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

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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.