Nuclear hard X-ray luminosities (Lx,nuc) for a sample of 112 early type galaxies within a distance of 67 Mpc are used to investigate their relationship with the central galactic black hole mass Mbh, the inner galactic structure (using the parameters describing its cuspiness), the age of the stellar population in the central galactic region, the hot gas content and the core radio luminosity. Lx,nuc ranges from 10^{38} to 10^{42} erg/s, and the Eddington ratio Lx,nuc/Ledd from 10^{-9} to 10^{-4}. Lx,nuc increases on average with the galactic luminosity Lb and Mbh, with a wide variation by up to 4 orders of magnitude at any fixed Lb>6x10^9 Lb,sun or Mbh>10^7 Msun. This large range should reflect a large variation of the mass accretion rate dotMbh. On the circumnuclear scale, dotMbh at fixed Lb (or Mbh) could vary due to differences in the fuel production rate from the stellar mass return linked to the inner galactic structure; however, dotMbh should vary with cuspiness by a factor exceeding a few only in hot gas poor galaxies and for large differences in the core radius. Lx,nuc does not depend on age, but less luminous nuclei are found among galaxies with a younger stellar component. Lx,nuc is detected both in gas poor and gas rich galaxies, on average increases with the total galactic hot gas cooling rate L_{X,ISM}, but again with a large variation. The lack of a tight relationship between Lx,nuc and the circumnuclear and total gas content can be explained if the gas is heated by black hole feedback, and/or the mass effectively accreted can be largely reduced with respect to that entering the circumnuclear region. Differently from Lx,nuc, the 5 GHz VLA luminosity shows a trend with the inner galactic structure similar to that of the total soft X-ray emission; therefore they could both be produced by the hot gas.
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