Let $\mathcal{H}$ be a Hilbert space and suppose $G=\{U(\alpha) \}_{\alpha \in \mathbb{R}} $ is one parameter group of linear bounded maps $U(\alpha):\mathcal{H}\to \mathcal{H}$ under composition such that $U(0)=Id $. Now suppose that for a fixed $\psi\in \mathcal{H}$ we define $A_\psi:\mathbb{R}\to \mathcal{H}$ as $$\tau \mapsto \int_0^\tau U(\alpha)\psi \;d\alpha $$ My question is: Under what conditions does the following equation holds? for any $\epsilon\in \mathbb{R}$ $$\bigr(U(\epsilon )\circ A_{\psi}\bigr)(\tau)= \int_0^\tau U(\epsilon)\cdot U(\alpha)\psi\; d\alpha $$
Here I used the notation $U(\epsilon)\cdot U(\alpha)\psi\equiv \bigr(U(\epsilon) \circ U(\alpha)\bigr)(\psi)$