Description:
<jats:title>Abstract</jats:title><jats:p>In this paper we relate the generator property of an operator <jats:italic>A</jats:italic> with (abstract) generalized Wentzell boundary conditions on a Banach space <jats:italic>X</jats:italic> and its associated (abstract) Dirichlet‐to‐Neumann operator <jats:italic>N</jats:italic> acting on a “boundary” space <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/mana201800064-math-0001.png" xlink:title="urn:x-wiley:0025584X:media:mana201800064:mana201800064-math-0001" />. Our approach is based on similarity transformations and perturbation arguments and allows to split <jats:italic>A</jats:italic> into an operator <jats:italic>A</jats:italic><jats:sub>00</jats:sub> with Dirichlet‐type boundary conditions on a space <jats:italic>X</jats:italic><jats:sub>0</jats:sub> of states having “zero trace” and the operator <jats:italic>N</jats:italic>. If <jats:italic>A</jats:italic><jats:sub>00</jats:sub> generates an analytic semigroup, we obtain under a weak Hille–Yosida type condition that <jats:italic>A</jats:italic> generates an analytic semigroup on <jats:italic>X</jats:italic> if and only if <jats:italic>N</jats:italic> does so on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/mana201800064-math-0002.png" xlink:title="urn:x-wiley:0025584X:media:mana201800064:mana201800064-math-0002" />. Here we assume that the (abstract) “trace” operator <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/mana201800064-math-0003.png" xlink:title="urn:x-wiley:0025584X:media:mana201800064:mana201800064-math-0003" /> is bounded that is typically satisfied if <jats:italic>X</jats:italic> is a space of continuous functions. Concrete applications are made to various second order differential operators.</jats:p>