Paper VI: Generator defects and functional calculus

E3 Generator defects and functional calculus

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Let \(X_f\) and \(X_c\) be complex Banach spaces. Let \(A_f\in\mathcal B(X_f)\), \(A_c\in\mathcal B(X_c)\), and \(C\in\mathcal B(X_f,X_c)\). Define

\[ \Delta=CA_f-A_cC \]

and

\[ \mathfrak D(t)=C{\mathrm e}^{tA_f}-{\mathrm e}^{tA_c}C. \]

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Theorem E3.1: Duhamel defect identity

For every \(t\geq0\),

\[ \mathfrak D(t) = \int_0^t {\mathrm e}^{(t-s)A_c}\Delta{\mathrm e}^{sA_f} {\,\mathrm d} s. \]

\[ {\left\lVert{\mathrm e}^{tA_f}\right\rVert}\leq M_f{\mathrm e}^{\omega_f t}, \qquad {\left\lVert{\mathrm e}^{tA_c}\right\rVert}\leq M_c{\mathrm e}^{\omega_c t}, \]

then

\[ {\left\lVert\mathfrak D(t)\right\rVert} \leq M_fM_c{\left\lVert\Delta\right\rVert} \int_0^t {\mathrm e}^{\omega_c(t-s)+\omega_fs} {\,\mathrm d} s. \]

For contraction semigroups,

\[ {\left\lVert\mathfrak D(t)\right\rVert}\leq t{\left\lVert\Delta\right\rVert}. \]

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Proof of Theorem E3.1

For fixed \(t\), set

\[ F(s)={\mathrm e}^{(t-s)A_c}C{\mathrm e}^{sA_f}. \]

Then

\[ F'(s)={\mathrm e}^{(t-s)A_c}\Delta{\mathrm e}^{sA_f}. \]

Integrating from zero to \(t\) gives the identity. The estimates follow by taking norms under the integral.

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Proposition E3.2: Infinitesimal recovery of the defect

The generator defect is the first order semigroup defect:

\[ \lim_{t\to0} \frac{\mathfrak D(t)}{t} = \Delta \]

in operator norm. Consequently,

\[ C{\mathrm e}^{tA_f}={\mathrm e}^{tA_c}C \quad\hbox{for every }t \]

if and only if \(CA_f=A_cC\).

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Proof of Proposition E3.2

The operator norm expansions

\[ {\mathrm e}^{tA_f}=I+tA_f+o(t), \qquad {\mathrm e}^{tA_c}=I+tA_c+o(t) \]

give the limit. The equivalence follows from this limit and Theorem E3.1.

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Theorem E3.3: Resolvent defect identity

Let \(z\) belong to the resolvent sets of \(A_f\) and \(A_c\). With

\[ R_f(z)=(zI-A_f)^{-1}, \qquad R_c(z)=(zI-A_c)^{-1}, \]

one has

\[ CR_f(z)-R_c(z)C = R_c(z)\Delta R_f(z). \]

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Proof of Theorem E3.3

Multiply the proposed identity on the left by \(zI-A_c\) and on the right by \(zI-A_f\). Both sides reduce to

\[ (zI-A_c)C-C(zI-A_f)=\Delta. \]

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Theorem E3.4: Holomorphic defect formula

Let \(\Gamma\) be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that \(\Gamma\) lies in the common resolvent set. Thus its winding number is one about every spectral point. If \(g\) is holomorphic on a neighborhood of the closure of the union, then

\[ Cg(A_f)-g(A_c)C = \frac{1}{2\pi{\mathrm i}} \int_\Gamma g(z)R_c(z)\Delta R_f(z) {\,\mathrm d} z. \]

In particular,

\[ \begin{aligned} {\left\lVert Cg(A_f)-g(A_c)C\right\rVert} \leq{}& \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}{\left\lvert g(z)\right\rvert} \sup_{z\in\Gamma}{\left\lVert R_c(z)\right\rVert} \sup_{z\in\Gamma}{\left\lVert R_f(z)\right\rVert} {\left\lVert\Delta\right\rVert}. \end{aligned} \]

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Proof of Theorem E3.4

Apply the Dunford integral formula to both operators and use Theorem E3.3 under the contour integral. The norm estimate is immediate.

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Corollary E3.5: Exact naturality

If \(CA_f=A_cC\), then

\[ Cg(A_f)=g(A_c)C \]

for every holomorphic function covered by Theorem E3.4. The conclusion includes exponentials, resolvents, and polynomials.

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Proof of Corollary E3.5

Set \(\Delta=0\) in Theorem E3.4.

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A finite semigroup defect and a direct Duhamel bound. The bound is designed to be uniform rather than sharp in this example.

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