Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$.
Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A \right)v, w \right> \right|\stackrel{t \to 0}{\rightarrow} 0$ for all $v, w \in \mathcal{H}$, where $A$ is some bounded operator on $\mathcal{H}$. Can we somehow conclude that $$ \left\| \left(\frac{U(t)-I}{t}-A\right)w \right\| \to 0 \ \ \forall_{w \in \mathcal{H}}, \ \mbox{ as } t \to 0?$$
If not, can we assume something more about $\mathcal{U}$ or $A$ to get this strong operator convergence?
Thank you for any hints, answers or good references.