Weak Lp-stability of a linear semigroup on a Hilbert space implies exponential stability

AbstractWe prove that a strongly continuous semigroup of linear operators on a Hilbert space is weakly Lp-stable for some p ϵ [1, ∞) if and only if the semigroup is exponentially stable. As an application, we prove that the Cauchy problems associated with the semigroup are well posed on the infinite time interval (− ∞, 0] if and only if the semigroup is exponentially stable