Cosine families of operators in a class of Fréchet spaces

Rachid Ameziane Hassani, Aziz Blali, Abdelkhalek El Amrani, Khalil Moussaouja
2018 Proyecciones  
M. sova [10] proved that the infinitesimal generator of all uniformly continuous cosine family, of operators in Banach space, is a bounded operator. We show by counter-example that the result mentioned above is not true in general on Fréchet spaces, and we prove that the infinitesimal generator of an uniformly continuous cosine family of operators in a class of Fréchet spaces (quojection) is necessarily continuous. Keywords : Strongly continuous cosine families. Semi-groups of operators.
more » ... convex space. Quojection Fréchet space. Cosine families of operators in locally convex space : Let X be a locally convex Hausdorff space, and Γ X a system of continuous semi-norms determining the topology of X. The strong topology τ s in the space L(X), of all continuous linear operators from X into itself, is determined by the family of seminorms : for each x ∈ X and q ∈ Γ X , L(X) equipped with this topology is noted L s (X).
doi:10.4067/s0716-09172018000100103 fatcat:5ijvkdesr5aenplzli5er2nm4e