TOWARDS A CHARACTERIZATION OF EVENTUALLY NORM CONTINUOUS SEMIGROUPS ON BANACH SPACES

摘要

As shown earlier on Hilbert spaces this condition is equivalent to the norm continuity of the semigroup for t > 0. Starting with the Hille-Yosida theorem it is a fundamental problem in the theory of one-parameter semigroups to characterize properties of the semigroup in terms of its generator and its resolvent, see [2], [3]. In this short note we make a contribution towards the still unsolved characterization of norm continuous semigroups on Banach spaces. In fact, P. You showed recently that for an exponentially stable semigroup (T(t))_(t ≥0) with generator A on a Hilbert space the norm continuity for t > 0 is equivalent to condition (E). In [1] we gave a much simpler and more straightforward proof of this result. Extending the methods used there we characterize in this note condition (E) for a semigroup generator A on an arbitrary Banach space in terms of the associated semigroup.