A note on uniformly bounded Hilbert space semigroups

Abstract

According to a theorem of Langer 111 and Sz-Nagy and Foias 121, a Hilbert space contraction can be uniquely decomposed into a disjoint direct sum of a unitary operator and a completely nonunitary (cnu) contraction-i.e., one which does not have any nontrivial unitary “part.” This result can, of course, be applied to Hilbert space contraction semigroups [2,3]. In this note we wish to explore possible extensions of the above results to noncontractive uniformly bounded semigroups over a Hilbert space. A decomposition of the Hilbert space associated with a uniformly bounded semigroup is obtained in Section 2. This decomposition is then shown to result in a condition for a uniformly bounded semigroup to be completely nonunitary. Moreover, it also yields the Nagy-Foias decomposition for contraction semigroups. A condition for a uniformly bounded semigroup to admit the Nagy-Foias decomposition is then obtained, and it will be shown that such a condition is satisfied by the class of uniformly bounded normal semigroups. Finally another type of decompsition of uniformly bounded semigroups-with respect to weak convergence property-is derived. Relationships between this and a decomposition of contraction semigroups due to Foguel [4] are then discussed.