We discuss a simple working scenario for the growth of supermassive black holes (BHs) at the center of spheroidal stellar systems. In particular, we assess the hypotheses that (1) star formation in spheroids and BH fueling are proportional to one another, and (2) the BH accretion luminosity stays near the Eddington limit during luminous quasar phases. With the aid of this simple picture, we are able to interpret many properties of the QSO luminosity function, including the puzzling steep decline of the characteristic luminosity from redshift z ≈ 2 to z ≈ 0: indeed the residual star formation in spheroidal systems is today limited to a small number of bulges, characterized by stellar velocity dispersions a factor of 2-3 smaller than those of the elliptical galaxies hosting QSOs at z approx> 2. A simple consequence of our hypotheses is that the redshift evolution of the QSO emissivity and of the star formation history in spheroids should be roughly parallel. We find this result to be broadly consistent with our knowledge of the evolution of both the global star formation rate and the QSO emissivity, but we identify interesting discrepancies at both low and high redshifts, to which we offer tentative solutions. Our hypotheses allow us to present a robust method to derive the duty cycle of QSO activity, based on the observed QSO luminosity function and the present-day relation between the masses of supermassive BHs and those of their spheroidal host stellar systems. The duty cycle is found to be substantially less than unity, with characteristic values in the range (3-6) x 10~(-3), and we compute that the average bolometric radiative efficiency is ε ≈ 0.07. Finally, we find that the growth in mass of individual BHs at high redshift (z approx> 2) can be dominated by mergers and is therefore not necessarily limited by accretion.
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