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$\gamma$-bounded $C_0$-semigroups and power $\gamma$-bounded operators : characterizations and functional calculi

par Dagdoug Mehdi - publié le

Mercredi 24 février 2021
Loris Arnold
(LmB, Université de Franche-Comté)

First we study $\gamma$-bounded $C_0$-semigroups on Banach spaces. We will able to generalize Gomilko Shi-Feng Theorem in Banach settings. This generalization gives us a characterization of $\gamma$-bounded $C_0$-semigroups. Further, in this context, we study the derivative bounded functional calculus introduced by Batty Haase and Mubeen.
Then we study operators which satisfy a condition called discrete Gomilko Shi-Feng condition. We show that this condition is equivalent to various bounded functional calculi. We also study power $\gamma$-bounded operators and we characterize them in a similar way as for $\gamma$-bounded $C_0$-semigroups.
Finally, we focus on $C_0$-semigroups on Hilbert space. Our goal is to construct a bounded functional calculus on a new algebra $\mathcal{A}\left(\mathbb{C}_+\right)$ inspired by Figa-Talamanca-Herz algebras. We show that this bounded functional calculus improves existing results.