Paper VI: Packet resolved semigroups

E2 Packet resolved semigroups

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Let \(X\) be a complex Banach space. A finite packet system is a family of bounded idempotents

\[ {\mathcal P}=(P_\alpha)_{\alpha\in I} \]

such that

\[ P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I_X. \]

No orthogonality is needed in this section.

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Definition E2.1: Packet resolved transfer system

A packet resolved transfer system is a triple

\[ (X,A,{\mathcal P}) \]

consisting of a Banach state space, a bounded generator, and a finite packet system. A marked transfer system also includes an initial state or source \(s\in X\). The topology and all limiting operations are supplied by the ambient Banach space structure.

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Definition E2.2: Packet blocks

For \(B\in\mathcal B(X)\), define

\[ B_{\beta\alpha}=P_\beta BP_\alpha. \]

If \(A\in\mathcal B(X)\) and \(T(t)={\mathrm e}^{tA}\), write

\[ T_{\beta\alpha}(t)=P_\beta T(t)P_\alpha. \]

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Theorem E2.3: Semigroup and packet convolution

For every bounded operator \(A\), the family \(T(t)={\mathrm e}^{tA}\) is a uniformly continuous group and

\[ T(s+t)=T(s)T(t). \]

For every \(\alpha,\beta\in I\),

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

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Proof of Theorem E2.3

The exponential series converges in operator norm. Since every power of \(A\) commutes with every other power, the Cauchy product gives the group law. Inserting \(I_X=\sum_\gamma P_\gamma\) between the two factors gives the block formula.

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Proposition E2.4: Block differential system

The packet blocks are differentiable in operator norm and satisfy

\[ \frac{{\,\mathrm d}}{{\,\mathrm d} t}T_{\beta\alpha}(t) = \sum_{\gamma\in I} A_{\beta\gamma}T_{\gamma\alpha}(t) = \sum_{\gamma\in I} T_{\beta\gamma}(t)A_{\gamma\alpha}. \]

Their initial values are

\[ T_{\beta\alpha}(0)=\delta_{\beta\alpha}P_\alpha. \]

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Proof of Proposition E2.4

Differentiate \(T(t)\) in operator norm and insert the packet resolution between \(A\) and \(T(t)\), then between \(T(t)\) and \(A\).

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Theorem E2.5: Laplace resolvent and packet transform

Let \(A\in\mathcal B(X)\), let \(T(t)={\mathrm e}^{tA}\), and let

\[ \operatorname{Re}z>{\left\lVert A\right\rVert}. \]

Then the Bochner integral converges in operator norm and

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

For every pair of packets,

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

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Proof of Theorem E2.5

The estimate \({\left\lVert T(t)\right\rVert}\leq{\mathrm e}^{t{\left\lVert A\right\rVert}}\) gives absolute convergence of the operator valued integral. Let

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

Since \(T'(t)=AT(t)=T(t)A\), integration by parts gives

\[ (zI-A)R(z) = -\int_0^\infty \frac{{\,\mathrm d}}{{\,\mathrm d} t} \bigl({\mathrm e}^{-zt}T(t)\bigr) {\,\mathrm d} t =I. \]

The same calculation on the other side gives \( R(z)(zI-A)=I \). Multiplication by \(P_\beta\) and \(P_\alpha\) may be passed under the norm convergent integral, which gives the packet formula.

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Corollary E2.6: Resolvent expansion over packet paths

With \(A=D+N\) as in the directed path decomposition, let

\[ R_D(z)=(zI-D)^{-1}. \]

\[ {\left\lVert R_D(z)N\right\rVert}<1, \]

then

\[ (zI-A)^{-1} = \sum_{m=0}^\infty \bigl(R_D(z)N\bigr)^mR_D(z). \]

After multiplication by \(P_\beta\) and \(P_\alpha\), the term of order \(m\) is a sum over directed packet paths of length \(m\) from \(\alpha\) to \(\beta\).

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Proof of Corollary E2.6

Factor

\[ zI-A=(zI-D)(I-R_D(z)N) \]

and expand the second inverse by its norm convergent Neumann series. The diagonal resolvent preserves each packet and every factor \(N\) changes the packet along an active transfer edge.

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Theorem E2.7: Reducing packet criterion

The following conditions are equivalent.

\(AP_\alpha=P_\alpha A\) for every \(\alpha\).
\(A_{\beta\alpha}=0\) whenever \(\beta\neq\alpha\).
\(T_{\beta\alpha}(t)=0\) whenever \(\beta\neq\alpha\) and \(t\in{\mathbb R}\).
Every packet space \(P_\alpha X\) is invariant under \(T(t)\) for every \(t\).

When these conditions hold,

\[ T(t)|_{P_\alpha X} = {\mathrm e}^{tA_{\alpha\alpha}}. \]

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Proof of Theorem E2.7

The first two conditions are equivalent because the packet resolution gives

\[ AP_\alpha=\sum_\beta A_{\beta\alpha}, \qquad P_\alpha A=\sum_\gamma A_{\alpha\gamma}. \]

A block diagonal generator has block diagonal powers, so the second condition implies the third. Differentiating the off diagonal blocks at zero proves the converse. The equivalence with invariance follows from the definition of the blocks. The final identity follows by restricting the exponential series.

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Packet block norms at three times. Transfer appears in the off diagonal blocks as soon as the generator couples the two packet spaces. The three panels are shown in consecutive figures.

Packet block norms at time 0.8. The same fixed packet spaces and norm scale are used in the preceding panels.

Packet block norms at time 2.0. The same fixed packet spaces and norm scale are used in the preceding panels.

E2.1 Directed path expansion

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Set

\[ D=\sum_{\alpha\in I}P_\alpha AP_\alpha, \qquad N=A-D. \]

The operator \(D\) preserves every packet and \(N\) contains only transfers between distinct packets.

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Theorem E2.8: Dyson expansion over packet paths

For \(t\geq0\),

\[ \begin{aligned} {\mathrm e}^{tA} ={}&{\mathrm e}^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} {\mathrm e}^{(t-s_m)D}N{\mathrm e}^{(s_m-s_{m-1})D} \cdots N{\mathrm e}^{s_1D} {\,\mathrm d} s_1\cdots{\,\mathrm d} s_m. \end{aligned} \]

The series converges absolutely in operator norm. After multiplication by \(P_\beta\) and \(P_\alpha\), the term of order \(m\) is a sum over directed packet paths of length \(m\) from \(\alpha\) to \(\beta\).

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Proof of Theorem E2.8

Write the variation of constants identity

\[ {\mathrm e}^{tA} = {\mathrm e}^{tD} + \int_0^t{\mathrm e}^{(t-s)D}N{\mathrm e}^{sA}{\,\mathrm d} s. \]

Iterating the identity gives the displayed series. The norm of the term of order \(m\) is at most

\[ {\mathrm e}^{t{\left\lVert D\right\rVert}}\frac{(t{\left\lVert N\right\rVert})^m}{m!}. \]

The majorant is summable. Inserting the packet resolution around every factor \(N\) leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.

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Corollary E2.9: Termination on an acyclic transfer graph

Suppose the directed graph with an edge \(\alpha\to\beta\) whenever \(P_\beta NP_\alpha\neq0\) is acyclic. If its longest directed path has length \(q\), then every Dyson term of order greater than \(q\) vanishes.

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Proof of Corollary E2.9

Every nonzero term of order \(m\) contains a directed transfer path of length \(m\). An acyclic graph has no path longer than \(q\).

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