The most important thing is ψ≥0 ⇒ Uψ≥0. (The stochastician in me would be content with a substochastic semigroup (Markov processes might die), and the Hilbert spaced part of me would add: well, that's simply the condition of the generator being negative definite.)

It seems the definite treatment of ψ≥0 ⇒ Uψ≥0 is theorem 1.6 in

Wolfgang Arendt, Kato's Inequality: A Characterisation of Generators of Positive Semigroups, Proc. R. Ir. Acad. Vol. 84A No. 2 (1984), 155-174

   ~ . ~ . ~

( Arendt also has the semigroup domination theorem 4.3 plus important ramifications. Until yesterday I called it the Kato-Simon-Shigekawa criterion. But no, according to Arendt he learned it from Kato himself. I've studied a bit of this stuff ca. 1995 (told Shigekawa about Simon's proof, and still (2011) have the most elegant verification of Kato's inequalities for general manifolds...) - but never did I come across Arendt's paper! (Well, these studies were laying so dormant in my ol brains that John's question of April 2011 couldn't shake them fully awake. It took a second hit from somewhere else.) Mestupid did quite some digging this week (plus, 3 failed attempts of proof of Arendt's theorem (in $L^1$) plus a total recall of all my higher analysis from gone times) to finally hit the paper... )

The most important thing is ψ≥0 ⇒ U(t)ψ≥0. (The stochastician in me would be content with a substochastic semigroup (Markov processes might die), and the Hilbert spaced part of me would add: well, that's simply the condition of the generator being negative definite.)

It seems the definite treatment of ψ≥0 ⇒ U(t)ψ≥0 is theorem 1.6 in

Wolfgang Arendt, Kato's Inequality: A Characterisation of Generators of Positive Semigroups, Proc. R. Ir. Acad. Vol. 84A No. 2 (1984), 155-174

   ~ . ~ . ~

( Arendt also has the semigroup domination theorem 4.3 plus important ramifications. Until yesterday I called it the Kato-Simon-Shigekawa criterion. But no, according to Arendt he learned it from Kato himself. I've studied a bit of this stuff ca. 1995 (told Shigekawa about Simon's proof, and still (2011) have the most elegant verification of Kato's inequalities for general manifolds...) - but never did I come across Arendt's paper! (Well, these studies were laying so dormant in my ol brains that John's question of April 2011 couldn't shake them fully awake. It took a second hit from somewhere else.) Mestupid did quite some digging this week (plus, 3 failed attempts of proof of Arendt's theorem (in $L^1$) plus a total recall of all my higher analysis from gone times) to finally hit the paper... )

   ~ . ~ . ~

This comment dedicated in memoriam Johann Schneidermeier

The most important thing is ψ≥0 ⇒ Uψ≥0. (The stochastician in me would be content with a substochastic semigroup (Markov processes might die), and the Hilbert spaced part of me would add: well, that's simply the condition of the generator being negative definite.)

It seems the definite treatment of ψ≥0 ⇒ Uψ≥0 is theorem 1.6 in

Wolfgang Arendt, Kato's Inequality: A Characterisation of Generators of Positive Semigroups, Proc. R. Ir. Acad. Vol. 84A No. 2 (1984), 155-174

   ~ . ~ . ~

( Arendt also has the semigroup domination theorem 4.3. plus important ramifications. Until yesterday I called it the Kato-Simon-Shigekawa criterion. But no, according to Arendt he learned it from Kato himself. I've studied a bit of this stuff ca. 1995 (told Shigekawa about Simon's proof, and still (2011) have the most elegant verification of Kato's inequalities for general manifolds...) - but never did I come across Arendt's paper! It took me quite some digging this week (plus, 3 failed attempts of proof of Arendt's theorem (in $L^1$) plus a total recall of all my higher analysis from gone times) to finally hit the paper... )

The most important thing is ψ≥0 ⇒ Uψ≥0. (The stochastician in me would be content with a substochastic semigroup (Markov processes might die), and the Hilbert spaced part of me would add: well, that's simply the condition of the generator being negative definite.)

It seems the definite treatment of ψ≥0 ⇒ Uψ≥0 is theorem 1.6 in

Wolfgang Arendt, Kato's Inequality: A Characterisation of Generators of Positive Semigroups, Proc. R. Ir. Acad. Vol. 84A No. 2 (1984), 155-174

   ~ . ~ . ~

( Arendt also has the semigroup domination theorem 4.3 plus important ramifications. Until yesterday I called it the Kato-Simon-Shigekawa criterion. But no, according to Arendt he learned it from Kato himself. I've studied a bit of this stuff ca. 1995 (told Shigekawa about Simon's proof, and still (2011) have the most elegant verification of Kato's inequalities for general manifolds...) - but never did I come across Arendt's paper! (Well, these studies were laying so dormant in my ol brains that John's question of April 2011 couldn't shake them fully awake. It took a second hit from somewhere else.) Mestupid did quite some digging this week (plus, 3 failed attempts of proof of Arendt's theorem (in $L^1$) plus a total recall of all my higher analysis from gone times) to finally hit the paper... )