KW-Stability in Semigroup of Bounded Linear Operators

Source abstract

In this paper, we investigate the stability in semigroups of bounded linear operators on Banach spaces in the sense of Koch and Wallace (1956). A semigroup S is said to be stable if for all a b S , ,  the inclusions aS abS  and Sa Sab  imply equalities. We show that every strongly continuous semigroup  ( )t 0 T t  of bounded linear operators on a Banach space is stable. This result is established using the commutativity of one-parameter operator semigroups. Consequences for Green’s relations, in particular, the equivalence of the D − and J − relations are derived. Applications are discussed in the context of spectral theory of the generator of semigroups, as well as their role in evolution equations.