On a criterion for the uniform boundedness of a <i>C</i><sub>0</sub>-semigroup of operators in a Hilbert space

  • I. Wróbel
  • A. M. Gomilko
  • J. Zemanek


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: $$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds < ∞$$ forall $x ∈ H$, where $T^{*}$ is the adjoint operator.
How to Cite
Wróbel, I., A. M. Gomilko, and J. Zemanek. “On a Criterion for the Uniform Boundedness of a <i>C</i><sub>0</Sub&gt;-Semigroup of Operators in a Hilbert Space”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, no. 6, June 2007, pp. 853-8, https://umj.imath.kiev.ua/index.php/umj/article/view/3350.
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