Paper VI: Refinement towers and limiting evolution

E4 Refinement towers and limiting evolution

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How to read the result

The limiting argument names its topology, reconstruction maps, and summability weights. Convergence of reconstructed trajectories is distinct from closure under the semigroup law. The range-leakage condition identifies the additional obstruction to that closure.

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The exact inverse limit and the approximate completion problem require different hypotheses. Exact intertwining preserves the inverse limit coordinate by coordinate. Approximate intertwining gives a Cauchy family only after reconstruction maps are chosen.

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E4.1 Exact inverse limits

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Let

\[ X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots \]

be Banach spaces with bounded connecting maps. Define the bounded inverse limit

\[ X_\infty = \left\{ (x_\ell)_{\ell\geq0}: C_\ell x_{\ell+1}=x_\ell, \quad \sup_\ell{\left\lVert x_\ell\right\rVert}<\infty \right\} \]

with the supremum norm.

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Lemma E4.1: Completeness of the bounded inverse limit

The space \(X_\infty\) is Banach.

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Proof of Lemma E4.1

The product of the spaces \(X_\ell\), equipped with the bounded supremum norm, is Banach. The equations \(C_\ell x_{\ell+1}=x_\ell\) define a closed linear subspace.

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Theorem E4.2: Exact projective limit semigroup

For each \(\ell\), let \(T_\ell(t)\) be a strongly continuous semigroup on \(X_\ell\). Assume

\[ C_\ell T_{\ell+1}(t)=T_\ell(t)C_\ell \]

for every \(\ell\) and \(t\geq0\), and assume the uniform bound

\[ \sup_\ell{\left\lVert T_\ell(t)\right\rVert}\leq M{\mathrm e}^{\omega t}. \]

Define

\[ T_\infty(t)(x_\ell)_\ell = (T_\ell(t)x_\ell)_\ell. \]

Then \(T_\infty(t)\) is a bounded semigroup on \(X_\infty\). It is strongly continuous if

\[ \lim_{t\downarrow0} \sup_\ell {\left\lVert T_\ell(t)x_\ell-x_\ell\right\rVert}=0 \]

for every \((x_\ell)_\ell\in X_\infty\).

If \(T_\ell(t)={\mathrm e}^{tA_\ell}\), the generators are bounded, and

\[ \sup_\ell{\left\lVert A_\ell\right\rVert}<\infty, \qquad C_\ell A_{\ell+1}=A_\ell C_\ell, \]

then the continuity condition holds and

\[ A_\infty(x_\ell)_\ell=(A_\ell x_\ell)_\ell \]

is the bounded generator of \(T_\infty\).

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Proof of Theorem E4.2

Exact intertwining shows that \((T_\ell(t)x_\ell)_\ell\) remains compatible. The uniform growth bound makes the coordinatewise operator bounded on the supremum norm. The semigroup law holds in every coordinate. The stated continuity condition is exactly strong continuity in the supremum norm.

For bounded generators with common bound \(M_A\),

\[ \sup_\ell {\left\lVert({\mathrm e}^{tA_\ell}-I)x_\ell\right\rVert} \leq ({\mathrm e}^{tM_A}-1)\sup_\ell{\left\lVert x_\ell\right\rVert}. \]

This tends to zero with \(t\). Exact generator intertwining makes \((A_\ell x_\ell)_\ell\) compatible, and coordinatewise differentiation identifies the generator.

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Remark E4.3

Strong continuity on each level does not by itself imply strong continuity in the supremum norm of the inverse limit. The additional uniform condition in Theorem E4.2 cannot be omitted without replacing the topology.

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E4.2 Summable adjacent defects

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Assume now that each \(C_\ell:X_{\ell+1}\to X_\ell\) has a bounded right inverse

\[ R_\ell:X_\ell\to X_{\ell+1}, \qquad C_\ell R_\ell=I_{X_\ell}. \]

For \(m<n\), set

\[ C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m. \]

The empty products are

\[ C_{m,m}=I_{X_m}, \qquad R_{m,m}=I_{X_m}. \]

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Definition E4.4: Reconstructed fine evolution

For a semigroup \(T_n(t)\) on \(X_n\), define the evolution seen from level \(m\) by

\[ U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}. \]

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Theorem E4.5: Weighted summable defect completion

Define the adjacent semigroup defect

\[ E_n(t)=C_nT_{n+1}(t)-T_n(t)C_n. \]

Fix a coarse level \(m\) and a time horizon \(T>0\). For \(n\geq m\), set

\[ w_{m,n} = {\left\lVert C_{m,n}\right\rVert}\, {\left\lVert R_nR_{n,m}\right\rVert}. \]

\[ \sum_{n=m}^{\infty} w_{m,n} \sup_{0\leq t\leq T}{\left\lVert E_n(t)\right\rVert}<\infty, \]

then \(U_{m,n}(t)\) converges in operator norm uniformly for \(0\leq t\leq T\). Its limit is denoted by \(U_m(t)\), and

\[ \sup_{0\leq t\leq T} {\left\lVert U_m(t)-U_{m,n}(t)\right\rVert} \leq \sum_{j=n}^{\infty}w_{m,j} \sup_{0\leq t\leq T}{\left\lVert E_j(t)\right\rVert}. \]

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Proof of Theorem E4.5

Since \(C_nR_n=I\),

\[ \begin{aligned} U_{m,n+1}(t)-U_{m,n}(t) ={}& C_{m,n} \bigl(C_nT_{n+1}(t)-T_n(t)C_n\bigr) R_nR_{n,m}. \end{aligned} \]

Therefore

\[ {\left\lVert U_{m,n+1}(t)-U_{m,n}(t)\right\rVert} \leq w_{m,n}{\left\lVert E_n(t)\right\rVert}. \]

The weighted summability hypothesis makes the sequence uniformly Cauchy. Summing the tail estimates gives the stated bound.

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Corollary E4.6: Contractive summable defect completion

Assume

\[ {\left\lVert C_\ell\right\rVert}\leq1, \qquad {\left\lVert R_\ell\right\rVert}\leq1, \qquad {\left\lVert T_\ell(t)\right\rVert}\leq1 \]

for every \(\ell\) and \(t\geq0\). If

\[ \sum_{\ell=0}^{\infty} \sup_{0\leq t\leq T}{\left\lVert E_\ell(t)\right\rVert}<\infty, \]

then the conclusion of Theorem E4.5 holds, and

\[ \sup_{0\leq t\leq T} {\left\lVert U_m(t)-U_{m,n}(t)\right\rVert} \leq \sum_{\ell=n}^{\infty} \sup_{0\leq t\leq T}{\left\lVert E_\ell(t)\right\rVert}. \]

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Proof of Corollary E4.6

Every weight \(w_{m,n}\) is at most one.

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Corollary E4.7: Generator defect criterion

Assume

\[ {\left\lVert C_\ell\right\rVert}\leq1, \qquad {\left\lVert R_\ell\right\rVert}\leq1, \]

and suppose \(T_\ell(t)={\mathrm e}^{tA_\ell}\) are contraction semigroups with bounded generators. Set

\[ \Delta_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell. \]

\[ \sum_{\ell=0}^{\infty}{\left\lVert\Delta_\ell\right\rVert}<\infty, \]

then the conclusion of Corollary E4.6 holds on every bounded time interval, and

\[ \sup_{0\leq t\leq T} {\left\lVert U_m(t)-U_{m,n}(t)\right\rVert} \leq T\sum_{\ell=n}^{\infty}{\left\lVert\Delta_\ell\right\rVert}. \]

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Proof of Corollary E4.7

Apply the contraction estimate in Theorem E3.1 to every adjacent pair.

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The limit in Theorem E4.5 need not satisfy the semigroup law. The obstruction comes from leaving the reconstructed fine subspace between two time steps.

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Theorem E4.8: Asymptotic closure criterion

Assume the weighted convergence in Theorem E4.5 holds for every bounded time interval, and define

\[ P_{m,n}=R_{n,m}C_{m,n} \]

on \(X_n\). Suppose that for every \(T>0\),

\[ \rho_{m,n}(T) = \sup_{0\leq s,t\leq T} {\left\lVert C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m} \right\rVert} \longrightarrow0. \]

Then

\[ U_m(s+t)=U_m(s)U_m(t) \]

for all \(s,t\geq0\). If the convergence in Theorem E4.5 is locally uniform and every \(T_n\) is norm continuous, then \(U_m\) is a norm continuous semigroup.

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Proof of Theorem E4.8

At a finite level,

\[ \begin{aligned} U_{m,n}(s)U_{m,n}(t)-U_{m,n}(s+t) ={}& C_{m,n}T_n(s) (P_{m,n}-I) T_n(t)R_{n,m}. \end{aligned} \]

Its norm is at most \(\rho_{m,n}(T)\) for \(s,t\leq T\). Pass to the limit using uniform convergence. Norm continuity follows from a uniform limit of norm continuous functions on compact intervals.

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Remark E4.9

Summable generator defects control consistency between adjacent levels. The quantity \(\rho_{m,n}(T)\) controls closure of the reconstructed fine range under the fine evolution. These are different requirements.

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Example E4.10: Summable defects do not force a semigroup limit

Let \(X_0={\mathbb R}\) and let \(X_n={\mathbb R}^2\) for every \(n\geq1\). Set

\[ C_0(x,y)=x, \qquad R_0x=(x,0), \]

and use identity connecting maps and reconstructions above level one. Let \(T_0(t)=I\) and let every higher level carry the rotation group

\[ T_n(t) = \begin{pmatrix} \cos t&-\sin t\\ \sin t&\cos t \end{pmatrix}. \]

Only the first adjacent defect is nonzero, so every defect sum is finite. Yet for \(n\geq1\),

\[ U_{0,n}(t)=C_0T_n(t)R_0=\cos t. \]

The limiting family is not a semigroup because in general

\[ \cos(s+t)\neq\cos s\cos t. \]

The failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.

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Proposition E4.11: Range leakage criterion

Under the contractive assumptions of Corollary E4.6, define

\[ \eta_{m,n}(T) = \sup_{0\leq t\leq T} {\left\lVert(I-P_{m,n})T_n(t)R_{n,m}\right\rVert}. \]

Then

\[ \rho_{m,n}(T)\leq\eta_{m,n}(T). \]

Consequently, \(\eta_{m,n}(T)\to0\) for every \(T>0\) is sufficient for the limiting family \(U_m\) to satisfy the semigroup law.

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Proof of Proposition E4.11

For \(s,t\leq T\), the contraction assumptions give

\[ \begin{aligned} &{\left\lVert C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m} \right\rVert}\\ &\qquad\leq {\left\lVert(I-P_{m,n})T_n(t)R_{n,m}\right\rVert}. \end{aligned} \]

Take the supremum and apply Theorem E4.8.

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Theorem E4.12: Generator control of range leakage

Let \(T(t)={\mathrm e}^{tA}\) on a Banach space \(X\), let \(P\) be a bounded projection, and set \(Q=I-P\). Assume

\[ {\left\lVert T(t)\right\rVert}\leq M{\mathrm e}^{\omega t} \]

and

\[ {\left\lVert{\mathrm e}^{t(QAQ)|_{QX}}\right\rVert} \leq M_Q{\mathrm e}^{\omega_Qt}. \]

Then

\[ {\left\lVert QT(t)P\right\rVert} \leq M M_Q{\left\lVert P\right\rVert}^2{\left\lVert QAP\right\rVert} \int_0^t {\mathrm e}^{\omega_Q(t-s)+\omega s} {\,\mathrm d} s. \]

In particular, for orthogonal projections and contraction bounds,

\[ \sup_{0\leq t\leq T} {\left\lVert QT(t)P\right\rVert} \leq T{\left\lVert QAP\right\rVert}. \]

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Proof of Theorem E4.12

For \(x\in PX\), write

\[ p(t)=PT(t)x, \qquad q(t)=QT(t)x. \]

The hidden component satisfies

\[ q'(t)=QAQq(t)+QAPp(t), \qquad q(0)=0. \]

Variation of constants gives

\[ q(t)= \int_0^t {\mathrm e}^{(t-s)(QAQ)|_{QX}} QAPp(s) {\,\mathrm d} s. \]

Since \( {\left\lVert p(s)\right\rVert}\leq{\left\lVert P\right\rVert}{\left\lVert T(s)\right\rVert}{\left\lVert x\right\rVert} \) and \(x=Px\), the first estimate follows. The contraction case is immediate.

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Corollary E4.13: A usable closure condition for refinement towers

Assume the contractive summability hypothesis of Corollary E4.6 holds on every bounded time interval. Suppose every \(P_{m,n}\) is an orthogonal projection on a Hilbert space and \(T_n(t)={\mathrm e}^{tA_n}\) is contractive. If, for every fixed \(m\),

\[ {\left\lVert(I-P_{m,n})A_nP_{m,n}\right\rVert} \longrightarrow0, \]

then the limiting family \(U_m\) is a semigroup.

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Proof of Corollary E4.13

A bounded generator of a contraction semigroup on a Hilbert space is dissipative. If \(Q_{m,n}=I-P_{m,n}\) and \(y\in Q_{m,n}X_n\), then

\[ \operatorname{Re}{\left\langle Q_{m,n}A_nQ_{m,n}y,y\right\rangle} = \operatorname{Re}{\left\langle A_ny,y\right\rangle} \leq0. \]

Hence the compressed hidden semigroup is contractive. Apply Theorem E4.12 to \(P_{m,n}\). Since \(P_{m,n}R_{n,m}=R_{n,m}\),

\[ \eta_{m,n}(T) \leq T{\left\lVert(I-P_{m,n})A_nP_{m,n}\right\rVert} {\left\lVert R_{n,m}\right\rVert}. \]

The right side tends to zero. Use Proposition E4.11.

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E4 Refinement towers and limiting evolution

How to read the result

E4.1 Exact inverse limits

Lemma E4.1: Completeness of the bounded inverse limit

Proof of Lemma E4.1

Theorem E4.2: Exact projective limit semigroup

Proof of Theorem E4.2

Remark E4.3

E4.2 Summable adjacent defects

Definition E4.4: Reconstructed fine evolution

Theorem E4.5: Weighted summable defect completion

Proof of Theorem E4.5

Corollary E4.6: Contractive summable defect completion

Proof of Corollary E4.6

Corollary E4.7: Generator defect criterion

Proof of Corollary E4.7

Theorem E4.8: Asymptotic closure criterion

Proof of Theorem E4.8

Remark E4.9

Example E4.10: Summable defects do not force a semigroup limit

Proposition E4.11: Range leakage criterion

Proof of Proposition E4.11

Theorem E4.12: Generator control of range leakage

Proof of Theorem E4.12

Corollary E4.13: A usable closure condition for refinement towers

Proof of Corollary E4.13