Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f

Green operators on an extension-dependent spectral domain

Green operators on an extension-dependent spectral domain · lines 709–745

Let $B$ be any extension in the restricted ledger. By Theorem thm:positivity, $B$ is bounded below. Choose $b > 0$ so that $B + b \geq 1$ , and define

H^{\infty}_{B} = k = 0 ⋂^{\infty} dom (B^{k}), p_{k} (v) = (B + b)^{k} v .

This is a Fréchet space; its topology is independent of the admissible shift $b$ . Put

D_{tc}_{B} = C_{c}^{\infty} (R; H^{\infty}_{B}), E_{B} = C^{\infty} (R; H^{\infty}_{B}) .

The first space has its usual inductive-limit topology over compact time supports; it is not asserted to be a Fréchet space.

For real $λ$ set

w_{t} (λ) = ⎩ ⎨ ⎧ sin (t λ) / λ, t, sinh (t - λ) / - λ, λ > 0, λ = 0, λ < 0, W_{B} (t) = w_{t} (B) .

The spectral multiplier is bounded for each fixed $t$ because $B$ is bounded below. Functional calculus, rather than convergence of an operator power series on arbitrary smooth vectors, is the definition.

Displayed formulas in this passage

unlabeled · line 715
$H^{\infty}_{B} = k = 0 ⋂^{\infty} dom (B^{k}), p_{k} (v) = (B + b)^{k} v .$
unlabeled · line 722
$D_{tc}_{B} = C_{c}^{\infty} (R; H^{\infty}_{B}), E_{B} = C^{\infty} (R; H^{\infty}_{B}) .$
unlabeled · line 732
$w_{t} (λ) = ⎩ ⎨ ⎧ sin (t λ) / λ, t, sinh (t - λ) / - λ, λ > 0, λ = 0, λ < 0, W_{B} (t) = w_{t} (B) .$

Green operators on an extension-dependent spectral domain

Displayed formulas in this passage

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