Infinite Differentiability of Hermitian and Positive <i>C</i>-Semigroups and <i>C</i>-Cosine Functions

Abstract

Let C be a bounded linear operator which is not necessarily injective. The following statements are proved: (1) hermitian C-semigroups are infinitely differentiable in operator norm on (0, ∞); (2) hermitian C-cosine functions are norm continuous at either non or all of points in [0, ∞); (3) positive C-semigroups which dominate C are infinitely differentiable in opetator norm on [0, ∞); (4) positive C-cosine functions are infinitely differentiable in operator norm on [0, ∞).