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The resolvent star algebra is dense in C_0(R)

Statement

Fix $z \notin R$ and let $r (x) = (x - z)^{- 1}$ . Let $A$ be the $*$ -algebra generated by $r$ in $C_{0} (R)$ , that is, the uniform closure of the linear span of the products $r^{k} {\overline{r}}^{j}$ with $k + j \geq 1$ . Then $A = C_{0} (R)$ . Under Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), consequently, if self-adjoint $A_{n}, A$ satisfy $R_{A_{n}} (z) \to R_{A} (z)$ strongly (respectively in norm) at one nonreal $z$ , then the same convergence holds at every nonreal $z^{'}$ , in particular at $\overline{z}$ .

Facts & Assumptions

[A1]

$A$ is a self-adjoint algebra of continuous functions vanishing at infinity, $r$ is injective on $R$ , and $r$ does not vanish at any point of $R$ ; the one-point compactification $R^{*} = R \cup {\infty}$ is a compact Hausdorff space and $C_{0} (R) = {f \in C (R^{*}) : f (\infty) = 0}$ (The one-point (Alexandroff) compactification $X^{*} = X \cup {\infty}$ , whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$ , $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Continuity of a map of topological spaces at a point and globally).

[A2]

Let $X$ be a compact Hausdorff space and let $B \subseteq C (X, C)$ be a self-adjoint complex function algebra containing the constants, separating points, with no common zero. Then $B$ is uniformly dense in $C (X, C)$ (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense).

[A3]

Under Countable Choice, with $D_{z} : = R_{A} (z) - R_{A^{'}} (z)$ for self-adjoint $A, A^{'}$ and nonreal $z$ , the resolvent identity gives $D_{z^{'}} = [I + (z^{'} - z) R_{A} (z)]^{- 1} D_{z} [I - (z^{'} - z) R_{A^{'}} (z^{'})]$ for nonreal $z^{'}$ , the first factor being $I + (z - z^{'}) R_{A} (z^{'})$ and the second $I + (z - z^{'}) R_{A^{'}} (z^{'})$ ; these affine transforms of resolvents are bounded with norms at most $1 + ∣ z^{'} - z ∣ / ∣ Im z^{'} ∣$ (Resolvent and spectrum of an unbounded operator, Resolvent of a self-adjoint operator: nonreal resolvents and the estimate, Norm and strong resolvent convergence, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: $z \notin R$ , $r (x) = (x - z)^{- 1}$ and the $*$ -algebra $A$ generated by $r$ .

1.1A1

Extend $r$ to $R^{*}$ by $r (\infty) = 0$ ; then $r \in C_{0} (R) \subseteq C (R^{*})$ and $r$ separates the points of $R^{*}$ : it is injective on $R$ , and $r (x) \neq 0 = r (\infty)$ .

1.2A3

Parameter independence: for nonreal $z^{'}$ the difference $D_{z^{'}} = R_{A} (z^{'}) - R_{A^{'}} (z^{'})$ factors as in [A3] through $D_{z}$ , with both outer factors of norm at most $1 + ∣ z^{'} - z ∣ / ∣ Im z^{'} ∣$ , since they are $I + (z - z^{'}) R_{A} (z^{'})$ and $I + (z - z^{'}) R_{A^{'}} (z^{'})$ and $∥ R_{A} (z^{'}) ∥, ∥ R_{A^{'}} (z^{'}) ∥ \leq 1 / ∣ Im z^{'} ∣$ ; hence $D_{z^{'}} \to 0$ in norm whenever $D_{z} \to 0$ in norm, and $D_{z^{'}} x \to 0$ for every $x$ whenever $D_{z} y \to 0$ for every $y$ , because bounded operators preserve both modes of convergence.

2.1A1A2step 1.1

The algebra $B : = {f + c 1 : f \in A, c \in C} \subseteq C (R^{*})$ contains the constants and $r, \overline{r}$ , is self-adjoint, separates points by step 1.1, and has no common zero because of the constant function $1$ ; hence $B$ is uniformly dense in $C (R^{*})$ by [A2].

3.1A1step 2.1

Therefore $A = C_{0} (R)$ : given $f \in C_{0} (R)$ and $ε > 0$ , density of $B$ gives $f = c 1 + g$ with $g \in A$ and $∥ f - c 1 - g ∥_{\infty} < ε$ ; evaluating at $\infty$ , where $f (\infty) = 0$ , gives $∣ c ∣ < ε$ , so $∥ f - g ∥_{\infty} < 2 ε$ and $g \in A$ .

4.1

The density claim is step 3.1 and the consequence is step 1.2, which also covers $z^{'} = \overline{z}$ . ∎

Depends on

Used by

Dependency tree · two levels

56 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Mathematical Methods in Quantum Mechanics, second edition (standard reference, not scraped)
Roland Schnaubelt, Evolution Equations (lecture notes) (standard reference, not scraped)