Isolated Points of Some Sets of Bounded Cosine Families, Bounded Semigroups, and Bounded Groups on a Banach Space

We show that if the set of all bounded strongly continuous cosine families on a Banach space X is treated as a metric space under the metric of the uniform convergence associated with the operator norm on the space L (X) of all bounded linear operators on X, then the isolated points of this set are precisely the scalar cosine families. By definition, a scalar cosine family is a cosine family whose members are all scalar multiples of the identity operator. We also show that if the sets of all bounded cosine families and of all bounded strongly continuous cosine families on an infinitedimensional separable Banach space X are viewed as topological spaces under the topology of the uniform convergence associated with the strong operator topology on L (X), then these sets have no isolated points. We present counterparts of all the above results for semigroups and groups of operators, relating to both the norm and strong operator topologies.