Evolution operators in empathy with a semigroup (Q1824843)

The paper considers two Banach spaces X, Y, E(t) a semigroup of bounded linear operators in Y and S(t): \(Y\to X\) a family of bounded linear operators such that \[ S(t+s)=S(t)E(s)=S(s)E(t),\quad (empathy). \] The authors derive conditions to construct a linear operator B: \(X\to Y\) such that \(E(t)=BS(t)\). Essentially these conditions ask E(t) to be of class (A), involve the invertibility of the Laplace transform of S(t) and the extent of the non-singularity of S(t) at \(t=0\). The paper is strongly related with previous results of the authors.