle\geq0\ \forall x\in K\}$ and separation statements under stated hypotheses	Treating a numerically near-conic point as exactly feasible
Conic primal–dual pair	$\min\{\langle c,x\rangle:Ax=b,\ x\in K\}$ and $\max\{\langle y,b\rangle:c-A^*y\in K^*\}$	Weak duality from the pairing; strong duality or attainment only with an applicable theorem and qualification	Inferring zero gap or attainment merely from convexity
Semidefinite program	A conic problem with $K=\mathbb S_+^n$ and pairing $\langle X,Y\rangle=\operatorname{Tr}(XY)$	Matrix-positivity certificates for the declared finite problem	Promoting a finite exclusion certificate into an existence theorem for a physical theory
Banach or Hilbert space	A complete normed space; in a Hilbert space the norm comes from a positive inner product	Limits live in the declared completion; Hilbert dual functionals have Riesz representatives	Using Riesz representation in an arbitrary Banach space
Bounded operator	A linear map defined on the whole normed space with $\lVert Tx\rVert\leq C\lVert x\rVert$	Continuity and a finite operator norm	Assuming boundedness merely because matrix elements look finite on a convenient basis
Compact operator	A bounded operator that maps bounded sets to relatively compact sets	Riesz–Schauder-type spectral structure; for compact self-adjoint operators, an orthonormal eigenvector description	Assuming that every compact non-normal operator has an orthonormal eigenbasis
Unbounded operator	The pair $(\mathcal D(A),A)$ with $\mathcal D(A)$ usually a proper dense subspace	Graph, closure, adjoint, and extension questions become well-typed	Treating the same differential expression on different domains as the same operator
Symmetric or self-adjoint operator	$A\subset A^\dagger$ for symmetric; $A=A^\dagger$ including equality of domains for self-adjoint	Self-adjointness licenses real spectrum, spectral calculus, and unitary evolution under the relevant theorem	Replacing equality of domains by an integration-by-parts calculation
Resolvent and spectrum	$R_A(z)=(A-zI)^{-1}$ when the inverse exists everywhere and is bounded; $\sigma(A)=\mathbb C\setminus\rho(A)$	A domain-aware classification of spectral parameters	Identifying the spectrum with eigenvalues only
Spectral measure	A projection-valued measure $E_A$ for a self-adjoint operator	$A=\int\lambda\,dE_A(\lambda)$ and a measurable functional calculus on its proper domain	Assigning a projection-valued spectral resolution to an arbitrary closed non-normal operator
Trace-class perturbation	$K$ has summable singular values	A basis-independent trace and the ordinary Fredholm determinant $\det(I+K)$	Defining a trace or determinant for every compact operator, or calling it the renormalized QFT determinant
Locally convex or nuclear space	A topology generated by a separating family of seminorms; nuclearity is an additional property	Continuity and tensor or kernel theorems appropriate to the named topology	Confusing a nuclear topological vector space with a nuclear operator
Rigged Hilbert space	A specified dense continuous embedding $\Phi\subset H\subset\Phi^\times$ with a declared dual or antidual convention	A setting in which an applicable theorem may represent generalized vectors	Claiming that the rigging alone produces a complete generalized eigenbasis
Positive functional and GNS data	A positive normalized linear functional $\omega$ on a unital $C^*$-algebra, followed by the GNS quotient and completion	A cyclic triple $(\pi_\omega,H_\omega,\Omega_\omega)$ with $\omega(a)=\langle\Omega_\omega,\pi_\omega(a)\Omega_\omega\rangle$	Assuming every state is normal, tracial, or represented by a density matrix in a preselected Hilbert space

For a closed densely defined operator, the spectral subdivision tests the map $A-zI$:

Spectral part	Kernel	Range	Inverse behavior
Point spectrum	Nonzero	Any	No inverse
Continuous spectrum	Zero	Dense but not all of $H$	The inverse on the range is unbounded
Residual spectrum	Zero	Not dense	No everywhere-defined inverse

This convention is declared because terminology varies slightly among sources. The residual part can occur for a general closed non-normal operator. It is empty for a self-adjoint operator; that special conclusion must not be exported backward to the general case.

Shared conventions and invariant checks

These chapter-level choices keep formulas compatible. Leaf-specific hypotheses still belong on the leaf that invokes a theorem.

Issue	Convention	Invariant check
Complex inner product	$\langle u,v\rangle$ is conjugate-linear in $u$ and linear in $v$	$\langle au,v\rangle=a^*\langle u,v\rangle$ and $\langle u,av\rangle=a\langle u,v\rangle$
Riesz representation	A continuous linear functional has the form $f(v)=\langle u_f,v\rangle$	With this slot convention, $u_f\mapsto f$ is conjugate-linear
Banach dual versus antidual	$X^*$ denotes continuous linear functionals unless a leaf explicitly declares an antidual	A rigged triple states which dual is used before writing $\Phi^\times$
Hilbert-space adjoint	For densely defined $A$, $v\in\mathcal D(A^\dagger)$ exactly when there is a unique $w\in H$ such that $\langle v,Au\rangle=\langle w,u\rangle$ for every $u\in\mathcal D(A)$; then $A^\dagger v=w$	Equivalently, $u\mapsto\langle v,Au\rangle$ is bounded in the ambient $H$ norm, not merely in the graph norm
Finite-dimensional dual map	In the real conic route, $A^*:W^*\to V^*$ is fixed by the dual pairings through $\langle A^*y,x\rangle=\langle y,Ax\rangle$	The weak-duality gap reduces to the pairing of an element of $K^*$ with an element of $K$
Algebra involution	$a^*$ is the star-algebra involution; an operator adjoint may be written $A^\dagger$ when confusion is possible	A star-representation obeys $\pi(a^*)=\pi(a)^\dagger$
Resolvent sign	$R_A(z)=(A-zI)^{-1}$	Poles, distances to the spectrum, and earlier Green-operator conventions use the same sign
Spectrum	Unless stated otherwise, spectrum is taken over the complexification or a complex Hilbert space	“No eigenvalue” does not imply “in the resolvent set”
Positive-semidefinite pairing	On Hermitian matrices, $\langle X,Y\rangle=\operatorname{Tr}(XY)$ and $\mathbb S_+^n$ is self-dual	Weak duality becomes a nonnegative cone pairing
Exact versus numerical statements	Feasibility, positivity, and separation are exact claims; residuals, gaps, and eigenvalue margins are finite-precision diagnostics	Scaling, tolerances, conditioning, and independent verification accompany any numerical conclusion
Trace and determinant	Trace class is stronger than compactness; the unregularized Fredholm determinant is attached to $I+K$ with $K$ trace class	A zeta, heat-kernel, or other regularized determinant is named separately
Topology	Continuity, convergence, boundedness, and duality always refer to a declared topology	Replacing a family of seminorms by one norm requires a theorem, not notation

In particular, the same symbol written as a matrix, a bounded operator, an unbounded realization, or an algebra element carries different admissible operations. The type and domain are part of every claim.

Topics in this chapter

Convex Cones, Separation, Conic Duality, and Semidefinite Programs

The convex/conic page answers the question: When does a finite-dimensional linear feasibility problem admit a separating certificate, and what do primal-dual conic and semidefinite formulations actually prove? It covers closed convex and dual cones, separation and Farkas alternatives, declared primal and dual conic forms, weak duality, qualified strong duality and attainment, the positive-semidefinite cone, linear matrix inequalities, and exact versus approximate certificates.

Forms and adjoints plus finite-dimensional spectral decomposition are hard preparation. This finite-dimensional method is theorem-led rather than a solver tutorial: small residuals are not exact feasibility, and strong duality is not claimed without a named constraint qualification such as an applicable Slater condition. Continue to From Crossing Equations to Convex Optimization for the physical positivity and crossing problem.

Banach and Hilbert Spaces, Completion, and Riesz Representation

The Hilbert-space page answers the question: How do norm, completeness, inner products, and dual representation organize infinite-dimensional state spaces? It is the base of the operator branch: metric and normed spaces lead to Banach completion, Hilbert geometry, closed subspaces, orthogonal projection, and Riesz representation with the site’s slot convention.

There is no hard prerequisite; $L^p$ spaces and inequalities are recommended. This is the foundation entry page. Continue to bounded/compact operators for continuous kernels, to unbounded operators for Hamiltonians and differential expressions, or directly to one of the topology and algebra bridges. Its first physical continuation is One-Particle States: Mass, Spin, and Relativistic Normalization.

Bounded, Compact, and Integral Operators

The bounded/compact page answers the question: Which operators are continuous or compact, and when does an integral kernel define one? It separates boundedness from compactness, identifies useful integral-operator criteria, and states which spectral conclusions require compactness, self-adjointness, or both.

Banach and Hilbert spaces are hard preparation. Continue to trace ideals when summability, traces, or Fredholm determinants are needed; continue to spectra and resolvents when compact resolvents or continuum spectrum are the target. Its first physical continuation is Spectral Decomposition of Two-Point Functions.

Unbounded Operators, Domains, Closure, and Adjoints

The unbounded-operator page answers the question: Why is the domain part of an unbounded operator, and how do closure and adjoints change the analysis? It treats a densely defined operator as its rule together with its domain, uses the graph to distinguish closed and closable operators, and derives adjoint domains rather than inferring them from a formal expression.

Banach and Hilbert spaces are hard preparation. Continue to self-adjointness when the question concerns extensions, boundary conditions, or unitary evolution; continue directly to spectra and resolvents for the general closed-operator classification. Its first physical continuation is Canonical Quantization: Algebra, Representation, and State.

Self-Adjointness, Extensions, and Unitary Evolution

The self-adjointness page answers the question: When does a symmetric operator admit a self-adjoint realization and generate unitary time evolution? It distinguishes symmetry, self-adjointness, essential self-adjointness, and extension data, and connects self-adjoint generators to strongly continuous unitary evolution under the appropriate theorem.

Unbounded operators, domains, closure, and adjoints are hard preparation. This is an advanced theorem page. Continue to spectral measures and functional calculus for observable and evolution functions of the self-adjoint realization. The physical continuation is Hilbert Positivity and Unitary Evolution.

Spectra, Resolvents, Spectral Measures, and Functional Calculus

The spectral page answers the question: How do resolvents and spectral measures distinguish general closed operators from self-adjoint ones? It covers the general and self-adjoint spectrum, the residual-spectrum caveat, resolvent identities, spectral measures, and functional calculus.

Unbounded operators are hard preparation; self-adjointness is recommended. This is an advanced concept page. Its first physical continuation is the Källén–Lehmann Representation, where QFT assumptions turn operator spectral information into a spectral representation of correlators.

Trace Ideals and Fredholm Determinants

The trace-ideal page answers the question: When are traces and determinants defined for infinite-dimensional operators without a separate regularization scheme? It places Hilbert–Schmidt and trace-class operators inside the compact operators, states basis-independent trace criteria, and defines the ordinary Fredholm determinant only under its operator-ideal hypotheses.

Bounded, compact, and integral operators are hard preparation. This is an advanced method page. Continue to Integrating Out Heavy Fields for physical determinant factors, and to heat-kernel or zeta methods only when an explicitly regularized determinant is intended.

Locally Convex, Nuclear, and Rigged Hilbert Spaces

The locally convex page answers the question: Why do finer topologies and nuclearity control test functions, tensor products, kernels, and generalized eigenvectors? It introduces seminorm topologies, Fréchet-type examples, nuclearity, continuous duals, and rigged Hilbert spaces while keeping theorem hypotheses attached to kernel and generalized spectral claims.

Banach and Hilbert spaces plus test-function distributions are both hard preparation. This is an advanced concept page. Continue to Wightman Fields, Domains, and Axioms for operator-valued tempered distributions and their physical axioms.

Operator Algebras and Positive Functionals: a Bridge

The operator-algebra bridge answers the question: Which algebraic and positivity notions must a reader know before entering algebraic QFT? It gives a bounded prerequisite map for normed star-algebras, positivity, states, representations, and the statement and construction pattern of GNS.

Banach and Hilbert spaces are hard preparation; bounded/compact operators are recommended. This is an advanced concept page. It stops before primary operator-algebraic definitions, local nets, representation comparison, folia, and AQFT theorems. Those begin with States, GNS Representations, and Folia and the surrounding Mathematical QFT chapter.

Two threads across the chapter

The chapter is intentionally not forced into one example. A finite certificate and an unbounded continuum operator expose different hypotheses. Together they show why type, topology, and conclusion must travel with every calculation.

A finite exclusion problem

Suppose a finite truncation produces a linear constraint $\mathcal A(X)=b$ with a Hermitian matrix variable $X\succeq0$. The convex route proceeds in six controlled steps:

1. declare the finite-dimensional real vector spaces and the trace pairing;
2. identify the positive-semidefinite cone and its dual;
3. write a primal and dual problem with one fixed sign convention;
4. derive weak duality directly from a nonnegative cone pairing;
5. state the exact qualification, if any, used for zero gap or attainment; and
6. separate exact certificates from floating-point evidence by reporting residuals, duality gap, PSD margin, scaling, and conditioning.

In a conformal-bootstrap application, this machinery can certify exclusion for the declared finite approximation under its positivity assumptions. It does not prove that a point not excluded by the finite problem corresponds to an existing CFT. Conformal Bootstrap develops the physical crossing equation, truncation, positivity gates, and claim ceiling; an executable solver and independent certificate verification require a separate numerical workflow.

One differential expression, several operators

Consider the expression $-\frac{d^2}{dx^2}$ on $(0,\ell)$. The operator thread changes the question at each page:

1. Hilbert space. Completion places square-integrable wave functions in $L^2(0,\ell)$, but the Hilbert space alone does not select boundary data.
2. Bounded and compact maps. A Green or resolvent kernel, when defined, may give a bounded or compact integral operator. The differential expression itself remains unbounded.
3. Domain and closure. Starting on $C_c^\infty(0,\ell)$ produces a densely defined symmetric operator. Its closure, adjoint, and maximal domain are separate objects.
4. Self-adjoint realization. Dirichlet, Neumann, Robin, or coupled boundary conditions select different realizations. Vanishing of the boundary form on the chosen domain is necessary for operator symmetry, while equality with the adjoint domain is decisive for self-adjointness.
5. Spectrum and calculus. A selected self-adjoint realization has a real spectral measure; its resolvent, time evolution, and functions such as $e^{-itA}$ follow from calculus for that realization.
6. Trace ideals. In one dimension a suitable resolvent may be trace class, permitting $\det(I+K)$ for a trace-class perturbation $K$. This does not define the formal determinant of the differential expression itself.
7. Rigging. A test space $\Phi$ with a finer topology can sit densely in $L^2(0,\ell)$ and admit a continuous dual. Generalized vectors arise only after the relevant spectral or kernel theorem is checked.
8. Positive functionals. Passing from a concrete Hilbert-space operator to an abstract observable algebra changes the primary object. A positive functional can then generate a cyclic representation by GNS, but local nets and folia require the Mathematical QFT treatment.

The expression stays visually the same while the space, domain, topology, and admissible conclusions change. That is the chapter’s central discipline.

Chapter synthesis

The infinite-dimensional route begins with a typed operator

$$A:\mathcal D(A)\subset H\longrightarrow H.$$

For a genuinely unbounded closed operator, $\mathcal D(A)$ is a proper subspace: the closed graph theorem would make an everywhere-defined closed operator on a Banach space bounded. If $z$ belongs to the resolvent set, then

$$R_A(z)=(A-zI)^{-1}:H\longrightarrow\mathcal D(A)\subset H$$

exists everywhere and is bounded as an $H$-valued map. If $A$ is self-adjoint, the spectral theorem provides a projection-valued measure and

$$A=\int_{\mathbb R}\lambda\,dE_A(\lambda), \qquad f(A)=\int_{\mathbb R}f(\lambda)\,dE_A(\lambda)$$

on the domain determined by $f$. These formulas are special conclusions, not definitions available for every closed operator.

The finite-dimensional route begins instead with a cone and a pairing. For the declared primal and dual forms,

$$\begin{aligned} p^\star &=\inf\{\langle c,x\rangle:Ax=b,\ x\in K\},\\ d^\star &=\sup\{\langle y,b\rangle:c-A^*y\in K^*\}, \end{aligned}$$

feasibility gives

$$\langle c,x\rangle-\langle y,b\rangle =\langle c-A^*y,x\rangle\geq0.$$

That identity proves weak duality. Equality of optimal values, existence of optimizers, or a robust numerical conclusion requires more.

The chapter’s reusable conclusions are:

• completion changes the space and therefore which limits count as states or vectors;
• boundedness, compactness, self-adjointness, and trace class answer different questions and none implies all the others;
• an unbounded operator is inseparable from its domain, graph, closure, and adjoint domain;
• symmetry is not self-adjointness, and unitary dynamics requires the appropriate self-adjoint generator statement;
• the spectrum of a general closed operator need not be real and can contain a residual part, while the self-adjoint case has real spectrum and no residual spectrum;
• projection-valued spectral measures and Borel functional calculus require normal or self-adjoint hypotheses appropriate to the statement;
• compactness alone does not define a trace, and an ordinary Fredholm determinant is not a synonym for a regularized functional determinant;
• locally convex topology, nuclearity, a rigged Hilbert triple, and a nuclear operator are distinct structures;
• a positive functional need not be a density matrix in a preselected representation, and GNS supplies a cyclic representation rather than the full structure of algebraic QFT; and
• finite-dimensional separation and conic duality state exactly what a certificate proves, while strong duality and finite-precision conclusions require additional hypotheses and checks.

Teschl 2014, Part 0 and Chapters 1–6 develops the Hilbert-space, compact-operator, unbounded-operator, self-adjointness, spectral-theorem, and quantum-dynamics spine. Boyd and Vandenberghe 2004, Chapters 2, 4, and 5, PDF supplies the finite-dimensional convex, semidefinite, and duality framework; the page narrows its use to exact certificate logic and explicitly separates it from solver evidence. Hunter and Nachtergaele 2001, Chapters 5, 6, and 8–10 provides a second teaching route through Banach and Hilbert spaces, bounded and compact operators, spectra, and differential-operator domains.

For the locally convex and nuclear branch, compare Chiba 2011, §3.1, Garrett 2017, §§1–3, PDF, and Vogt 2000, §§1–4, PDF. Fewster and Rejzner 2020, §2, PDF fixes the algebraic-QFT boundary of the short operator-algebra bridge. Reed and Simon 1980, Chapters II–VIII and Simon 2005, Chapters 1–3 provide broader operator-theoretic references, while Jansson 2009, pp. 337–363 supports the distinction between a numerical solver result and a verified conic conclusion.

Review the chapter

Route selection. A reader wants to verify a semidefinite exclusion, a Hamiltonian’s unitary evolution, and a Fredholm determinant. Give the minimum route for each. Success keeps the conic branch independent; sends the Hamiltonian through Hilbert spaces, unbounded operators, and self-adjointness; and sends the determinant through Hilbert spaces, bounded/compact operators, and trace ideals. Repair with the route table above.

Conic derivation and claim limit. Starting from the primal and dual forms in the synthesis, derive weak duality and identify the one nonnegative pairing. Then explain why neither a small residual nor a small duality gap is an exact certificate by itself. Success fixes the adjoint and sign conventions, names an applicable constraint qualification before invoking strong duality, and reports PSD margins, scaling, conditioning, and independent verification for numerical evidence. Repair with the convex/conic page.

Domain diagnosis. Two authors write the same differential expression $-d^2/dx^2$ but impose Dirichlet and Neumann boundary conditions. Are they describing the same operator, and does formal symmetry prove either one self-adjoint? Success says that the domains define different operators, computes or cites the adjoint domain, and tests equality of domains. Repair with Unbounded Operators and Self-Adjointness.

Spectrum classification. Let $A$ be a closed densely defined operator and let $z$ not be an eigenvalue. List the remaining tests needed before declaring $z\in\rho(A)$. Success checks whether the range of $A-zI$ is dense, whether it equals all of $H$, and whether the inverse is bounded. It permits continuous or residual spectrum in the general case and removes residual spectrum only after self-adjointness is known. Repair with Spectra and Resolvents.

Representation change. On $L^2([0,1])$, let $(M_xf)(x)=x f(x)$. Replace the nonexistent orthonormal eigenvector sum by the appropriate spectral representation. Success notes that $M_x$ has spectrum $[0,1]$ but no eigenvalues, defines $(E(B)f)(x)=\mathbf 1_B(x)f(x)$, and writes $M_x=\int_{[0,1]}\lambda\,dE(\lambda)$. It attributes this projection-valued measure to self-adjointness rather than to closedness alone. Repair with Spectra and Resolvents.

Trace and determinant check. Diagnose the claim “$K$ is compact, so $\operatorname{Tr}K$ and $\det K$ exist.” Success distinguishes compact, Hilbert–Schmidt, and trace-class hypotheses, defines the ordinary Fredholm determinant for $I+K$ with $K$ trace class, and refuses to identify it with a zeta- or heat-regularized determinant. As a counterexample, the diagonal operator $Ke_n=n^{-1}e_n$ on $\ell^2$ is compact and Hilbert–Schmidt but not trace class, so neither a naive $\operatorname{Tr}K$ nor the ordinary $\det(I+K)$ is licensed. A formal $\operatorname{Tr}\log$ does not repair the missing hypothesis. Repair with Trace Ideals and Fredholm Determinants.

Topology and duality check. A generalized eigenvector is written as $F\in\Phi^\times$. State what must be declared before this notation has content. Success names the locally convex topology on $\Phi$, its continuous embedding into $H$, the linear-dual or antidual convention, and the theorem that produces or completes the generalized spectral representation. It does not infer the result from a rigging alone. Repair with Locally Convex, Nuclear, and Rigged Hilbert Spaces.

GNS boundary check. Given a positive normalized functional $\omega$ on a unital $C^*$-algebra, state the output of GNS and three properties not implied by positivity alone. Success produces a cyclic representation, Hilbert space, and cyclic vector reproducing $\omega$; it does not assume that $\omega$ is normal, tracial, or a density-matrix state in a fixed prior representation. It also leaves nets, folia, and representation comparison to Mathematical QFT. Repair with the operator-algebra bridge.

Chapter-scale synthesis. A scalar theory uses a finite bootstrap truncation, an unbounded Hamiltonian, a continuous mass spectrum, and a formal one-loop determinant. For each phrase, name the chapter object and the extra hypothesis that prevents overclaiming. Success uses conic qualification and certificate scope for the truncation; a domain and self-adjointness for the Hamiltonian; spectral measures rather than an eigenvector sum for the continuum; and trace-class or an explicitly named regularization for the determinant. Repair each item at its corresponding leaf route.

QFT and theorem handoffs

This chapter stops when physical axioms, interpretation, or specialist frameworks become the main question.

• From Crossing Equations to Convex Optimization develops the conformal crossing equation, positivity assumptions, finite reduction, and physical exclusion claim.
• One-Particle States: Mass, Spin, and Relativistic Normalization uses Hilbert-space completion and inner products for relativistic state spaces.
• Canonical Quantization: Algebra, Representation, and State supplies the controlled physical first application of Hamiltonians and operator domains.
• Hilbert Positivity and Unitary Evolution develops the physical role of positivity and unitary time evolution.
• Spectral Decomposition of Two-Point Functions and The Källén–Lehmann Representation turn completeness, positivity, and spectral measures into physical correlator statements.
• Integrating Out Heavy Fields develops the QFT meaning of determinant factors and their low-energy expansion.
• Wightman Fields, Domains, and Axioms uses test-function topology and common invariant domains for operator-valued distributions.
• States, GNS Representations, and Folia develops representations, folia, and inequivalence beyond the bounded GNS bridge.

For boundary realizations, compact resolvents, and heat evolution, return to Elliptic Boundary Problems and Heat Kernels. For kernel theorems and distributions on manifolds, continue to Distributional Kernels and Distributions on Manifolds. To choose another route in this volume, return to Mathematical Methods.

References

• Stephen Boyd and Lieven Vandenberghe, Convex Optimization, PDF, Chapters 2, 4, and 5, Cambridge University Press, 2004. This is the teaching and structural source for convex sets and cones, semidefinite programs, Lagrange and conic duality, Slater qualifications, and primal–dual certificates. The chapter guide separates these exact results from finite-precision solver claims. Its real symmetric PSD notation is translated here to Hermitian matrices regarded as a real vector space with the trace pairing.
• Hayato Chiba (2011), “A Spectral Theory of Linear Operators on Rigged Hilbert Spaces under Analyticity Conditions”, §3.1, arXiv:1107.5858. Only the definition and dual-topology discussion of a rigged Hilbert space are used here; the paper’s later analytic-continuation and resonance theory is outside this overview. Chiba’s first-slot-linear convention and continuous antidual are translated to the site’s first-slot-conjugate-linear notation.
• Christopher J. Fewster and Kasia Rejzner, “Algebraic Quantum Field Theory — an Introduction”, PDF, §2, Progress of Theoretical and Experimental Physics 2020, 103I01. This is the QFT-facing structural source for star-algebras, positive functionals, states, representations, and GNS, and for the point at which the short bridge hands off to algebraic QFT.
• Paul Garrett, “Nuclear Spaces, Schwartz’ Kernel Theorem”, PDF, §§1–3, 2017. This is a structural source for Hilbert–Schmidt maps, nuclear Fréchet spaces, and the topology behind the Schwartz kernel theorem. Its first-slot-linear inner-product formulas are translated to the site’s first-slot-conjugate-linear convention.
• John K. Hunter and Bruno Nachtergaele (2001), Applied Analysis, Chapters 5, 6, and 8–10. This is an accessible teaching source for Banach and Hilbert spaces, bounded and compact operators, bounded-operator spectra, and differential-operator domains and Green functions.
• Christian Jansson, “On Verified Numerical Computations in Convex Programming”, Japan Journal of Industrial and Applied Mathematics 26 (2009), 337–363. This is the numerical reliability source for rounding error, conditioning, verified bounds, and the difference between an approximate solver result and a rigorous conic conclusion.
• Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Chapters II–VIII, revised and enlarged edition, Academic Press, 1980. These chapters develop Hilbert and Banach spaces, locally convex spaces, bounded and trace-class operators, the spectral theorem, domains, adjoints, self-adjointness, and Stone’s theorem.
• Barry Simon, Trace Ideals and Their Applications, second edition, Mathematical Surveys and Monographs 120, American Mathematical Society, 2005. This edition develops Schatten ideals, traces, determinants, and their operator-theoretic hypotheses.
• Gerald Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, second edition, Part 0 and Chapters 1–6, American Mathematical Society, 2014. This is the main structural source for Hilbert spaces, compact and trace-class operators, unbounded operators, self-adjointness, spectral measures, functional calculus, Stone’s theorem, and quantum dynamics. Teschl uses the site’s inner-product slot convention; his $A^*$ is written $A^\dagger$ here when it must be distinguished from an algebra involution.
• Dietmar Vogt (2000), Lectures on Fréchet Spaces, PDF, §§1–4. This is a structural reference for locally convex and Fréchet topology, nuclear maps, and nuclear spaces; it also makes clear that nuclearity of a space is not the same definition as nuclearity of one operator.