On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space

• Date: 2007
• Main Authors: Wróbel, I., Gomilko, A. M., Zemanek, J., Врубель, И., Гомилко, А. М., Земанек, Я.
• Format: Article
• Language: Russian; English
• Published: Institute of Mathematics, NAS of Ukraine 2007
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/3350
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

Let $T(t),\quad t ≥ 0$, be a $C_0$-semigroup of linear operators acting in a Hilbert space $H$ with norm $‖·‖$. We prove that $T(t)$ is uniformly bounded, i.e., $‖T(t)‖ ≤ M, \quad t ≥ 0$, if and only if the following condition is satisfied: $$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds...