E14 Schur Feshbach elimination
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Let \(X=X_P\oplus X_Q\) and write
For \(zI-A_{QQ}\) invertible, define
and
Theorem E14.1: Schur Feshbach resolvent
Assume \(zI-A_{QQ}\) is invertible. Then \(zI-A\) is invertible if and only if \(F_P(z)\) is invertible. In that case,
In finite dimension,
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Proof of Theorem E14.1
Set
and
Direct multiplication gives \(zI-A=L_zS_z\). The first factor is invertible. The claims follow from triangular inversion and the determinant of the second factor. This is the finite block form of the Feshbach reduction [Feshbach1958].
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E14.1 Time domain memory
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For the homogeneous equation \(\dot x=Ax\), variation of constants gives
Substitution into the retained equation gives
where
Theorem E14.2: Memory and self energy
If \(z\) lies to the right of the spectrum of \(A_{QQ}\), then
If, for constants \(M\geq0\) and \(\gamma>0\),
then
and
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Proof of Theorem E14.2
If \(X_Q=\{0\}\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \(B=A_{QQ}\), which is a bounded operator on the hidden Banach space, and put \(s(B)=\max\operatorname{Re}{\operatorname{Spec}}(B)\). Choose a real \(\beta\) with \(s(B)<\beta<\operatorname{Re}z\). Spectral mapping and the spectral-radius formula give \(r({\mathrm e}^B)={\mathrm e}^{s(B)}<{\mathrm e}^\beta\), hence \({\left\lVert{\mathrm e}^{nB}\right\rVert}\leq M_\beta{\mathrm e}^{\beta n}\) for every integer \(n\geq0\), after increasing \(M_\beta\) to cover the finitely many initial values. Write \(t=n+r\), where \(0\leq r<1\). Then
Thus \({\mathrm e}^{-zt}{\mathrm e}^{tB}\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \(R=\int_0^\infty{\mathrm e}^{-zt}{\mathrm e}^{tB}{\,\mathrm d} t\), so \(R=(zI-B)^{-1}\). This is the bounded-generator growth principle in [EngelNagel2000], Chapter IV, Corollary 3.12. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.
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Theorem E14.3: Criterion for exact local closure
Assume \(x_Q(0)=0\) and \(d=\dim X_Q<\infty\). The following conditions are equivalent.
For every retained initial state,
\[ P{\mathrm e}^{tA}P = {\mathrm e}^{tA_{PP}}P \]for all \(t\geq0\).
\(\Sigma_P(z)=0\) for every sufficiently large \(z\).
\({\mathcal K}(t)=0\) for every \(t\geq0\).
If \(d=\dim X_Q\), then
\[ A_{PQ}A_{QQ}^kA_{QP}=0 \]for \(0\leq k<d\).
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Proof of Theorem E14.3
By Theorem E2.5, the Laplace transform of the projected propagator is the projected resolvent. By Theorem E14.1, the first condition is equivalent to
for all sufficiently large \(z\), which is equivalent to the second condition. The second and third conditions are equivalent by Theorem E14.2 and uniqueness of the Laplace transform.
For large \(z\),
Thus the second condition implies the moment identities. Conversely, the first \(d\) identities imply all later ones by the Cayley Hamilton theorem, so the series vanishes.
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Remark E14.4
A finite hidden block produces a strictly proper rational self energy. It tends to zero as \(z\) tends to infinity. Hence a nonzero constant self energy cannot arise from hidden dynamics alone. A constant correction must already be included in the retained generator.