Domination for subordinated positive semigroups

We first give a necessary and sufficient condition for -f(B) to be dominated by -f(A), where f is a completely Bernstein function, B and A are C0-semigroup generators. Then we prove that there is no domination relationship between a semigroup and the subordinated semigroup if additional conditions are satisfied.

Spectrum and convergence of eventually positive operator semigroups

An intriguing feature of positive $$C_0$$ C 0 -semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups. In particular, the convergence of semigroup operators (strongly or in the operator norm) as time tends to infinity can be characterized by a set of simple spectral and compactness conditions. In the present paper, we show that similar theorems remain true for the larger class of (uniformly) eventually positive semigroups—which recently arose in the study of various concrete differential equations. A major step in one of our characterizations is to show a version of the famous Niiro–Sawashima theorem for eventually positive operators. Several proofs for positive operators and semigroups do not work in our setting any longer, necessitating different arguments and giving our approach a distinct flavour.

Positive Semigroups Using Resolvent Estimate Bounds on Sharp Growth Rates

In this paper, we study growth rates for strongly continuous semigroups. We fixate that a growth rate for the resolvent estimate on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroupis positive and the underlying space is an -space or a space of continuous functions. Also proved variations of the main results on fractional domains; these are valid on more general Banach spaces by Jan Rozendaal and Mark Veraar. In the second part apply the main theorem to prove optimality in a classical example of a perturbed wave equation which shows unusual sequence of spectral behavior.

Hadamard Completely Positive Semigroups

We review the Gorini, Kossakowski, Sudarshan derivation of the generator of a completely positive norm-continuous semigroup when the constituent maps act on density matrices according to the Hadamard product rule.