  For small values of the dimensionless viscosity parameter, namely
$\alpha\lesssim 0.1$, the dynamics of non-radiating accretion flows is
dominated by convection; convection strongly suppresses the accretion of matter
onto the central object and transports a luminosity $\sim 10^{-3}-10^{-2} \dot
M c^2$ from small to large radii in the flow. A fraction of this convective
luminosity is likely to be radiated at large radii via thermal bremsstrahlung
emission. We show that this leads to a correlation between the frequency of
maximal bremsstrahlung emission and the luminosity of the source, $\nu_{\rm
peak} \propto L^{2/3}$. Accreting black holes with X-ray luminosities $10^{-4}
L_{Edd}\gtrsim L_X(0.5-10{\rm keV}) \gtrsim 10^{-7}L_{Edd}$ are expected to
have hard X-ray spectra, with photon indices $\Gamma\sim2$, and sources with
$L_X\lesssim 10^{-9}L_{Edd}$ are expected to have soft spectra, with
$\Gamma\sim3.5$. This is testable with {\it Chandra} and {\it XMM}.
