Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $H$ be a complex Hilbert space. Adjoints of elements of $\mathcal{L}(H)$ exist and lie in $\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint.

1. (Resolvent)¶ Let $A\in\mathcal{L}(H)$ be self-adjoint and let $y\ne0$ be real. The resolvent of $A$ at $y$ is $R_{y}(A)=(A-iyI)^{-1}$, the inverse of the bijection $A-iyI$ of $H$ onto $H$; it exists and belongs to $\mathcal{L}(H)$ by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §invertible.

2. (Resolvent transform)¶ Let $d\in\mathbb{N}$, with initial segment $[d]$, and let $a=(a_{1},\dots,a_{d})$ be a $d$-tuple of self-adjoint elements of $\mathcal{L}(H)$. The resolvent transform of $a$ is the $2d$-tuple $\mathbf{R}(a)$ in $\mathcal{L}(H)$ with

For a single self-adjoint $A\in\mathcal{L}(H)$, $\mathbf{R}(A)$ is the resolvent transform of the $1$-tuple $(A)$.