Kato’s Inequality

The aim of this article is to show some interesting consequences of Kato’s inequality. First we show three striking properties of Schrödinger semigroups on $$L_1(\mathbb {R}^d)$$ L 1 ( R d ) (holomorphy and closedness of $$-\varDelta + V$$ - Δ + V , the test functions are a core) with the same elegant argument Kato gave, but extending the results to possibly non-symmetric elliptic operators. In the second part, we consider the Dirichlet problem $$ (- \varDelta + V) u = 0 , \quad u|_{\partial \varOmega } = \varphi , \quad u \in C(\overline{\varOmega }) , $$ ( - Δ + V ) u = 0 , u | ∂ Ω = φ , u ∈ C ( Ω ¯ ) , where $$V \in L_\infty (\varOmega ,\mathbb {R})$$ V ∈ L ∞ ( Ω , R ) and $$\varOmega $$ Ω is a bounded Wiener regular set. Well-posedness has been studied in a recent paper (Arendt and ter Elst, Annales de l’Institut Fourier, 2019, [9]). Here we investigate when the maximum principle holds.