KW-Stability in Semigroup of Bounded Linear Operators
Otobong J. Tom; Otobong G. Udoaka; E.S. Udofia
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.
Keywords
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