Infinite Differentiability of Hermitian and Positive $C$-Semigroups and $C$-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 $(0,\infty )$; (2) hermitian $C$-cosine functions are norm continuous at either non or all of points in $[0,\infty )$; (3) positive $C$-semigroups which dominate $C$ are infinitely differentiable in opetator norm on $[0,\infty )$; (4) positive $C$-cosine functions are infinitely differentiable in operator norm on $[0,\infty )$.