We use ROSAT HRI spatial data and ASCA spectral measurements for a sample of seven nearby, early type spiral galaxies, to address the question of whether a low-luminosity Active Galactic Nucleus (LLAGN) is present in galaxies that have a LINER 2 classification. The brightest discrete X-ray source in the ROAST HRI observations is invariably found to be positionally coincident with the optical galactic nucleus, and in most cases its flux dominates the X-ray emission from the central region of the galaxy. All seven galaxies have X-ray spectra consistent with a two-component, soft thermal plus hard power-law, spectral form. If we exclude the two galaxies with relatively hard X-ray spectra, NGC 3628 and NGC 4594, for which there is supporting evidence for a LLAGN (or alternatively in the case of NGC 3628 a dominant ultra-luminous X-ray binary), then the remaining galaxies show surprisingly similar X-ray spectral properties. Specifically the flux ratio F_x(0.5-1)/F_x(2-5), which measures the relative strengths of the thermal and non-thermal emission components, shows little scatter about a mean of 0.66, a value very similar to that measured in the classic starburst galaxy NGC 253. Since there is no obvious reason why the luminosity of the hard power-law continuum emanating from a putative LLAGN should be very closely correlated with the thermal emission of the surrounding region, this suggests that that the broad-band (0.5-5 keV) X-ray emission from these LINER 2 galaxies may originate in a common set of processes probably associated with the starburst phenomenon. Conversely, it appears that in many, perhaps the majority, of LINER 2 galaxies, the nuclear X-ray luminosity does not derive directly from the presence of a LLAGN.
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