# Gamma-boundedness of $C_0$-semigroups and their $H^\infty$-functional calculi

## TL;DR

This paper explores gamma-boundedness in $C_0$-semigroups and their $H^$-functional calculi, establishing new characterizations and extending classical results like Gearhart-Pruss to $K$-convex spaces.

## Contribution

It introduces the notion of gamma-$H^$-bounded calculus, characterizes gamma-bounded $C_0$-semigroup generation in $K$-convex spaces, and extends the Gearhart-Pruss theorem.

## Key findings

- Established connection between gamma-$H^$-bounded calculus and semigroup generation.
- Provided a characterization of gamma-bounded $C_0$-semigroups in $K$-convex spaces.
- Extended the Gearhart-Pruss theorem to $K$-convex spaces.

## Abstract

We discuss the notion of $\gamma$-$H^{\infty}$-bounded calculus, strong $\gamma$-$m$-$H^{\infty}$-bounded calculus on half-plane and weak-$\gamma$-Gomilko-Shi-Feng condition and give a connection between them. Then we state a characterization of generation of $\gamma$-bounded $C_0$-semigroup in $K$-convex space, which leads to a version of Gearhart-Pr\"uss on $K$-convex space.