Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the Lipschitz Bound, and Extension to Self-Adjoint Vectors
lemmalem:resolvent-tracial-l2-2026aAnalysisIn a tracial W*-probability space: product bounds on vacuum vectors, resolvents stay in the algebra, resolvents are L^2-Lipschitz, and the resolvent at 1 extends uniquely to self-adjoint L^2 vectors.Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation
lemmalem:resolvent-calculus-complex-2026aAnalysisAdjoints and commutation of resolvents, the resolvent identity, the recovery identity A - y^2 Re R_y(A) = A^3 R_y R_{-y}, stepping between resolvent parameters, and uniform polynomial approximation of resolvents.Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple
definitiondef:resolvent-transform-complex-2026aAnalysisDefines the resolvent (A - iyI)^{-1} of a bounded self-adjoint operator and the resolvent transform of a self-adjoint tuple: the real and imaginary parts of the resolvents at y = 1.The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A
lemmalem:neumann-series-resolvent-complex-2026aAnalysisNeumann series with a rate, a powers estimate, inverses of invertible elements of a double commutant, and invertibility of A - iyI for bounded self-adjoint A with inverse norm at most 1/|y|.Scaling a Plan-Jet Viscosity Subsolution of an Equation with a Quadratic Hamiltonian Gives a Strict Subsolution
lemmalem:nc-quadratic-hamiltonian-scaling-2026aAnalysisPDEIf u is a plan-jet subsolution for a quadratic Hamiltonian with convex Lipschitz remainder, then theta u with theta<1 satisfies the subsolution inequality with a gain proportional to the squared momentum, up to a small constant.Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws
definitiondef:nc-quadratic-hamiltonian-2026aAnalysisPDEA Hamiltonian is quadratic with a convex Lipschitz remainder if it is one half the squared momentum plus a remainder that is convex and Lipschitz with linear growth in the momentum and bounded above at zero momentum.Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples
definitiondef:nc-law-function-lift-2026aAnalysisPDEThe lift of a function on square-integrable noncommutative laws to a tracial W*-probability space: evaluate the function at the law of the tuple.Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space
definitiondef:l2-tuple-pairing-2026aAnalysisDefines sums, real multiples and the real pairing of square-integrable self-adjoint tuples, as the vector operations and inner product of the direct-sum Hilbert space.Comparison and Uniqueness for the Linear-Quadratic Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
corollarycor:nc-lq-comparison-2026aAnalysisPDEComparison for bounded semicontinuous plan-jet sub- and supersolutions of the linear-quadratic equation, and uniqueness of bounded continuous plan-jet viscosity solutions.The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure
lemmalem:nc-lq-hamiltonian-conditions-2026aAnalysisPDEWith bounded Lipschitz drift coefficients and a bounded uniformly continuous running cost, the linear-quadratic Hamiltonian lifts to the expected formula, satisfies the structure condition, and is quadratic with a convex Lipschitz remainder.The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws
equationeq:nc-lq-hamilton-jacobi-2026aAnalysisPDEThe linear-quadratic Hamiltonian, one half the squared momentum minus the pairing with a law-dependent affine drift minus a running cost, and its discounted Hamilton-Jacobi equation on noncommutative laws.Comparison Principle for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
theoremthm:nc-plan-comparison-2026aAnalysisPDEFor a Hamiltonian satisfying the Crandall-Ishii-Lions structure condition that is either Lipschitz in the momentum with linear growth or quadratic with a convex Lipschitz remainder, every bounded usc plan-jet subsolution lies below every bounded lsc plan-jet supersolution.Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws
lemmalem:nc-plan-doubling-jets-2026aAnalysisPDEAt an Ekeland point of the doubled difference over joint laws, with a logarithmic confinement, the shared doubling momentum gives plan superjets of the first function and plan subjets of the second; the confinement momenta are small, and the doubling distance is controlled by the…Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions
lemmalem:nc-plan-viscosity-jets-2026aAnalysisPDETouching transfers plan jets from the test function to the function, so plan-jet viscosity solutions can be tested equivalently on the plan jets of the function itself.Plan-Jet Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
definitiondef:nc-plan-viscosity-solution-2026aAnalysisPDEPlan-jet viscosity sub- and supersolutions: whenever a test function touches from above (below) and has a plan super- (sub-)differential with slack, the equation holds as an inequality up to a momentum perturbation of the size of the slack.The Discounted Stationary Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
equationeq:nc-discounted-hamilton-jacobi-2026aAnalysisPDEThe discounted stationary Hamilton-Jacobi equation on square-integrable noncommutative laws, with the Hamiltonian evaluated on the joint law of position and gradient.Hamiltonians on Phase-Space Noncommutative Laws that are Lipschitz in the Momentum with Linear Growth
definitiondef:nc-hamiltonian-momentum-lipschitz-2026aAnalysisPDEA Hamiltonian on phase-space laws is Lipschitz in the momentum with linear growth if changing the momentum by Q changes it by at most a constant times one plus the position norm times the norm of Q.The Structure Condition for a Hamiltonian on Phase-Space Noncommutative Laws
definitiondef:nc-hamiltonian-structure-condition-2026aAnalysisPDEThe structure condition of Crandall-Ishii-Lions on phase-space laws, local in the positions: at the doubling momentum alpha(X-Y), moving the base point from X to Y raises the Hamiltonian by little when alpha|X-Y|^2+|X-Y| is small.Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws
definitiondef:nc-plan-jets-2026aAnalysisPDEA plan is a plan superdifferential (subdifferential) of a function on laws, with slack, if the first-order upper (lower) expansion along the momentum holds uniformly over every tuple jointly realised with the plan; the plan jets are the sets of these.- A plan at a law is a joint law of position and momentum whose first marginal is that law.