Manuscript En01-A@1 · En01-A.tex · theorem · 558b972b

Theorem: Inversion and operator localization

Operator-valued Möbius hierarchy and three orders · lines 1599–1635 · proof included or linked

En01-A@1#thm:localization, En01-A@1#eq:localization

For every $J$ ,

Δ_{J} ρ = K \subseteq J \sum Z_{K} .

For nonempty $J$ ,

Z_{J} = (- 1)^{∣ J ∣} r \neq = r^{'} ordered J \subseteq D (r, r^{'}) \sum p_{r} p_{r^{'}} ∣ Φ_{r} ⟩ ⟨ Φ_{r^{'}} ∣.

Equivalently,

Z_{J} = j \in J \prod (Δ_{{j}} - id) (ρ) .

Proof. For inversion, interchange the subset sums. The coefficient of $Δ_{L} ρ$ is

K : L \subseteq K \subseteq J \sum (- 1)^{∣ K ∣ - ∣ L ∣},

which is one when $L = J$ and zero otherwise.

For localization, the diagonal terms cancel when $J \neq = \emptyset$ . A cross term with difference set $D$ receives the coefficient

K \subseteq J ∖ D \sum (- 1)^{∣ J ∣ - ∣ K ∣} = (- 1)^{∣ J ∣} [J \subseteq D] .

The product formula follows by expanding the commuting factors.

Displayed formulas in this passage

En01-A@1#eq:localization · line 1606
$Z_{J} = (- 1)^{∣ J ∣} r \neq = r^{'} ordered J \subseteq D (r, r^{'}) \sum p_{r} p_{r^{'}} ∣ Φ_{r} ⟩ ⟨ Φ_{r^{'}} ∣.$
unlabeled · line 1602
$Δ_{J} ρ = K \subseteq J \sum Z_{K} .$
unlabeled · line 1614
$Z_{J} = j \in J \prod (Δ_{{j}} - id) (ρ) .$
unlabeled · line 1622
$K : L \subseteq K \subseteq J \sum (- 1)^{∣ K ∣ - ∣ L ∣},$
unlabeled · line 1629
$K \subseteq J ∖ D \sum (- 1)^{∣ J ∣ - ∣ K ∣} = (- 1)^{∣ J ∣} [J \subseteq D] .$