f
\longrightarrow
\exp(-i\varepsilon tH_P)\,Pf\,, \label{product-s} \\
&&[P(1/n)\,\exp(-i\varepsilon tH/n) ]^n f
\longrightarrow
\exp(-i\varepsilon tH_P)\,Pf\,, \label{product-ns}\\
&&[\,\exp(-i\varepsilon tH/n)\,P(1/n)]^n f
\longrightarrow
\exp(-i\varepsilon tH_P)\,Pf\,, \label{product-ns2}
 \end{aligned}$$ in the topology of $L_\mathrm{loc}^2({\mathbb{R}}; {\cal H})$ as $\:n\rightarrow \infty$.

Note that $H_P$ differs in general from the operator $PHP$, which may not be self-adjoint in ${{\mathcal H}}$, nor even closed, because $PH$ is not necessarily closed, though $HP$ is. $H_P$ is a self-adjoint extension of $PHP$. The requirement of the theorem that $H_P$ is densely defined in ${{\mathcal H}}$ means nothing else but that the domain $D[H^{1/2}P]$ of the quadratic form in question is dense in ${{\mathcal H}}$.

Note also that for $\varepsilon =1$, the theorem concerns a nonnegative self-adjoint operator $\varepsilon H = H$, while for $\varepsilon =-1$, we get product formulae for the non-positive self-adjoint operator $\varepsilon H = -H$. Moreover, the result is preserved when $H$ is replaced with a shifted operator $H+cI$, i.e. for any semi-bounded self-adjoint operator in a separable Hilbert space.

An important particular case, most often met in the applications, concerns the situation when the projection-valued function is constant.

\[cor1\] Let $H$ be a self-adjoint operator bounded from below in a separable Hilbert space ${{\mathcal H}}$ and $P$ an orthogonal 