Smallest sampling frequency to fully reconstruct a signal
Your understanding is not correct for complex sampling:

For the specific case of a complex bandpass (analytic) signal whose spectrum is by definition zero for the negative frequencies such as:

$$ H(\Omega) =  \begin{cases}  0 &amp;, \text{ by def. for } \Omega &lt; 0 \\ A_\Omega &amp;, \text{ nonzero for } \Omega_1 &lt; \Omega &lt; \Omega_2 \\ \end{cases}$$ 

then the necessary minimum sampling rate which would avoid aliasing (spectral overlap) is given by $$ \Omega_s &gt; ( \Omega_2 - \Omega_1 ) $$ Note that if the signal were real with a symmetric bandwidth then the minimum sampling rate would be twice that of the complex case. Also then the allowed range of valid sampling rates would be found differently. 

The reconstrcution filter that would yield the original complex signal $x(t)$ back will be a complex bandpass filter, not a real one. You can obtain the complex bandpass filter $H_+(\Omega)$ from the real bandpass filter $H_r(\Omega)$ using a Hilbert transformer such as:

$$ h_+(t) = h_r(t) + j \mathcal{H} \{h_r(t) \} $$ 

Note that for a given sampling rate you will be getting $F_s$ complex samples per second which is equivalent to $2 \times  F_s$ real samples per second. Hence the apparent advantage of complex sampling is actually not realized , as the total number of samples per second will be the same in both cases.