Functional calculus and dilation for $C_0$-groups of polynomial growth
Résumé
Let $U (t) = e^{itB}$ be a $C_0$-group on a Banach space $X$. Let further $\phi \in C^\infty_c(\mathbb R)$ satisfy $\sum_{n \in \mathbb Z} \phi(\cdot - n) \equiv 1$. For $\alpha \geq 0$, we put $E^\alpha_\infty = \{f \in C_b(\mathbb R) :\: \|f\|_{E^\alpha_\infty} = \sum_{n \in \mathbb Z} (1 + |n|)^\alpha \|f * [\phi(\cdot− n)]\check{\phantom{i}} \|_{L^\infty(\mathbb R)} < \infty \}$, which is a Banach algebra. It is shown that $\|U(t)\|\leq C(1 + |t|)^\alpha$ for all $t \in \mathbb R$ if and only if the generator $B$ has a bounded $E^\alpha_\infty$ functional calculus, under some minimal hypothesis, which exclude simple counterexamples. A third equivalent condition is that $U(t)$ admits a dilation to a shift group on some space of functions $\mathbb R \to X$. In the case $U (t) = A^it$ with some sectorial operator $A$, we use this calculus to show optimal bounds for fractions of the semigroup generated by $A$, resolvent functions and variants of it. Finally, the $E^\alpha_\infty$ calculus is compared with Besov functional calculi as considered in [4, 16].
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