In Theorem 2.1.10 we saw that if A is the infinitesimal generator of a C0-semigroup T(t), the solution of the abstract homogeneous Cauchy initial value problem $$\begin{array}{*{20}{c}}{\dot z\left( t \right) = Az\left( t \right),}&{t \geqslant 0,}&{z\left( 0 \right) = {z_0} \in D\left( A \right)}\end{array}$$ is given by $$z\left( t \right) = T\left( t \right){z_0}.$$