  Assume that a block operator of the form
$\left(\begin{smallmatrix}A_{1}\\A_{2}\quad 0\end{smallmatrix}\right)$, acting
on the Banach space $X_{1}\times X_{2}$, generates a contraction
$C_{0}$-semigroup. We show that the operator $A_{S}$ defined by
$A_{S}x=A_{1}\left(\begin{smallmatrix}x\\SA_{2}x\end{smallmatrix}\right)$ with
the natural domain generates a contraction semigroup on $X_{1}$. Here, $S$ is a
boundedly invertible operator for which $\epsilon\ide-S^{-1}$ is dissipative
for some $\epsilon>0$. With this result the existence and uniqueness of
solutions of the heat equation can be derived from the wave equation.
