Almost periodic regularized groups, semigroups, and cosine functions (Q1916702)

Let \(T(t)\), \(t\in {\mathbf J}\), be a strongly continuous family of linear bounded operators over a Banach space \(X\) such that \[ T(0) = C, \] and \[ T(t + s)C = T(t)T(s), \quad\forall t,s\in {\mathbf J}, \] where \(C\) -- called regulator operator -- is a linear bounded operator in \(X\). Then \(T(t)\) is called a C-regularized semigroup in case \({\mathbf{J}} = {\mathbf{R}}_{+}\), and C-regularized group in the case \({\mathbf{J}} = {\mathbf{R}}\). This paper studies the almost periodicty of regularized groups, semigroups, cosine functions, and the corresponding sine functions. The authors obtain a characterization of the generator of an almost periodic regularized group and give conditions under which weak almost periodicity (resp. almost periodicity) of a regularized group implies its almost periodicty (resp. uniform almost periodicity). Almost periodic regularized semigroups as well as regularized cosine functions and sine functions are then characterized. Finally, periodicity of regularized groups, cosine functions and sine functions are characterized. The case in which the regulator operator \(C\) is some power of the generator is also studied as well the applications to distribution semigroup.