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),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, t ≥ 0, if and only if the following condition is satisfied: _t > 0 1t ∫_0^t∥(T(s)+T^∗(s))x ∥^2ds < ∞ forall x ∈ H, where T^* is the adjoint operator.