On the accretivity and m-accretivity of Laplacians and porous medium-type operators on graphs

Abstract

We study the accretivity and m-accretivity of Laplacian and porous medium-type operators on weighted graphs. In particular, we give several conditions that imply these properties for maximal operators and investigate when these operators agree with various restrictions. For porous medium-type operators on $\ell^1$ and for Laplacians on $\ell^p$ for $p \in [1,\infty)$, we show that there always exists a dense subset of the domain on which the maximal operator is m-accretive. As a consequence, we establish that accretivity, m-accretivity and injectivity of the shifted operator are all equivalent for these maximal operators. Under additional conditions on the graph, we then prove that the maximal operators are m-accretive on the entire domain, not just a dense subset. We also investigate minimal operators and show that they are m-accretive if and only if the minimal and maximal operators agree and the maximal operator is accretive. We then give some conditions that imply this agreement. Furthermore, for the minimal Laplacian on $\ell^p$, we show that accretivity and m-accretivity are not equivalent. For the $\ell^2$ case, we give connections to Markov uniqueness and essential self-adjointness. For the $\ell^\infty$ case, we establish the equivalence of stochastic completeness at infinity, m-accretivity for the maximal Laplacian on $\ell^\infty$, and m-accretivity of the minimal Laplacian on $\ell^1$.