Paper VI: Transfer paths and product-topology limits

R1 Transfer paths and product-topology limits

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Let \(T(t)\) be an arbitrary strongly continuous semigroup on a complex Banach space \(X\), with generator \(A\), and suppose \({\left\lVert T(t)\right\rVert}\leq M e^{\omega t}\) for \(M\geq1\) and \(\omega\in\mathbb R\). Let \((P_\alpha)_{\alpha\in I_0}\) be a finite family of bounded pairwise orthogonal idempotents summing to \(I\), and put \(T_{\beta\alpha}(t)=P_\beta T(t)P_\alpha\). The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each \(x\in X\) and take its Bochner integral in \(X\). It is not asserted to be a Bochner integral in the operator-norm space \(\mathcal B(X)\). The general Laplace formula is [EngelNagel2000], Chapter II, Theorem 1.10.

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Theorem R1.1: Packet convolution

For all \(s,t\geq0\) and all packets \(\alpha,\beta\),

\[ \boxed{ T_{\beta\alpha}(s+t) = \sum_{\gamma\in I_0} T_{\beta\gamma}(s)T_{\gamma\alpha}(t). } \]

The original semigroup is reconstructed from its blocks by

\[ T(t)=\sum_{\alpha,\beta\in I_0}T_{\beta\alpha}(t). \]

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Proof of Theorem R1.1

Insert the identity between the two semigroup factors:

\[ \begin{aligned} P_\beta T(s+t)P_\alpha &=P_\beta T(s)\left(\sum_\gamma P_\gamma\right)T(t)P_\alpha\\ &=\sum_\gamma P_\beta T(s)P_\gamma T(t)P_\alpha. \end{aligned} \]

The reconstruction formula follows by inserting the packet identity on both sides of \(T(t)\).

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Proposition R1.2: Packet resolvent convolution

Let \(z\) belong to the resolvent set of \(A\). If \(\operatorname{Re}z\) is larger than a growth bound for the semigroup, then

\[ P_\beta(zI-A)^{-1}P_\alpha = \int_0^{\infty}{\mathrm e}^{-zt}T_{\beta\alpha}(t){\,\mathrm d} t. \]

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Proof of Proposition R1.2

The standard Laplace representation of the resolvent is

\[ (zI-A)^{-1} = \int_0^{\infty}{\mathrm e}^{-zt}T(t){\,\mathrm d} t. \]

Multiplication by the bounded packet projections may be passed through the Bochner integral.

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For the path-order statement, now assume \(A\) is bounded as in Part II. A directed edge from \(\alpha\) to \(\beta\) means that \(P_\beta A P_\alpha\) is nonzero, with \(\alpha\ne\beta\); a packet always has the length-zero path to itself.

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Corollary R1.3: Minimum transfer order

Suppose every directed packet path from \(\alpha\) to \(\beta\) uses at least \(m_0\) off diagonal transfers. Then

\[ P_\beta{\mathrm e}^{tA}P_\alpha =O(t^{m_0}) \quad\text{as }t\downarrow0. \]

If there is no directed path from \(\alpha\) to \(\beta\), the block vanishes for all \(t\).

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Proof of Corollary R1.3

Every nonzero term in Theorem E2.8 contains at least \(m_0\) time integrations. The simplex \(0<r_1<\cdots<r_m<t\) has volume \(t^m/m!\), so the first possible order is \(t^{m_0}\). If there is no path, every packet term vanishes.

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Let \(C_j:X_{j+1}\to X_j\) be bounded linear maps, and put \(C_{\ell,m}=C_\ell\cdots C_{m-1}\) for \(\ell<m\), with \(C_{j,j}=I\). No common ambient space is assumed.

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Theorem R1.4: Several level telescoping identity

Let \(T_j\) be bounded operators on \(X_j\), and set

\[ E_j=C_jT_{j+1}-T_jC_j. \]

Then

\[ \boxed{ C_{\ell,m}T_m-T_\ell C_{\ell,m} = \sum_{j=\ell}^{m-1} C_{\ell,j}E_jC_{j+1,m}. } \]

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Proof of Theorem R1.4

Expand each summand as

\[ C_{\ell,j+1}T_{j+1}C_{j+1,m} - C_{\ell,j}T_jC_{j,m}. \]

The intermediate terms cancel. The first surviving term is \(C_{\ell,m}T_m\), and the last is \(-T_\ell C_{\ell,m}\).

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Corollary R1.5: Summable generator defects

Suppose \(C_j\) are contractions and \((T_j(t))\) are contraction semigroups with bounded generators \(A_j\). Set

\[ \delta_j={\left\lVert C_jA_{j+1}-A_jC_j\right\rVert}. \]

Then

\[ {\left\lVert C_{\ell,m}T_m(t)-T_\ell(t)C_{\ell,m}\right\rVert} \leq t\sum_{j=\ell}^{m-1}\delta_j. \]

If \(\sum_{j\geq0}\delta_j<\infty\), the right side tends to zero as \(\ell\to\infty\), uniformly in \(m>\ell\) and uniformly for \(t\) in a fixed bounded interval.

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Proof of Corollary R1.5

By Theorem E3.1, the interlevel semigroup defect has norm at most \(t\delta_j\). Apply Theorem R1.4 and use the contraction bounds on all coarsening maps.

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Definition R1.6: Projective limit space

Let \((X_j,C_j)\) be a sequence of Banach spaces and bounded linear maps \(C_j:X_{j+1}\to X_j\). Its projective limit is

\[ X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}. \]

We equip \(X_\infty\) with the subspace topology inherited from the product topology.

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Theorem R1.7: Exact projective limit semigroup

Suppose each \(X_j\) carries a strongly continuous semigroup \((T_j(t))_{t\geq0}\), and

\[ C_jT_{j+1}(t)=T_j(t)C_j \]

for every \(j\) and every \(t\geq0\). Then

\[ T_\infty(t)(x_j)_{j\geq0} = (T_j(t)x_j)_{j\geq0} \]

defines a strongly continuous semigroup on \(X_\infty\).

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Proof of Theorem R1.7

Exact intertwining gives

\[ C_jT_{j+1}(t)x_{j+1} = T_j(t)C_jx_{j+1} = T_j(t)x_j. \]

Thus the coordinatewise evolution remains in the projective limit. The semigroup law holds in every coordinate. Strong continuity in the product topology follows because every coordinate map

\[ t\longmapsto T_j(t)x_j \]

is continuous.

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Corollary R1.8: Coordinate generators

Assume the setting of Theorem R1.7. If \(x=(x_j)\) satisfies \(x_j\in\operatorname{Dom}(A_j)\) and the sequence \((A_jx_j)\) belongs to \(X_\infty\), then \(x\) lies in the generator domain of \(T_\infty\) and

\[ A_\infty x=(A_jx_j)_{j\geq0}. \]

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Proof of Corollary R1.8

The difference quotients converge in every coordinate to \(A_jx_j\). The assumed compatibility places the coordinatewise limit in \(X_\infty\), which is exactly convergence in the product topology.

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The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement.

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