Exponential Decay of Semigroups in Hilbert Space

We give simple bounds for the exponential decay of a strongly continuous semigroup V (t) = exp(At) in a Hilbert space. The bounds are expressed by means of the solution X of the Ljapunov equation A X + XA = ?I and the quantity sup 0th kV (t)k for small h. Our main bound is attained on any normal semigroup and can therefore not be improved in general. Let V (t) be a strongly continuous semigroup in a complex Hilbert space H. Then it is exponentially bounded i.e. kV (t)k Me t (1) for some M 1 and some real. The set of all such semigroups is denoted by G(M;). 1 We say that V (t) decays exponentially, if (1) holds with a negative. The aim of this paper is to give sharp bounds for the exponential decay and in particular for the number (type of the semigroup) (2) A rst result connecting exponential decay with the Ljapunov equation in the Hilbert space was provided by Datko 1] which we partly resume for convenience: 2 Theorem 1 (Datko) The following are equivalent: (i) V () decays exponentially, (ii) The strong integral X =