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

• Autorzy: Adam Bobrowski , Wojciech Chojnacki
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47D09: Operator sine and cosine functions and higher-order Cauchy problems
• 34G10: Linear equations
• 60J65: Brownian motion

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2013
• Tom: 217
• Numer: 3
• Strony: 219-241

Daty

wydano

2013

Twórcy

autor

Adam Bobrowski

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland

autor

Wojciech Chojnacki

• School of Computer Science, The University of Adelaide, Adelaide, SA 5005, Australia
• Wydział Matematyczno-Przyrodniczy, Szkoła Nauk Ścisłych, Uniwersytet Kardynała Stefana Wyszyńskiego, Dewajtis 5, 01-815 Warszawa, Poland

Identyfikatory

DOI

10.4064/sm217-3-2