three of the above statements is taken as the definition in certain parts of the literature). That the Lp conditions are equivalent to exponential stability is called the Datko-Pazy theorem. In case X is a Hilbert space there is another condition that is equivalent to exponential stability in terms of the resolvent operator of the generator: all λ with positive real part belong to the resolvent set of A and the resolvent operator is uniformly bounded on the right half plane, i.e. (λI − A)−1 belongs to the Hardy space . This is called the Gearhart-Pruss theorem. The spectral bound of an operator A is the constant , with the convention that s(A) = −∞ if the spectrum of A is empty. The growth bound of a semigroup and the spectral bound of its generator are related by s(A) ≤ ω0(T ). There are examples where s(A) < ω0(T ). If s(A) = ω0(T ), then T is said to satisfy the spectral determined growth condition. Eventually norm-continuous semigroups satisfy the spectral determined growth condition. This gives another equivalent characterization of exponential stability for these semigroups: An eventually norm-continuous semigroup is exponentially stable if and only if s(A) < 0. Note that eventually compact, eventually differentiable, analytic and uniformly continuous semigroups are eventually norm-continuous so that the spectral determined growth condition holds in particular for those semigroups. Strong stability A strongly continuous semigroup T is called strongly stable or asymptotically stable if for all x ∈ X: . Exponential stability implies strong stability, but the converse is not generally true if X is infinite-dimensional (it is true for X finite-dimensional). The following sufficient condition for strong stability is called the Arendt–Batty–Lyubich–Phong theorem: Assume that T is bounded: there exists a M ≥ 1 such that , A has not residual spectrum on the imaginary axis, and The spectrum of A located on the imaginary axis is countable. Then T is strongly stable. If X is reflexive then the conditions simplify: if T is bounded, A has no eigenvalues on the imaginary axis and the spectrum of A located on the imaginary axis is countable, then T is strongly stable. See also Hille–Yosida theorem Lumer–Phillips theorem Trotter–Kato theorem Analytic semigroup Contraction semigroup Matrix exponential Strongly continuous family of operators Abstract differential equation Notes References E Hille, R S Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1975. R F Curtain, H