Proof of The Law of the Resolvent Transform is a Lipschitz Function of the Law, and Reconstruction of a Square-Integrable Tuple from a Bounded Tuple with the Law of Its Transform

Each result cited is universally quantified over the data in its own statement. Evaluation of polynomials at tuples of bounded operators is linear and multiplicative, respects adjoints at self-adjoint tuples, and turns substitution into composition, by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution. For $m\in\mathbb{N}$, $\kappa_{m}:\Sigma_{m}\to\Sigma^{2}_{m}$ is the canonical map of Square-Integrable Noncommutative Laws: Standing Notation §laws; it is an isometry from $(\Sigma_{m},W_{2})$, by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry, and $\widehat{W}_{2}$ is a metric by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §metric.

Step B1 (bounded law determination). Let $a$ be a self-adjoint $d$-tuple in $M$ and $u$ a self-adjoint $d$-tuple in the operators $M_{1}$ of a tracial W-probability space $(H_{1},M_{1},\Omega_{1})$, with $\lambda_{a}=\lambda_{u}$. Then $\lambda_{\mathbf{R}(a)}=\lambda_{\mathbf{R}(u)}$.* Choose real $r>0$ with $\lVert a_{j}\rVert_{\mathrm{op}}\le r$ and $\lVert u_{j}\rVert_{\mathrm{op}}\le r$ for every $j$. For $\ell\in\mathbb{N}$, Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §polynomial gives $q_{\ell}\in\mathcal{P}_{1}$ with $\lVert R_{1}(C)-q_{\ell}(C)\rVert_{\mathrm{op}}\le\ell^{-1}$ for every self-adjoint $C$ of norm at most $r$ on any complex Hilbert space. Let $\sigma^{j}=\sigma_{(x_{j})}:\mathcal{P}_{1}\to\mathcal{P}_{d}$ and let $Q_{\ell}$ be the $2d$-tuple in $\mathcal{P}_{d}$ with

Since $\sigma^{j}(q_{\ell})(a)=q_{\ell}(a_{j})$ and $\sigma^{j}(q_{\ell})^{*}(a)=q_{\ell}(a_{j})^{*}$, the entries of the tuple $Q_{\ell}(a)=(Q_{\ell,1}(a),\dots,Q_{\ell,2d}(a))$ differ from those of $\mathbf{R}(a)$ (Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple §transform) by operators of norm at most $\ell^{-1}$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint); all these entries have norm at most $2$. For a word $w=w_{1}\cdots w_{k}$ of length $k$ in the letters $1,\dots,2d$, with $T_{w}$ the product along $w$, distributivity gives

(empty products being $I$), so $\lVert Q_{\ell}(a)_{w}-\mathbf{R}(a)_{w}\rVert_{\mathrm{op}}\le k2^{k-1}\ell^{-1}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations; so $\langle\Omega,Q_{\ell}(a)_{w}\Omega\rangle\to\langle\Omega,\mathbf{R}(a)_{w}\Omega\rangle=\lambda_{\mathbf{R}(a)}(x_{w})$, and likewise for $u$. But $Q_{\ell}(a)_{w}=x_{w}(Q_{\ell}(a))=(\sigma_{Q_{\ell}}x_{w})(a)$, so

Letting $\ell\to\infty$ (Uniqueness of Limits in a Metric Space) gives $\lambda_{\mathbf{R}(a)}(x_{w})=\lambda_{\mathbf{R}(u)}(x_{w})$ for every word $w$, and both laws are linear, so they agree on $\mathcal{P}_{2d}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension.

Step B2 (bounded contraction). Let $a$ be a self-adjoint $d$-tuple in $M$ and $b$ one in $N$. Then $W_{2}(\lambda_{\mathbf{R}(a)},\lambda_{\mathbf{R}(b)})\le W_{2}(\lambda_{a},\lambda_{b})$. Here $\lambda_{a},\lambda_{b}\in\Sigma_{d}$ and $\lambda_{\mathbf{R}(a)},\lambda_{\mathbf{R}(b)}\in\Sigma_{2d}$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law and Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §membership. Let $\gamma\in\Pi(\lambda_{a},\lambda_{b})$. By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound and Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, $\gamma\in\Sigma_{2d}$, so $(\mathcal{H}_{\gamma},\mathcal{M}_{\gamma},\Omega_{\gamma})$ is a tracial W*-probability space by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star. The operators $L_{x_{i}}$ ($i\in[2d]$) lie in $\mathcal{M}_{\gamma}$ (The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra) and are self-adjoint (Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint, $x_{i}^{*}=x_{i}$), and $\lambda_{L_{x}}=\gamma$ for $L_{x}=(L_{x_{1}},\dots,L_{x_{2d}})$ by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law. Put $u=(L_{x_{1}},\dots,L_{x_{d}})$ and $v=(L_{x_{d+1}},\dots,L_{x_{2d}})$. For $p\in\mathcal{P}_{d}$, $p(u)=(\iota^{1}p)(L_{x})$, so $\lambda_{u}(p)=\gamma(\iota^{1}p)=\lambda_{a}(p)$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling; likewise $\lambda_{v}=\lambda_{b}$. By Step B1, $\lambda_{\mathbf{R}(u)}=\lambda_{\mathbf{R}(a)}$ and $\lambda_{\mathbf{R}(v)}=\lambda_{\mathbf{R}(b)}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling applied to the self-adjoint $2d$-tuples $\mathbf{R}(u)$ and $\mathbf{R}(v)$, then The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent, The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §lipschitz, and Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling again with $\lambda_{(u,v)}=\lambda_{L_{x}}=\gamma$,

As $\gamma$ was arbitrary, The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance and Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field give the claim.

Claim 1. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §limit choose self-adjoint $d$-tuples $s^{k}$ in $M$ and $t^{k}$ in $N$ with $s^{k}\Omega\to X$ and $t^{k}\Psi\to X'$ entrywise, so $\lVert s^{k}\Omega-X\rVert_{2}\to0$ and $\lVert t^{k}\Psi-X'\rVert_{2}\to0$. By The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent and The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §lipschitz, $\lVert\mathbf{R}(s^{k})\Omega-\mathbf{R}(X)\Omega\rVert_{2}\le\lVert s^{k}\Omega-X\rVert_{2}$, and by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz

and likewise for $t^{k}$ and $X'$. Hence, by the triangle inequality for $\widehat{W}_{2}$, the isometry of $\kappa_{2d}$ and $\kappa_{d}$, and Step B2,

with $\varepsilon_{k}=\lVert s^{k}\Omega-X\rVert_{2}+\lVert t^{k}\Psi-X'\rVert_{2}\to0$. Letting $k\to\infty$ (Order Properties of Limits of Real Sequences) gives the inequality. If $\mathrm{law}(X)=\mathrm{law}(X')$, then $W_{2}(\lambda_{\mathbf{R}(X)},\lambda_{\mathbf{R}(X')})=0$, and $\lambda_{\mathbf{R}(X)}=\lambda_{\mathbf{R}(X')}$ by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §separation.

Claim 2. Fix $j\in[d]$, let $\sigma^{(j)}=\sigma_{(x_{2j-1},x_{2j})}:\mathcal{P}_{2}\to\mathcal{P}_{2d}$, and note that $(\mathbf{R}(X)_{2j-1},\mathbf{R}(X)_{2j})=\mathbf{R}(X_{j})$ by The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple §transform. For $n,l\in\mathbb{N}$ put $q=\sigma^{(j)}(P_{n}-P_{l})\in\mathcal{P}_{2d}$, so that $q(s)=P_{n}(s_{2j-1},s_{2j})-P_{l}(s_{2j-1},s_{2j})$ and $q(\mathbf{R}(X))=P_{n}(\mathbf{R}(X_{j}))-P_{l}(\mathbf{R}(X_{j}))$ by the substitution rule. Since $s$ and $\mathbf{R}(X)$ are self-adjoint tuples (The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent), $q(s)^{*}q(s)=(q^{*}q)(s)$ and likewise at $\mathbf{R}(X)$, so

By The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §recovery, $(P_{n}(\mathbf{R}(X_{j}))\Omega)_{n}$ converges to $X_{j}$, hence is Cauchy; therefore $(P_{n}(s_{2j-1},s_{2j})\Psi)_{n}$ is Cauchy and converges in the complex Hilbert space $K$ to some $\widehat{X}_{j}$. Each $P_{n}(s_{2j-1},s_{2j})$ is a self-adjoint element of $N$ ($P_{n}\in\mathcal{P}_{2,\mathrm{sa}}$, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport with $\mathcal{A}=N$), so its vector is fixed by the conjugation of $(K,N,\Psi)$, and so is the limit $\widehat{X}_{j}$, by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint. Thus $\widehat{X}$ is an $L^{2}$ $d$-tuple of $(K,N,\Psi)$.

For the law, let $Q_{n}=(\sigma^{(1)}P_{n},\dots,\sigma^{(d)}P_{n})$, a $d$-tuple in $\mathcal{P}_{2d}$, and let $t^{n}=Q_{n}(s)$ and $r^{n}=Q_{n}(\mathbf{R}(X))$ be the self-adjoint $d$-tuples in $N$ and $M$ of their values; then $t^{n}\Psi\to\widehat{X}$ and $r^{n}\Omega\to X$ entrywise by the previous paragraph. For $p\in\mathcal{P}_{d}$, $p(t^{n})=(\sigma_{Q_{n}}p)(s)$ and $p(r^{n})=(\sigma_{Q_{n}}p)(\mathbf{R}(X))$, so

By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §law, $\mathrm{law}(\widehat{X})$ and $\mathrm{law}(X)$ are the limits of the same sequence $(\kappa_{d}(\lambda_{t^{n}}))=(\kappa_{d}(\lambda_{r^{n}}))$, so they coincide by Uniqueness of Limits in a Metric Space.