On a criterion for the uniform boundedness of a C ₀-semigroup of operators in a Hilbert space

Abstract

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:

$$\mathop {\sup }\limits_{t > 0} \frac{1}{t}\int\limits_0^t {\left\| {(T(s) + T^ * (s))x} \right\|^2 ds < \infty for all x \in H,} $$

, where T* is the adjoint operator.