The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $(H,M,\Omega)$ be a tracial W*-probability space with fixed vectors $H_{\mathrm{sa}}$ as in Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §spaces; adjoints of elements of $M$ are those of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §maps.

1. (Resolvent of a self-adjoint vector)¶ For $\xi\in H_{\mathrm{sa}}$, the resolvent $R_{1}(\xi)\in M$ of $\xi$ is the unique $T\in M$ of Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §extension: $(R_{1}(s_{k})\Omega)$ converges to $R_{1}(\xi)\Omega$ for every sequence $(s_{k})$ of self-adjoint elements of $M$ with $s_{k}\Omega\to\xi$. By the same clause, $R_{1}(s\Omega)=R_{1}(s)$ for every self-adjoint $s\in M$, so the notation agrees with Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple §resolvent.

2. (Resolvent transform)¶ For $d\in\mathbb{N}$ and an $L^{2}$ $d$-tuple $X=(X_{1},\dots,X_{d})$ of $(H,M,\Omega)$, the resolvent transform of $X$ is the $2d$-tuple $\mathbf{R}(X)$ in $M$ with

For $\xi\in H_{\mathrm{sa}}$, $\mathbf{R}(\xi)$ is the resolvent transform of the $L^{2}$ $1$-tuple $(\xi)$. The argument distinguishes this transform of a tuple of vectors from the transform $\mathbf{R}(s)$ of a tuple of operators in Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple §transform.