THE WEISS CONJECTURE FOR BOUNDED ANALYTIC SEMIGROUPS

Abstract

New results concerning the so-called Weiss conjecture on admissible operators for bounded analytic semigroups are given. Let \[ \left(T_t\right)_{t\geqslant 0} \] be a bounded analytic semigroup with generator $-A$ on some Banach space $X$. It is proved that if $A^{1/2}$ is admissible for $A$, that is, if there is an estimate \[ \int_{0}^{\infty^{\vphantom{-1}}}\|A^{1/2}e^{-tA}x\|^2\, dt\leqslant M^2\|x\|^2,\] then any continuous mapping $C : D\left(A\right)\longrightarrow Y$ valued in a Banach space $Y$ is admissible for $A$ provided that there is an estimate \[ \|\left(-{\rm Re}\left({\lambda}\right)\right)^{1/2}C\left(\lambda -A\right)^{-1}\|\leqslant K \] for $\lambda\in\mathbb{C}$, ${\rm Re}\left({\lambda}\right)<0$. This holds in particular if \[ \left(T_t\right)_{t\geqslant 0}\] is a contractive (analytic) semigroup on Hilbert space. In the converse direction, it is shown that this may happen for a bounded analytic semigroup on Hilbert space that is not similar to a contractive one. Applications in non-Hilbertian Banach spaces are also given.