Roughness of exponential dichotomy under unbounded perturbation in linear partial functional differential equations

This paper is concerned with the roughness of exponential dichotomies under unbounded perturbations of a class of linear partial functional differential equations \begin{equation}\label{pfde-000-1star} u’(t)=Au(t)+Bu_t, \end{equation} where $A$ is a linear operator on a Banach space $\mathbb{X}$ and $B$ is a linear operator from $C([-r,0],\mathbb{X})$ into $\mathbb{X}$, where $r>0$ is a given constant. To quantify the size of unbounded perturbations, we introduce the $\mathit{\text{Yosida}}\mathit{\text{distance}}$ between linear operators $U$ and $V$, defined by $d_Y(U,V):={\mathrm{lim\hspace{0.17em}sup}}_{\mu \to +\mathrm{\infty }}\mathrm{\parallel }{U}_{\mu }-V_{\mu }\mathrm{\parallel }$, where $U_{\mu }$ and $V_{\mu }$ are the Yosida approximations of $U$ and $V$, respectively. We show that if $d_Y(A,A_1)$ and $d_Y(B,B_1)$ are sufficiently small, then the perturbed equation \begin{equation}\label{pfde-000-2star} u’(t)=A_1u(t)+B_1u_t \end{equation} also admits an exponential dichotomy whenever \eqref{pfde-000-1star} admits one. The proofs are based on estimates of the Yosida distance between the generators of the solution semigroups associated with \eqref{pfde-000-1star} and \eqref{pfde-000-2star} in the phase space $C([-r,0],\mathbb{X})$, without assuming any relation between their domains.