Perturbations of local C-cosine functions

Abstract

We show that $\tA+\tB$ is a closed subgenerator of a local $\tC$-cosine function $\tT(\cdot)$ on a complex Banach space $\tX$ defined by
$$\tT(t)x=\sum\limits_{n=0}^\infty \tB^n\int_0^tj_{n-1}(s)j_n(t-s)\tC(|t-2s|)xds$$
for all $x\in\tX$ and
$0\leq t<T_0$, if $\tA$ is a closed subgenerator of
a local $\tC$-cosine function $\tC(\cdot)$ on $\tX$ and one of the following cases holds:
$(i)$ $\tC(\cdot)$ is exponentially bounded, and $\tB$ is a bounded
linear operator on $\overline{\tD(\tA)}$ so that $\tB\tC=\tC\tB$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$;
$(ii)$ $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ which commutes with
$\tC(\cdot)$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$;
$(iii)$ $\tB$ is a bounded linear operator on $\tX$ which commutes with
$\tC(\cdot)$ on $\tX$.
Here $j_n(t)=\frac{t^n}{n!}$ for all $t\in\Bbb R$, and
$$\int_0^tj_{-1}(s)j_0(t-s)\tC(|t-2s|)xds=\tC(t)x$$
for all $x\in\tX$ and $0\leq t<T_0$.