Generators with a closure relation

Abstract

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 $A_S$ defined by $A_{S}x=A_1\left(\begin{array}{c}x\\ S{A}_{2}x\end{array}\right)$ with the natural domain generates a contraction semigroup on $X_1$. Here, $S$ is a boundedly invertible operator for which $ε\ide-S^{-1}$ 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.

9 pages

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