The Adjoint of α -Times-Integrated C -Regularized Semigroups.

We consider an operator { S (t) } t ≥ 0 on a Banach space X with generator A, characterized by being an α -times-integrated C-regularized semigroup. The adjoint family S * (t) : X * → X * is introduced for analysis. { S * (t) } t ≥ 0 maintains the characteristics of an α -times-integrated C-regularized semigroup, though with strong continuity and Bochner integrals being substituted by weak* continuity and weak* integrals, respectively. Our investigation focuses on the closed subspace X ⊙ , where { S * (t) } t ≥ 0 exhibits strong continuity. Additionally, a comparison between the adjoint A * of A and the generator of the adjoint family is conducted. [ABSTRACT FROM AUTHOR]