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Black hole masses predicted from the Mbh-sigma relationship conflict withthose predicted from the Mbh-L relationship for the most luminous galaxies,such as brightest cluster galaxies (BCGs). This is because stellar velocitydispersion, sigma, increases only weakly with L for BCGs and other giantellipticals. The Mbh-L relationship predicts that the most luminous BCGs mayhave Mbh approaching 10^{10}M_sol, while the M-sigma relationship alwayspredicts Mbh<3X10^9M_sol. We argue that the Mbh-L relationship is a plausibleor even preferred description for BCGs and other galaxies of similarluminosity. If cores in central stellar density are formed by binary BHs, thenthe inner-core cusp radius, r_gamma, may be an independent witness of Mbh.Using structural parameters derived from a large sample of early-type galaxiesobserved by HST, we argue that L is superior to sigma as an indicator ofr_gamma in luminous galaxies. The observed r_gamma-Mbh relationship for 11 coregalaxies with measured Mbh appears to be consistent with the Mbh-L relationshipfor BCGs. BCGs have large cores appropriate for their large luminosities thatmay be difficult to generate with the modest BH masses inferred from theMbh-sigma relationship. Mbh~L may be expected to hold for BCGs, if they wereformed in dissipationless mergers, which should preserve ratio of BH to stellarmass. This picture appears to be consistent with the slow increase in sigmawith L and the more rapid increase in effective radii with L seen in BCGs. IfBCGs have large BHs commensurate with their luminosities, then the local BHmass function for Mbh>3X10^9M_sol may be nearly an order of magnitude richerthan that inferred from the Mbh-sigma relationship. The volume density of QSOsat earlier epochs may favor the predictions from the Mbh-L relationship.Comment: 62 pages, 18 figures, submitted to the Astrophysical Journal; revised  after initial revie

The Masses of Nuclear Black Holes in Luminous Elliptical Galaxies and Implications for the Space Density of the Most Massive Black Holes