Journal of Operator Theory

Volume 32, Issue 2, Fall 1994 pp. 299-309.

An equivalent description of non-quasianalyticity through spectral theory of C_0 groups

Authors: Senzhong Huang
Author institution:Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Summary: Consider a weight $\omega$ on $\mathbb R$ with the following property.
(ne) For any C_0-group $\mathcal T \coloneqq (T(t))_{t \in \mathbb R}$ on a Banach space E satisfying $\left\| {T(t)} \right\| \le \omega (t)$ for all $t \in \mathbb R$, there holds $\sigma (A) \ne \not 0$ for the generator A of $\mathcal T$.
It is well-known that a non-quasianalytic weight $\omega$ (i.e., $\int_{ - \infty }^{ + \infty } {\frac{{\log \omega (t)}}{{1 + t^2 }}} {\rm{ d}}t < + \infty $) shares (ne). Assuming that $\omega$ is not a non-quasianalytic weight, we construct a C_0-group $\mathcal T \coloneqq = (T(t))_{t \in \mathbb R}$ of translations on some weighted Hardy space such that $\left\| {T(t)} \right\| \le \omega (t)$ for all $t \in \mathbb R$, but $\sigma (A) \ne \not 0$ for the generator A of $\mathcal T$. This shown that (ne) is equivalent to the non-quasianalyticity of the weight.

Keywords: C_0-group, spectrum, non-quasianalitic weight, H^p spaces.

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