Contraction semigroups on $L_{\infty }(\mathbf{R})$

Abstract

If $X$ is a non-degenerate vector field on $\mathbf{R}$ and $H=-X^2$ we examine conditions for the closure of $H$ to generate a continuous semigroup on $L_{\infty }$ which extends to the $L_p$-spaces. We give an example which cannot be extended and an example which extends but for which the real part of the generator on $L_2$ is not lower semibounded.

14 pages