Gap-controlled correlation resolvent and positive coefficients

Cauchy–Schwarz in the finite weighted inner product and [F1] give $\mathrm{\mid }{C}_{A,B}(t)\mathrm{\mid }\le e^{-\gamma t}\mathrm{\parallel }A{\mathrm{\parallel }}_{\sigma }\mathrm{\parallel }B{\mathrm{\parallel }}_{\sigma }$. The right side is integrable on $[0,\mathrm{\infty })$, so the correlation integral converges absolutely.

The integral $Y:={\int }_0^{\mathrm{\infty }}{e}^{tG}Adt$ converges in the finite-dimensional invariant space ${\mathcal{M}}_0$. Since $\frac{d}{dt}{e}^{tG}A=G(e^{tG}A)$ and the decay in [F1] sends the endpoint to zero, integrating gives $G(Y)=-A$. If $GZ=0$ for centered $Z$, then [F1] gives $\mathrm{\parallel }Z{\mathrm{\parallel }}_{\sigma }=\mathrm{\parallel }{e}^{tG}Z{\mathrm{\parallel }}_{\sigma }\le e^{-\gamma t}\mathrm{\parallel }Z{\mathrm{\parallel }}_{\sigma }$, hence $Z=0$. Thus $-G{\mathrm{\mid }}_{{\mathcal{M}}_0}$ is invertible and $Y=(-G{\mathrm{\mid }}_{{\mathcal{M}}_0}{)}^{-1}A$, which proves the resolvent identity.

For $Y$ from step 1.2, $\mathrm{\Gamma }(A)=\mathrm{Re}⟨Y,-G(Y){⟩}_{\sigma }$. The term $\mathrm{Re}⟨Y,-K(Y){⟩}_{\sigma }$ vanishes because $K$ is anti-self-adjoint. The remaining term is $⟨Y,-D(Y){⟩}_{\sigma }\ge \gamma \mathrm{\parallel }Y{\mathrm{\parallel }}_{\sigma }^2\ge 0$ by [F1]–[F2]. Therefore $\mathrm{\Gamma }$ is a nonnegative real quadratic form on the real vector space of Hermitian centered observables; real polarization gives a positive-semidefinite symmetric matrix.