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We discuss a simple model for the growth of supermassive black holes (BHs) atthe center of spheroidal stellar systems. In particular, we assess thehypotheses that (1) star formation in spheroids and BH fueling are proportionalto one another, and (2) the BH accretion luminosity stays near the Eddingtonlimit during luminous quasar phases. With the aid of this simple model, we areable to interpret many properties of the QSO luminosity function, including thepuzzling steep decline of the characteristic luminosity from redshift z=2 to toz=0: indeed the residual star formation in spheroidal systems is today limitedto a small number of bulges, characterized by stellar velocity dispersions afactor of 2-3 smaller those of the elliptical galaxies hosting QSOs at z > 2. Asimple consequence of our hypotheses is that the redshift evolution of the QSOemissivity and of the star formation history in spheroids should be roughlyparallel. We find this result to be broadly consistent with our knowledge ofthe evolution of both the global star formation rate, and of the evolution ofthe QSO emissivity, but we identify interesting discrepancies at both low andhigh redshifts, to which we offer tentative solutions. Finally, our hypothesesallow us to present a robust method to derive the duty cycle of QSO activity,based on the observed QSO luminosity function, and on the present-day relationbetween the masses of supermassive BHs and those of their spheroidal hoststellar systems. The duty cycle is found to be substantially less than unity,with characteristic values in the range (3-6)x10^(-3), and we compute that theaverage bolometric radiative efficiency is epsilon=0.07. Finally, we find thatthe growth in mass of individual black holes at high redshift (z>2) can bedominated by mergers, and is therefore not necessarily limited by accretion.Comment: Submitted to ApJ, 26 preprint pages with 3 figure

Reasoning From Fossils: Learning from the Local Black Hole Population about the Evolution of Quasars