Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $H$ be a complex Hilbert space, let $A,B\in\mathcal{L}(H)$ be self-adjoint, and let $y,y'\ne0$ be real. The resolvents $R_{y}(A)$ and the resolvent transforms $\mathbf{R}(a)$ are those of Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple; inverses of bijections and commutants are as in The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A; $A^{3}=AAA$; and for a self-adjoint $C$ in $\mathcal{L}(K)$, $K$ a complex Hilbert space, and $p\in\mathcal{P}_{1}$, $p(C)$ is the value of $p$ at the $1$-tuple $(C)$.

1. (Adjoints and commutation)¶ $R_{y}(A)^{*}=R_{-y}(A)$. $R_{y}(A)$ commutes with $A$, with $R_{y'}(A)$, and with every element of $\mathcal{L}(H)$ that commutes with $A$. If $\mathcal{S}\subseteq\mathcal{L}(H)$ satisfies $\mathcal{S}=\mathcal{S}''$ and $A\in\mathcal{S}$, then $R_{y}(A)\in\mathcal{S}$.

2. (Entries of the transform)¶ Let $d\in\mathbb{N}$ and let $a$ be a $d$-tuple of self-adjoint elements of $\mathcal{L}(H)$. Every entry of $\mathbf{R}(a)$ is self-adjoint with operator norm at most $1$, and $R_{1}(a_{j})=\mathbf{R}(a)_{2j-1}+i\,\mathbf{R}(a)_{2j}$ for every $j\in[d]$.

3. (Resolvent identity)¶ $R_{y}(A)-R_{y}(B)=R_{y}(A)\,(B-A)\,R_{y}(B)$.

4. (Recovery at the bounded level)¶ If $y>0$, then

5. (Stepping)¶ If $y>0$, $y'>0$ and $|y'-y|<y$, then $\lVert(y'-y)R_{y}(A)\rVert_{\mathrm{op}}<1$ and

6. (Polynomial approximation)¶ For all real $r>0$ and $\varepsilon>0$ there is $p\in\mathcal{P}_{1}$ such that $\lVert R_{1}(C)-p(C)\rVert_{\mathrm{op}}\le\varepsilon$ for every complex Hilbert space $K$ and every self-adjoint $C\in\mathcal{L}(K)$ with $\lVert C\rVert_{\mathrm{op}}\le r$.

7. (Approximation through the transform)¶ For all real $s\ge1$ and $\varepsilon>0$ there is $p\in\mathcal{P}_{2}$ such that $\lVert R_{s}(C)-p(\mathbf{R}(C))\rVert_{\mathrm{op}}\le\varepsilon$ for every complex Hilbert space $K$ and every self-adjoint $C\in\mathcal{L}(K)$.