Etude De Certains Operateurs Dans L’espace De Banach, Exemple Les Semi-groupes

Résumé: In this thesis, we focus on the invertibility of operators and semigroups, the behavior of C0-semigroups, and the relationships between Lyapunov inequality and Sylvester equality. We present two versions of Arens’s theorem: the first is on reflexive Banach spaces, and the second is on Banach spaces isomorphic to a Hilbert space. We prove that there is an inverse semigroup in the same space for Banach spaces isomorphic to Hilbert spaces. In the first version, we establish that the operators (also semigroups) satisfying a lower bound condition have an inverse operator (inverse semigroup, respectively) in a large space with the same type of topology. Furthermore, in the second version, we establish that there is an inverse semigroup in the same space. We also study the invertibility of semigroups and the behaviors of the C0-semigroups, and examine their relationships with Lyapunov inequality and Sylvester equality. Our main contributions in this dissertation are: 1. We present two versions of Arens’s theorem: one for reflexive Banach spaces and one for Banach spaces isomorphic to Hilbert spaces. 2. We establish that operators (as well as semigroups) satisfying a lower bound condition have inverses (or inverse semigroups, respectively) in a sufficiently large space, maintaining the same topological properties. 3. We prove that there is an inverse semigroup in the same space for Banach spaces isomorphic to Hilbert spaces. 4. We study the invertibility of semigroups, along with the behaviors of C0- semigroups, and examine the relationships between Lyapunov inequality and Sylvester equality. 5. establish a result of uniformly boundedness of the resolvent inside and outside the left half-plane via relations with Lyapunov inequalities and Sylvester’s equality. Arens’s Theorem, Reflexive Banach Space Banach

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