  We prove a product formula which involves the unitary group generated by a
semibounded self-adjoint operator and an orthogonal projection $P$ on a
separable Hilbert space $\HH$, with the convergence in
$L^2_\mathrm{loc}(\mathbb{R};\HH)$. It gives a partial answer to the question
about existence of the limit which describes quantum Zeno dynamics in the
subspace \hbox{$\mathrm{Ran} P$}. The convergence in $\HH$ is demonstrated in
the case of a finite-dimensional $P$. The main result is illustrated in the
example where the projection corresponds to a domain in $\mathbb{R}^d$ and the
unitary group is the free Schr\"odinger evolution.
