In this paper we give a simplified proof of a recent result of P. You which states that an exponentially stable semigroup $(T(t))_{t\ge0}$ on a Hilbert space is norm continuous for $t>0$ if and only if the resolvent of its generator tends to zero along the imaginary axis. As an easy consequence of our proof we obtain a characterization of semigroups which are norm continuous for $t>t_0$ in terms of the growth of some power of the resolvent on the imaginary axis. The key tools in our approach are a complex representation formula for $T(t)$ and the Plancherel theorem for the Hilbert space valued Fourier Transform.
ON THE CHARACTERIZATION OF EVENTUALLY NORM CONTINUOUS SEMIGROUPS IN HILBERT-SPACES
ENGEL, KLAUS JOCHEN OTTO
1994-01-01
Abstract
In this paper we give a simplified proof of a recent result of P. You which states that an exponentially stable semigroup $(T(t))_{t\ge0}$ on a Hilbert space is norm continuous for $t>0$ if and only if the resolvent of its generator tends to zero along the imaginary axis. As an easy consequence of our proof we obtain a characterization of semigroups which are norm continuous for $t>t_0$ in terms of the growth of some power of the resolvent on the imaginary axis. The key tools in our approach are a complex representation formula for $T(t)$ and the Plancherel theorem for the Hilbert space valued Fourier Transform.File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
