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# Experiments in Kato's inequality

Created Experiments in Kato's inequality. Angle brackets don't render in my box. Those from blog do.

$$\langle \vert a\vert , H f\rangle \ge \langle (\sgn a)(Ka), f\rangle \quad \forall ..., f\ge0$$ Now me sleep, then work, then perhaps state an unsurprising theorem on stochastic semigroups, then sleep again, etc., then give proof of theorem. That's the plan.

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Sounds good!

Comment Source:Sounds good!
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Alas time is relative, as are plans... Got stuck in practical botany and soil arts, and then product integrals...

Changed "stochastic semigroup" to "positivity preserving". That's the knack point methinks - contractivity being secondary.

Comment Source:Alas time is relative, as are plans... Got stuck in practical botany and soil arts, and then product integrals... Changed "stochastic semigroup" to "positivity preserving". That's the knack point methinks - contractivity being secondary.
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edited October 2011

I'm baack... (too late to phone in sick). It seems the experiment as originally intended has been performed to sufficient precision by one Wolfgang Arendt in 1984...

So... I was sick, and had this 3rd idea of proof and ventured into combining Vitali's theorem on stochastic convergence with Banach-Alaoglu. And yet I didn't make it to the internets box (let alone boot my crazy offline laptop) to update said wiki page. At least, in the parallel universe of my micro paper works it is verified that I haven't saved Shigekawa's 1st preprint (incl. his proof, perhaps giving the idea, other than Barry Simon's simpler proof via resolventics.)...

Of all this a lemma remains (the only thing my mathematical experimentations produced) which questions the common definition of the signum functional (not always linear). So, the experiment might go on...

Will write up later, so the plan.

Comment Source:I'm baack... (too late to phone in sick). It seems the [experiment](http://mathoverflow.net/questions/61270/infinitesimal-generators-of-stochastic-processes) as originally intended has been performed to sufficient precision by one Wolfgang Arendt in 1984... So... I was sick, and had this 3rd idea of proof and ventured into combining Vitali's theorem on stochastic convergence with Banach-Alaoglu. And yet I didn't make it to the internets box (let alone boot my crazy offline laptop) to update said wiki page. At least, in the parallel universe of my micro paper works it is verified that I haven't saved Shigekawa's 1st preprint (incl. his proof, perhaps giving the idea, other than Barry Simon's simpler proof via resolventics.)... **Of all this** a lemma remains (the only thing my mathematical experimentations produced) which questions the common definition of the signum _functional_ (not always linear). So, the experiment might go on... Will write up later, so the plan.
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By the way, Martin - thanks for that reference over on MathOverflow. I haven't gotten around to reading it. I've actually become somewhat more accepting of nonsymmetric Dirichlet forms, despite their limitations. I'm enjoying that Springer book on nonsymmetric Dirichlet forms, whatever it's called. But I do want to come back to this!

Comment Source:By the way, Martin - thanks for that reference over on MathOverflow. I haven't gotten around to reading it. I've actually become somewhat more accepting of nonsymmetric Dirichlet forms, despite their limitations. I'm enjoying that Springer book on nonsymmetric Dirichlet forms, whatever it's called. But I do want to come back to this!
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edited November 2011

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.

Comment Source: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.