Abstract

We will give a necessary and sufficient condition for the infinitesimal generator of a strongly continuous cosine operator function $C(t)$, such that $\|C(t)\|\le1$ for all $t\in R$ on a reflexive, strictly convex (complex) Banach space with a G\^ateaux differentiable norm to be a spectral scalar type operator with the spectral family of hermitian bounded linear projectors.

Keywords: Cosine operator function, hermitian operator, strictly convex Banach space, G\^ateaux differentiable norm, spectral operator.

MSC: 47D09, 46C50, 47B40

Pages:  193$-$206     

Volume  60 ,  Issue  3 ,  2008