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

Volume 217 / 2013

Adam Bobrowski, Wojciech Chojnacki Studia Mathematica 217 (2013), 219-241 MSC: Primary 47D09, 34G10; Secondary 60J65. DOI: 10.4064/sm217-3-2

Abstract

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.

Authors

  • Adam BobrowskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland
    e-mail
  • Wojciech ChojnackiSchool 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
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image