Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\frac{d}{dt} E_t = -A_tE_t
\quad\qquad\qquad\qquad\qquad\qquad\qquad(1)
$$
that defines the *ordered exponential* of the family $A_t$.

If $A_t=A$ is independ of $t$, then the solution of the *ODE* is the usual exponential $E_t=e^{-tA}$.

Note that the above operators $E_t$ are ** bounded**.

I suspect that, if I put appropriate hypotheses on $A_t$, (such as having a common dense domain, depending continuously on $t$, whatever that might mean, etc.) the solution of (1) will also be bounded. Intuitively, it's kind of clear: $$ E_t = \lim_{N\to\infty} \Big(e^{-\frac t N A_t} \cdot e^{-\frac t N A_{t(1-1 /N)}} \cdot e^{-\frac t N A_{t(1-2 /N)}}\cdots \cdot e^{-\frac t N A_{t(3 /N)}} \cdot e^{-\frac t N A_{t(2 /N)}} \cdot e^{-\frac t N A_{t(1 /N)}}\Big) $$ and each little exponential in the above product has norm $\le 1$.

**Q:**
Which properties should I impose on the family $A_t$ in order for the solution of (1) to be well defined and bounded?