Theorem: Linear Yang–Feldman characterization

The following identities hold:

1. ${\mathcal{R}}_q=1-{\mathcal{R}}_q\mathsf{L}[q]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. It is the unique formal series with leading term $1$ satisfying this recursion. Equivalently, ${\hat{\varphi }}_q(c)=\hat{\varphi }(c)-{\hat{\varphi }}_q(\mathsf{L}[q]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c)$.

2. ${\hat{\varphi }}_q({\mathsf{E}}_g[q]v)=0$ for every $v\in \mathcal{D}(M)$.

3. If $\mathrm{supp}c\cap J^{+}(\mathrm{supp}g)=\varnothing$, then ${\hat{\varphi }}_q(c)=\hat{\varphi }(c)$.

4. The algebraic commutator is

   $$[{\hat{\varphi }}_q(c),{\hat{\varphi }}_q(d)]=i\hslash ⟨c,({{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}})d⟩1.$$

Proof. The inverse identity $(1+A{)}^{-1}=1-(1+A{)}^{-1}A$, with $A=\mathsf{L}[q]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, proves (i). Comparing coefficients determines $-{\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ at first order and ${\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{\mathsf{L}}_{2,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ at second order uniquely.

For compact $v$, ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{P}_{m}v=v$, so

Its class vanishes, proving (ii). The support condition in (iii) gives $\mathsf{L}[q]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c=0$.

For (iv), transposition gives ${{\mathcal{R}}_q}^{\mathsf{T}}=(1+{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}\mathsf{L}[q]{)}^{-1}=1-{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}}\mathsf{L}[q]$, while ${\mathcal{R}}_q=1-\mathsf{L}[q]{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}}$. Using the free commutator,

The operator in the middle equals

This proves the commutator identity.