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

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $d\in\mathbb{N}$, and let $(H,M,\Omega)$ and $(K,N,\Psi)$ be tracial W*-probability spaces. Resolvent transforms $\mathbf{R}(X)$ of $L^{2}$ tuples are those of The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple; $\lambda_{t}$ is the law of a self-adjoint tuple $t$ in $M$ or in $N$; $W_{2}$ is the noncommutative Wasserstein distance on $\Sigma_{2d}$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §couplings; and $p(t)$ is the value of $p\in\mathcal{P}_{n}$ at an $n$-tuple $t$ of operators as in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §polynomials.

1. (Law of the transform)¶ Let $X$ be an $L^{2}$ $d$-tuple of $(H,M,\Omega)$ and $X'$ an $L^{2}$ $d$-tuple of $(K,N,\Psi)$. Then

In particular, $\mathrm{law}(X)=\mathrm{law}(X')$ implies $\lambda_{\mathbf{R}(X)}=\lambda_{\mathbf{R}(X')}$.

2. (Reconstruction)¶ Let $(P_{n})_{n\in\mathbb{N}}$ be a sequence as in The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §recovery. Let $X$ be an $L^{2}$ $d$-tuple of $(H,M,\Omega)$ and let $s$ be a self-adjoint $2d$-tuple in $N$ with $\lambda_{s}=\lambda_{\mathbf{R}(X)}$. Then for every $j\in[d]$ the sequence $\bigl(P_{n}(s_{2j-1},s_{2j})\Psi\bigr)_{n\in\mathbb{N}}$ converges in $K$ to a vector $\widehat{X}_{j}$ fixed by the conjugation of $(K,N,\Psi)$, and $\widehat{X}=(\widehat{X}_{1},\dots,\widehat{X}_{d})$ is an $L^{2}$ $d$-tuple of $(K,N,\Psi)$ with $\mathrm{law}(\widehat{X})=\mathrm{law}(X)$.