TY - JOUR AU - I. Wróbel AU - A. M. Gomilko AU - J. Zemanek PY - 2007/06/25 Y2 - 2024/03/29 TI - On a criterion for the uniform boundedness of a <i>C</i><sub>0</sub>-semigroup of operators in a Hilbert space JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 59 IS - 6 SE - Short communications DO - UR - https://umj.imath.kiev.ua/index.php/umj/article/view/3350 AB - 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. ER -