Bi-Continuous C-Semigroups

In order to treat initial value problems for partial differential equations and some practical applications, mathematicians created operator semigroup theory in middle period of the last century. With the further process of the problems, semigroup theory is also developed gradually, we have already known: C₀ semigroups, n-times integrated semigroups and C semigroups on Banach spaces and on locally convex spaces. By studying some concrete semigroups, F. Kiihnemund established bi-continuous semigroups on Banach space endowed with an additional locally convex Hausdorff topology r which is coarser than the norm topology. In this paper, we introduce bi-continuous C-semigroups by combining bi-continuous semigroups and C-semigroups, and define their generators. On the basis of these concepts, the relation among bi-continuous C-semigroups, generators and their C-resolvents is investigated systematically. Then we conclude two important theorems of bi-continuous C-semigroups: generation theorem and approximation theorem.