Proof of The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the $L^2$ Lipschitz Bound and Universal Polynomial Recovery

Each result cited is universally quantified over the data in its own statement. Throughout, $(H,M,\Omega)$ is a tracial W*-probability space; $M=(M')'$ is closed under sums and complex multiples by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §algebra; norm bounds for sums, scalar multiples, composites and adjoints are those of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations and Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint; and for $\xi\in H_{\mathrm{sa}}$, $R_{1}(\xi)\in M$ with $\lVert R_{1}(\xi)\rVert_{\mathrm{op}}\le1$ by The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple §resolvent and Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §extension.

Step A (continuity of polynomials in $L^{2}$). Let $n\in\mathbb{N}$, let $t=(t_{1},\dots,t_{n})$ and $t^{k}=(t^{k}_{1},\dots,t^{k}_{n})$ ($k\in\mathbb{N}$) be $n$-tuples in $M$ whose entries have operator norm at most $1$, and suppose $t^{k}_{i}\Omega\to t_{i}\Omega$ for every $i\in[n]$. Then $p(t^{k})\Omega\to p(t)\Omega$ for every $p\in\mathcal{P}_{n}$. Indeed, for a word $w$ of length $\ell\in\mathbb{N}$, distributivity gives

the empty products being $I$, and by Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §products (first inequality, then second) and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §bound with $R=1$ each term applied to $\Omega$ has norm at most $\lVert(t^{k}_{w_{i}}-t_{w_{i}})\Omega\rVert$, which tends to $0$; for the empty word both sides equal $I$. Since $p$ is a finite linear combination of monomials and evaluation is linear (Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation), the claim follows.

Claim 1. Let $j\in[d]$ and $R=R_{1}(X_{j})\in M$. The entries $\tfrac12(R+R^{*})$ and $\tfrac{1}{2i}(R-R^{*})$ of $\mathbf{R}(X)$ lie in $M$, are self-adjoint because $(\tfrac12(R+R^{*}))^{*}=\tfrac12(R^{*}+R)$ and $(\tfrac{1}{2i}(R-R^{*}))^{*}=-\tfrac{1}{2i}(R^{*}-R)$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus), and have operator norm at most $\tfrac12(\lVert R\rVert_{\mathrm{op}}+\lVert R^{*}\rVert_{\mathrm{op}})\le1$. So $\mathbf{R}(X)$ is a self-adjoint $2d$-tuple in $M$ as in Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §tuple. If $s$ is a self-adjoint $d$-tuple in $M$, then $R_{1}(s_{j}\Omega)=R_{1}(s_{j})$ for every $j$ by Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §extension, so the entries of $\mathbf{R}(s\Omega)$ (The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple §transform) and of $\mathbf{R}(s)$ (Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple §transform) coincide. Finally, the $i$-th entry of the pair $(X,Y)$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations) is $X_{i}$ for $i\in[d]$ and $Y_{i-d}$ for $d<i\le d+m$; so for $i\in[d]$ the entries $2i-1,2i$ of $\mathbf{R}((X,Y))$ are the entries $2i-1,2i$ of $\mathbf{R}(X)$, and for $i=d+l$ with $l\in[m]$ the entries $2d+2l-1,2d+2l$ of $\mathbf{R}((X,Y))$ are the entries $2l-1,2l$ of $\mathbf{R}(Y)$.

Claim 2. By Claim 1 and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded, $\mathbf{R}(X)\Omega$ and $\mathbf{R}(Y)\Omega$ are $L^{2}$ $2d$-tuples. Fix $j\in[d]$ and put $\Delta_{j}=R_{1}(X_{j})-R_{1}(Y_{j})\in M$. Choose sequences $(s_{k})$, $(t_{k})$ of self-adjoint elements of $M$ with $s_{k}\Omega\to X_{j}$ and $t_{k}\Omega\to Y_{j}$ (Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §extension). By Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §lipschitz with $y=1$, $\lVert R_{1}(s_{k})\Omega-R_{1}(t_{k})\Omega\rVert\le\lVert s_{k}\Omega-t_{k}\Omega\rVert$; letting $k\to\infty$ with Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §extension and Order Properties of Limits of Real Sequences gives $\lVert\Delta_{j}\Omega\rVert\le\lVert X_{j}-Y_{j}\rVert$. The entries $2j-1,2j$ of $\mathbf{R}(X)-\mathbf{R}(Y)$ are $\tfrac12(\Delta_{j}+\Delta_{j}^{*})$ and $\tfrac{1}{2i}(\Delta_{j}-\Delta_{j}^{*})$, so by the identity of Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §products and Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples,

and the claim follows by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Claim 3. Construction. For $n\in\mathbb{N}$, Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §transform with $s=n$ and $\varepsilon=n^{-3}$ gives $p_{n}\in\mathcal{P}_{2}$ with $\lVert R_{n}(C)-p_{n}(\mathbf{R}(C))\rVert_{\mathrm{op}}\le n^{-3}$ for every complex Hilbert space $K$ and every self-adjoint $C\in\mathcal{L}(K)$. Put $P_{n}=\tfrac{n^{2}}{2}(p_{n}+p_{n}^{*})$; it belongs to $\mathcal{P}_{2,\mathrm{sa}}$ since the adjoint of The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §adjoint is additive and involutive (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts) and $n^{2}/2$ is real. The sequence $(P_{n})$ is fixed once and for all.

Bounded case. For self-adjoint $a\in M$ put $g_{n}(a)=\tfrac{n^{2}}{2}(R_{n}(a)+R_{-n}(a))\in M$ (Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §membership). The entries of $\mathbf{R}(a)$ are self-adjoint (Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §parts), so $p_{n}^{*}(\mathbf{R}(a))=p_{n}(\mathbf{R}(a))^{*}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint, and with $R_{-n}(a)=R_{n}(a)^{*}$ (Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §adjoint),

of operator norm at most $n^{2}n^{-3}=n^{-1}$. With Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §recovery this gives

Moreover, for self-adjoint $a,b\in M$, $g_{n}(a)-g_{n}(b)=\tfrac{n^{2}}{2}(\Delta+\Delta^{*})$ with $\Delta=R_{n}(a)-R_{n}(b)$, so by Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §products (the identity) and Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §lipschitz with $y=n$,

General case. Let $\xi\in H_{\mathrm{sa}}$ and choose self-adjoint $s_{k}\in M$ with $s_{k}\Omega\to\xi$ (Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §extension). For fixed $n$: by Claim 1, $\mathbf{R}(s_{k})=\mathbf{R}(s_{k}\Omega)$ for the $1$-tuple $(s_{k})$, and by Claim 2 its entries applied to $\Omega$ converge to those of $\mathbf{R}(\xi)$; all entries have operator norm at most $1$ (Claim 1), so Step A gives $P_{n}(\mathbf{R}(s_{k}))\Omega\to P_{n}(\mathbf{R}(\xi))\Omega$ as $k\to\infty$.

Let $\varepsilon>0$. Choose $m\in\mathbb{N}$ with $\lVert\xi-s_{m}\Omega\rVert\le\varepsilon$, and, by The Archimedean Property of the Real Numbers, $N\in\mathbb{N}$ with $N^{-1}\le\varepsilon$ and $\lVert s_{m}\rVert_{\mathrm{op}}^{3}N^{-2}\le\varepsilon$. For $n\ge N$ and every $k$, the two bounded estimates give

where the first inequality repeats the bounded-case computation with $\lVert(g_{n}(s_{k})-P_{n}(\mathbf{R}(s_{k})))\Omega\rVert\le n^{-1}$. Hence

for every $k$. Letting $k\to\infty$ (Order Properties of Limits of Real Sequences), the first and last terms tend to $0$ and $\lVert s_{k}\Omega-s_{m}\Omega\rVert\to\lVert\xi-s_{m}\Omega\rVert\le\varepsilon$, so $\lVert\xi-P_{n}(\mathbf{R}(\xi))\Omega\rVert\le4\varepsilon$ for every $n\ge N$. As $\varepsilon>0$ was arbitrary, $P_{n}(\mathbf{R}(\xi))\Omega\to\xi$; the sequence $(P_{n})$ does not depend on $(H,M,\Omega)$ or $\xi$.