Taiwanese Journal of Mathematics

ON THE COLLECTIVE COMPACTNESS OF STRONGLY CONTINUOUS SEMIGROUPS AND COSINE FUNCTIONS OF OPERATORS

Hernan R. Henriquez

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Abstract

Let $X$ be a complex Banach spcce, and denote by $T$ a strongly continuous semigroup of linear operators defined on $X$ and by $C$ a cosine function of operators with associated sine function $S$ defined on $X$. In this note we characterize in terms of spectral properties of the infinitesimal generator those semigroups $T$ and cosine functions $C$ such that $\{T(t) - I : t \geq 0 \}$, $\{C(t) - I : t \in {\Bbb R}\}\;$ and $\{S(t) : t \in {\Bbb R}\}$ are collectively compact sets of bounded linear operators.

Article information

Source
Taiwanese J. Math., Volume 2, Number 4 (1998), 497-509.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407020

Digital Object Identifier
doi:10.11650/twjm/1500407020

Mathematical Reviews number (MathSciNet)
MR1662950

Zentralblatt MATH identifier
0960.47023

Subjects
Primary: 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47D09: Operator sine and cosine functions and higher-order Cauchy problems [See also 34G10]

Keywords
semigroups of operators cosine functions of operators compact operators collectively compact operators almost periodic functions asymptotically almost periodic functions

Citation

Henriquez, Hernan R. ON THE COLLECTIVE COMPACTNESS OF STRONGLY CONTINUOUS SEMIGROUPS AND COSINE FUNCTIONS OF OPERATORS. Taiwanese J. Math. 2 (1998), no. 4, 497--509. doi:10.11650/twjm/1500407020. https://projecteuclid.org/euclid.twjm/1500407020


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