Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let Τ=(T(t)) _t≥0 to be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x∈Y, σ _u (A, C _x ), the set of all points λ∈i R to which (λ–A)^–1 Cx can not be extended holomorphically, is at most countable and σ _r (A)∩i R=∅, then Τ is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous semigroups. The first author was supported by the NSF of China. The second author was supported by TRAPOYT and the NSF of China (No. 10371046)