On the best approximation of the infinitesimal generator of a contraction semigroup in a Hilbert space

Let $A$ be the infinitesimal generator of a strongly continuous contraction semigroup in a Hilbert space $H$. We give an upper estimate for the best approximation of the operator $A$ by bounded linear operators with a prescribed norm in the space $H$ on the class $Q_2 = \{x\in \mathcal{D}(A^2) : \|A^2 x\| \leq 1\}$, where $\mathcal D(A^2)$ denotes the domain of $A^2$.