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Gamma-bounded C0-semigroups and power gamma-bounded operators : characterizations and functional calculi

Abstract : First and foremost we show that there exist bounded sectorial operators A of type 0 (respectively Ritt operators T ) such that the set {e^{-tA}: tgeq 0} is not gamma-bounded (respectively the set {T^n : n in N } is not gamma-bounded).In the second chapter, 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.The next chapter is dedicated to 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.In the final chapter, we focus on C_0-semigroups on Hilbert space. Our goal is to construct a bounded functional calculus on a new algebra A{C_+} inspired by Figa-Talamanca-Herz algebras. We show that this bounded functional calculus improves existing results. We also get results about bounded Fourier multipliers on the Hardy space H^1(R) which are useful for the study of A(C_+).
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Submitted on : Thursday, January 27, 2022 - 11:16:07 AM
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Loris Arnold. Gamma-bounded C0-semigroups and power gamma-bounded operators : characterizations and functional calculi. Functional Analysis [math.FA]. Université Bourgogne Franche-Comté, 2021. English. ⟨NNT : 2021UBFCD016⟩. ⟨tel-03545380⟩

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