## Taiwanese Journal of Mathematics

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

Hernan R. Henriquez

#### 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

https://projecteuclid.org/euclid.twjm/1500407020

Digital Object Identifier
doi:10.11650/twjm/1500407020

Mathematical Reviews number (MathSciNet)
MR1662950

Zentralblatt MATH identifier
0960.47023

#### 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