Dirichlet forms is 1st Kato inequality world (in essence - to get Markovian something more than 1st Kato is needed), where you have some quadratic thing and can work with sort of a square root of the generator (e.g. differential with Laplacian). I don't remember anything of the Röckner & Ma book. Should have had a look.

This 2nd Kato inequality thing is when there's no quadratic form available.

Alas I haven't yet studied Arendt's proof completely: The major part is in another paper. I'm thinking about translating it from the general Banach space setting to L^1. Plus I should write a little wikipedia article on Kato's inequalities before I forget the stuff again.

Maybe I find time this weekend. I'm a bit overworked these days.

This 2nd Kato inequality thing is when there's no quadratic form available.

Alas I haven't yet studied Arendt's proof completely: The major part is in another paper. I'm thinking about translating it from the general Banach space setting to L^1. Plus I should write a little wikipedia article on Kato's inequalities before I forget the stuff again.

Maybe I find time this weekend. I'm a bit overworked these days.