Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann operator

Abstract

In this paper we relate the generator property of an operator $A$ with (abstract) generalized Wentzell boundary conditions on a Banach space $X$ and its associated (abstract) Dirichlet-to-Neumann operator $N$ acting on a "boundary" space $\partial X$. Our approach is based on similarity transformations and perturbation arguments and allows to split $A$ into an operator $A_{00}$ with Dirichlet-type boundary conditions on a space $X_0$ of states having "zero trace" and the operator $N$. If $A_{00}$ generates an analytic semigroup, we obtain under a weak Hille--Yosida type condition that $A$ generates an analytic semigroup on $X$ if and only if $N$ does so on $\partial X$. Here we assume that the (abstract) "trace" operator $L:X\to\partial X$ is bounded what is typically satisfied if $X$ is a space of continuous functions. Concrete applications are made to various second order differential operators.