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 groupsAuthors: Senzhong Huang
Author institution: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Summary: Consider a weight ω on R with the following property.
(ne) For any C_0-group T\coloneqq(T(t))t∈R on a Banach space E satisfying ‖ 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.
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