Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space

Adam Bobrowski ¹   ; Wojciech Chojnacki ²

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland
² School of Computer Science The University of Adelaide Adelaide, SA 5005, Australia and Wydział Matematyczno-Przyrodniczy Szkoła Nauk Ścisłych Uniwersytet Kardynała Stefana Wyszyńskiego Dewajtis 5 01-815 Warszawa, Poland

Studia Mathematica, Tome 217 (2013) no. 3, pp. 219-241

Résumé

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 $\mathcal {L}(X)$ 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 infinite-dimensional separable Banach space $X$ are viewed as topological spaces under the topology of the uniform convergence associated with the strong operator topology on $\mathcal {L}(X)$, 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.

Détail

DOI : 10.4064/sm217-3-2

Keywords: set bounded strongly continuous cosine families banach space treated metric space under metric uniform convergence associated operator norm space mathcal bounded linear operators isolated points set precisely scalar cosine families definition scalar cosine family cosine family whose members scalar multiples identity operator sets bounded cosine families bounded strongly continuous cosine families infinite dimensional separable banach space viewed topological spaces under topology uniform convergence associated strong operator topology mathcal these sets have isolated points present counterparts above results semigroups groups operators relating norm strong operator topologies