Let \(X\) be a complex Banach space and \(L_p= L_p(R_+,X)\). The author considers the linear differential operator \[ {\mathcal L}= -{d\over dt}+ A(t): D({\mathcal L})\subset L_p\to L_p,\quad p\in[1,\infty] \] and studies its spectral properties under the assumption that the family of closed operators \(A(t): D(A(t))\subset X\to X\), \(t\geq 0\), generates a well-posed Cauchy problem. The results are closely related to those presented in [\textit{Ju. L. Daleckii} and \textit{M. G. Krein}, ``Stability of solutions of differential equations in Banach space'', AMS Translations of Math. Monographs (1974; Zbl 0286.34094)] and in [\textit{D. Henry}, ``Geometric theory of semilinear parabolic equations'' (1981; Zbl 0456.35001)].