Talk by Sahiba Arora

Extensive literature has been devoted to study the operators for which the maximum and/or the anti-maximum principle holds. Combining an idea of Takáč (1996) with those from the recent theory of eventually positive $C_0$-semigroups, we look at some necessary and sufficient conditions for (anti-)maximum principles to hold in an abstract setting of Banach lattices.

More precisely, if $A : \operatorname{dom}(A) \subseteq E\to E$ is a closed, densely defined, and real operator on a complex Banach lattice $E$ (or in particular, an $L^p$-space), then we consider the equation
\[
(\lambda-A)u =f
\]
for real numbers $\lambda$ in the resolvent set of $A$. We ask whether $f\geq 0$ implies $u\geq 0$ for $\lambda$ in a right neighbourhood of an eigenvalue. In this case, we say that the maximum principle is satisfied. Analogously, when the implication $f\geq 0$ implies $u\leq 0$ holds for $\lambda$ in a left neighbourhood of an eigenvalue, we say that the anti-maximum principle holds.

We will also see how these abstract results can be applied to various concrete differential operators and illustrate how several previously known results about (anti-)maximum principles can be proved via this theory. This is joint work with Jochen Glück.