Proof of The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure

Each result cited is universally quantified over the data in its own statement. Fix a tracial W*-probability space $(H,M,\Omega)$. By Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space, its $L^{2}$ $d$-tuples lie in the complex Hilbert space $H^{d}$ with its sums, real multiples, norm $\lVert\cdot\rVert_{2}$ and inner product $\langle\cdot,\cdot\rangle_{2}$, which is real, symmetric and real bilinear on $L^{2}$ $d$-tuples (conjugate symmetry, additivity and homogeneity in the second argument, conditions 1, 2 and 3 of Complex Inner Product Space, and realness); hence $\lVert P+Q\rVert_{2}^{2}=\lVert P\rVert_{2}^{2}+2\langle P,Q\rangle_{2}+\lVert Q\rVert_{2}^{2}$, $|\langle Z,W\rangle_{2}|\le\lVert Z\rVert_{2}\lVert W\rVert_{2}$ by Cauchy-Schwarz Inequality in a Complex Inner Product Space, and the triangle inequality and homogeneity hold by claim 2 of The Induced Norm is a Norm, and Induces a Metric. For an $L^{2}$ $d$-tuple $W$, $W=\sum_{i=1}^{d}\iota_{i}W_{i}$ with the coordinate inclusions of Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §inclusions, each isometric by Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert, so $\lVert W\rVert_{2}\le\sum_{i=1}^{d}\lVert W_{i}\rVert$; and $\lVert W_{j}\rVert\le\lVert W\rVert_{2}$ for each $j$ by the definition of $\lVert\cdot\rVert_{2}$ in Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples. Recall $\lVert\Omega\rVert=1$ (Tracial W*-Probability Spaces §space).

Estimate. Let $\mu,\nu\in\Sigma^{2}_{d}$ and let $X,Y$ be $L^{2}$ $d$-tuples. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, $(b_{\mu}X)_{i}=c(\mu)_{i}\Omega+\sum_{j=1}^{d}A(\mu)_{ij}X_{j}$. Hence, by the triangle inequality in $H$ and the hypotheses,

and, writing $A(\mu)_{ij}X_{j}-A(\nu)_{ij}Y_{j}=A(\mu)_{ij}(X_{j}-Y_{j})+(A(\mu)_{ij}-A(\nu)_{ij})Y_{j}$,

Clause 1. Let $X,P$ be $L^{2}$ $d$-tuples, $\pi=\mathrm{law}(X,P)$ and $\mu=\mathrm{law}(X)$. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling, $\mathrm{pr}^{1}_{\#}\pi=\mu$ and $\mathrm{pr}^{2}_{\#}\pi=\mathrm{law}(P)$, so $\widehat{M}(\mathrm{pr}^{2}_{\#}\pi)=\lVert P\rVert_{2}^{2}$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations and the entries of $S_{\mu}=(B,e)$ in The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws, the $2d$-tuple $S_{\mu}(X,P)$ has coordinates $e_{i}\Omega+\sum_{j=1}^{d}B_{ij}X_{j}+\sum_{j=1}^{d}B_{i,d+j}P_{j}=(b_{\mu}X)_{i}$ and $e_{d+i}\Omega+\sum_{j=1}^{d}B_{d+i,j}X_{j}+\sum_{j=1}^{d}B_{d+i,d+j}P_{j}=P_{i}$ for $i\in[d]$ (terms with coefficient $0$ vanish), so $S_{\mu}(X,P)=(b_{\mu}X,P)$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $(S_{\mu})_{\#}\pi=\mathrm{law}(b_{\mu}X,P)$, and by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments applied to the $2d$-tuple $(b_{\mu}X,P)$, $\mathrm{m}_{i,d+i}((S_{\mu})_{\#}\pi)=\langle(b_{\mu}X)_{i},P_{i}\rangle$. Summing over $i$ gives $\langle b_{\mu}X,P\rangle_{2}$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing). Substituting into The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws §hamiltonian and using Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts yields clause 1.

Clause 2. Let $R>0$ and $\eta>0$. By Uniformly Continuous Map Between Metric Spaces fix $r_{f}>0$ with $|f(\mu)-f(\nu)|<\eta/2$ whenever $\widehat{W}_{2}(\mu,\nu)<r_{f}$. Put $K_{R}=dL+d^{2}a+d^{2}LR$ and $r=\min\{r_{f},\;\eta/(2(K_{R}+1))\}>0$. Let $X,Y$ be $L^{2}$ $d$-tuples with $\lVert X\rVert_{2},\lVert Y\rVert_{2}\le R$, let $\alpha>0$ be real with $\alpha\lVert X-Y\rVert_{2}^{2}+\lVert X-Y\rVert_{2}<r$, put $P=\alpha(X-Y)$, so that $\lVert X-Y\rVert_{2}\lVert P\rVert_{2}=\alpha\lVert X-Y\rVert_{2}^{2}$, and let $\mu=\mathrm{law}(X)$, $\nu=\mathrm{law}(Y)$; then $\widehat{W}_{2}(\mu,\nu)\le\lVert X-Y\rVert_{2}$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz. By clause 1 the terms $\frac12\lVert P\rVert_{2}^{2}$ cancel and, by bilinearity,

By Cauchy--Schwarz and (2), $|\langle b_{\mu}X-b_{\nu}Y,P\rangle_{2}|\le K_{R}\lVert X-Y\rVert_{2}\lVert P\rVert_{2}=K_{R}\alpha\lVert X-Y\rVert_{2}^{2}<K_{R}r\le\eta/2$. Since $\widehat{W}_{2}(\mu,\nu)\le\lVert X-Y\rVert_{2}<r\le r_{f}$, $f(\mu)-f(\nu)<\eta/2$. Hence the difference is $<\eta$, which is The Structure Condition for a Hamiltonian on Phase-Space Noncommutative Laws §structure.

Clause 3. Let $\mathcal{H}_{0}:\Sigma^{2}_{2d}\to\mathbb{R}$ be $\mathcal{H}_{0}(\pi)=\mathcal{H}^{\mathrm{LQ}}(\pi)-\frac12\widehat{M}(\mathrm{pr}^{2}_{\#}\pi)$, and let $B_{f}\ge0$ be real with $|f|\le B_{f}$ (Bounded Real-Valued Function on a Set). Then Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §decomposition holds by construction, and by clause 1, for all $L^{2}$ $d$-tuples $X,P$ of any tracial W*-probability space, $\mathcal{H}_{0,M}(X,P)=-\langle b_{\mu}X,P\rangle_{2}-f(\mu)$ with $\mu=\mathrm{law}(X)$. Let $X,P,P',Q$ be $L^{2}$ $d$-tuples of one space and $0\le t\le1$; all terms below have the same $X$, hence the same $\mu$. By bilinearity, $\mathcal{H}_{0,M}(X,P+Q)-\mathcal{H}_{0,M}(X,P)=-\langle b_{\mu}X,Q\rangle_{2}$, whose absolute value is at most $d^{2}a(1+\lVert X\rVert_{2})\lVert Q\rVert_{2}$ by Cauchy--Schwarz and (1); so Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §lipschitz holds with $L=d^{2}a$ (Hamiltonians on Phase-Space Noncommutative Laws that are Lipschitz in the Momentum with Linear Growth §lipschitz). By bilinearity, $\mathcal{H}_{0,M}(X,tP+(1-t)P')=t\,\mathcal{H}_{0,M}(X,P)+(1-t)\,\mathcal{H}_{0,M}(X,P')$, which gives Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §convex. Finally $\mathcal{H}_{0,M}(X,0)=-f(\mu)\le B_{f}$, which is Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §bound with $C=B_{f}$. Hence $\mathcal{H}^{\mathrm{LQ}}$ is quadratic with a convex Lipschitz remainder.