The generator of a uniformly continuous semigroup is a bounded operator. Analytic semigroups Contraction semigroups Differentiable semigroups A strongly continuous semigroup T is called eventually differentiable if there exists a such that (equivalently: for all and T is immediately differentiable if for all . Every analytic semigroup is immediately differentiable. An equivalent characterization in terms of Cauchy problems is the following: the strongly continuous semigroup generated by A is eventually differentiable if and only if there exists a such that for all the solution u of the abstract Cauchy problem is differentiable on . The semigroup is immediately differentiable if t1 can be chosen to be zero. Compact semigroups A strongly continuous semigroup T is called eventually compact if there exists a t0 > 0 such that T(t0) is a compact operator (equivalently if T(t&hairsp;) is a compact operator for all t ≥ t0) . The semigroup is called immediately compact if T(t&hairsp;) is a compact operator for all t > 0. Norm continuous semigroups A strongly continuous semigroup is called eventually norm continuous if there exists a t0 ≥ 0 such that the map t → T(t&hairsp;) is continuous from (t0, ∞) to L(X). The semigroup is called immediately norm continuous if t0 can be chosen to be zero. Note that for an immediately norm continuous semigroup the map t → T(t&hairsp;) may not be continuous in t = 0 (that would make the semigroup uniformly continuous). Analytic semigroups, (eventually) differentiable semigroups and (eventually) compact semigroups are all eventually norm continuous. Stability Exponential stability The growth bound of a semigroup T is the constant It is so called as this number is also the infimum of all real numbers ω such that there exists a constant M (≥ 1) with for all t ≥ 0. The following are equivalent: There exist M,ω>0 such that for all t ≥ 0: The growth bound is negative: ω0 < 0, The semigroup converges to zero in the uniform operator topology: , There exists a t0 > 0 such that , There exists a t1 > 0 such that the spectral radius of T(t1) is strictly smaller than 1, There exists a p ∈ [1, ∞) such that for all x ∈ X: , For all p ∈ [1, ∞) and all x ∈ X: A semigroup that satisfies these equivalent conditions is called exponentially stable or uniformly stable (either of the first