Strong and uniform mean stability of cosine and sine operator functions (Q879064)

It is first observed that a uniformly bounded cosine operator function \(C(\cdot)\) and the associated sine function \(S(\cdot)\) are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, the authors prove some results concerning the strong mean stability and uniform mean stability of \(C(\cdot)\). Among them are: (1) \(C(\cdot)\) is strongly \((C, 1)\)-mean stable (or \((C, 2)\)-mean stable, or Abel-mean stable) if and only if \(0\in\rho(A)\cup \sigma_C(A)\); (2) \(C(\cdot)\) is uniformly \((C, 2)\)-mean stable if and only if \(S(\cdot)\) is uniformly \((C, 1)\)-mean stable, if and only if \(\|\int^t_0 S(s)\,ds\|= O(t)\) \((t\to\infty)\), if and only if \(\sup_{t> 0}\|\int^t_0 S(s)\,ds\|< \infty\), if and only if \(C(\cdot)\) is uniformly Abel-mean stable, if and only if \(S(\cdot)\) is uniformly Abel-mean stable, if and only if \(0\in\rho(A)\).