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

Adam Bobrowski, and Wojciech Chojnacki. "Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space." Studia Mathematica 217.3 (2013): 219-241. <http://eudml.org/doc/285524>.

@article{AdamBobrowski2013,
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 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 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.},
author = {Adam Bobrowski, Wojciech Chojnacki},
journal = {Studia Mathematica},
keywords = {cosine family; semigroup of operators; strong continuity; separability; isolated point; strong convergence},
language = {eng},
number = {3},
pages = {219-241},
title = {Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space},
url = {http://eudml.org/doc/285524},
volume = {217},
year = {2013},
}

TY - JOUR
AU - Adam Bobrowski
AU - Wojciech Chojnacki
TI - Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space
JO - Studia Mathematica
PY - 2013
VL - 217
IS - 3
SP - 219
EP - 241
AB - 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 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 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.
LA - eng
KW - cosine family; semigroup of operators; strong continuity; separability; isolated point; strong convergence
UR - http://eudml.org/doc/285524
ER -