Subordination for sequentially equicontinuous equibounded $C_0$-semigroups

TL;DR

This paper studies operators generating equicontinuous semigroups on locally convex spaces, showing that applying Bernstein functions to these operators preserves the semigroup properties, including fractional powers.

Contribution

It demonstrates that Bernstein functions applied to generators of equicontinuous semigroups produce new generators of the same class, extending the understanding of functional calculus in this setting.

Findings

Bernstein functions preserve semigroup generation properties

Fractional powers of generators are also generators

Extension of functional calculus for $C_0$-semigroups in locally convex spaces

Abstract

We consider operators $A$ on a sequentially complete Hausdorff locally convex space $X$ such that $-A$ generates a (sequentially) equicontinuous equibounded $C_0$-semigroup. For every Bernstein function $f$ we show that $-f(A)$ generates a semigroup which is of the same `kind' as the one generated by $-A$. As a special case we obtain that fractional powers $-A^{\alpha}$, where $\alpha \in (0,1)$, are generators.