Compactness results for \({\mathcal{L}}^{\infty}\)-solution operator of a linear evolution equation involving measures (Q2474182)

The author gives several sufficient conditions for the compactness in \(C([a, b]; X)\) of \({\mathcal L}^\infty\)-solutions operators \((\xi, g)\mapsto u\) associated to the nonhomogeneous Cauchy problem of the type \[ du = (Au)dt +dg, \qquad u(a) = \xi, \tag{1} \] where \(A : D(A) \subseteq X \to X\) generates a \(C_0\)-semigroup of contractions in a real Banach space \(X\), \(\xi \in X\) and \(g \in BV([a, b]; X)\). The proofs are accurately presented and contain sufficient details. The author provides also an illustrative example.