A stability theorem for a class of linear evolution systems

Dynamical systems of the form,u(0)=u₀, where,B is a densely defined linear operator mapping its domainD (B ) into — the infinitesimal generator of a semigroup of operatorsT (t, B) of classC₀ — are investigated, such that for each solutionu to, whereP is the spectral eigenprojection onto the null space ofB.It is shown that under some general hypotheses concerning spectral properties ofB the above stability condition is equivalent with the following situation: There exist (i) a normal generating coneK such thatT(t;B)KK fort0 and (ii) a strictly positive element in the dual coneK such that, whereB denotes the dual ofB. Condition (ii) implies the so called total concentration time preservation, i. e..