Manuscript I-B-VP@3 · I-B-VP-v3.tex · lemma · 93c04eb4

Lemma: Uniform positive Krein bound

Krein formula, infrared control, and nonnegative states · lines 1544–1567 · proof included or linked

I-B-VP@3#lem:Cbound

If $H_{W} \geq 0$ , then

0 \leq C_{W} (p) \leq S^{- 1}, p > 0,

as Hermitian matrices.

Proof. The inverse in (eq:CW) is bounded above by $(V^{†} Δ V)^{- 1}$ . Moreover

V (V^{†} Δ V)^{- 1} V^{†} \leq Δ^{- 1},

because conjugation by $Δ^{1/2}$ turns the left side into the orthogonal projection onto $ran (Δ^{1/2} V)$ . Consequently

C_{W} \leq \frac{1}{2} E^{- 1} Δ^{- 1} E^{- 1} = S^{- 1} .

Displayed formulas in this passage

unlabeled · line 1547
$0 \leq C_{W} (p) \leq S^{- 1}, p > 0,$
unlabeled · line 1557
$V (V^{†} Δ V)^{- 1} V^{†} \leq Δ^{- 1},$
unlabeled · line 1564
$C_{W} \leq \frac{1}{2} E^{- 1} Δ^{- 1} E^{- 1} = S^{- 1} .$