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](http://en.wikipedia.org/wiki/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?



ordered exponential of unbounded operators
This is usually called the magnus expansion method and has a nice literature in numerical analysis. Kato also used this method to show the existence of solutions in the hyperbolic case. 


I would say that strong resolvent continuity and a sufficiently big common domain is sufficient in your case. See Section 5.3 in [Pazy](http://books.google.hu/books/about/Semigroups_of_Linear_Operators_and_Appli.html?id=sIAyOgM4R3kC&redir_esc=y).


I can also give a related [self-reference](http://arxiv.org/abs/1105.6372), where also the investigation of this product appears and some of the ideas are explained in a simpler situation.


**ADDED:** My answer concentrates on the method you propose to converge to the solution. To make the content of the references short: yes. A common dense domain and continuity of hte maps $t\mapsto A\_tx$ implies the convergence of the product to the solution of the differential equation.


