Generators with a closure relation

Assume that a block operator of the form $\left(\begin{array}{c}A_1\\ A_2\ 0\end{array}\right)$, acting on the Banach space $X_1 \times X_2$, generates a contraction $C_0$-semigroup. We show that the operator AS defined by $A_Sx = A_1 \left(\begin{array}{c}x\\ SA_2x \end{array}\right)$ with the natural domain generates a contraction semigroup on X1. Here, S is a boundedly invertible operator for which $\varepsilon I-S^{-1}$ is dissipative for some ε > 0. With this result the existence and uniqueness of solutions of the heat equation can be derived from the wave equation.