On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space - Wróbel, I.; Gomilko, A. M.; Zemanek, J. (Ukrains’kyi Matematychnyi Zhurnal, 2007)
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.