Growth order and stability of semigroups and cosine operator functions

Equivalent conditions for a trajectory of a C 0 -semigroup T ( ⋅ ) (resp. cosine function C ( ⋅ ) ) of operators to have the growth order O ( t α ) or o ( t α ) are expressed in terms of Cesàro and Abel means of the norm of the trajectory. We then deduce characterizations of growth order and stability for T ( ⋅ ) and C ( ⋅ ) . It is also shown that under some Tauberian condition the uniform boundedness (resp. strong convergence) of T ( ⋅ ) is equivalent to the uniform boundedness (resp. strong convergence) of its Abel means.