A condition of uniform exponential stability for semigroups

The aim of this paper is to prove that the uniform exponential stability of a strongly continuous semigroup {T(t)}(t >= 0) (acting on a complex Hilbert space H) can be derived as a consequence of the well behavior of its numerical range in a suitable Orlicz space. More precisely, assuming that there exists an Orlicz space E = (L-Phi ,rho(Phi)), over R+ such that [GRAPHICS] and [GRAPHICS] then the uniform growth bound w(0) of the semigroup verifies an estimate of the form w(0) <= M-beta := beta - (2M vertical bar vertical bar exp(-beta) vertical bar vertical bar(E*))(-1) < 0 for some positive number beta. As an application, the well posedness of an abstract infinite time Cauchy problem is discussed.