Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C(⋅) and the associated sine function S(⋅) are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(⋅). Among them are: (1) C(⋅) is strongly (C,1)-mean stable (or (C,2)-mean stable, or Abel-mean stable) if and only if 0∈ρ(A)∪σc(A); (2) C(⋅) is uniformly (C,2)-mean stable if and only if S(⋅) is uniformly (C,1)-mean stable, if and only if , if and only if , if and only if C(⋅) is uniformly Abel-mean stable, if and only if S(⋅) is uniformly Abel-mean stable, if and only if 0∈ρ(A).