We present results on low-resolution mid-infrared (MIR) spectra of 70 infrared-luminous galaxies obtained with the Infrared Spectrograph (IRS) onboard Spitzer. We selected sources from the European Large Area Infrared Survey (ELAIS) with S15 > 0.8 mJy and photometric or spectroscopic z > 1. About half of the sample are QSOs in the optical, while the remaining sources are galaxies, comprising both obscured AGN and starbursts. We classify the spectra using well-known infrared diagnostics, as well as a new one that we propose, into three types of source: those dominated by an unobscured AGN (QSOs), obscured AGN, and starburst-dominated sources. Starbursts concentrate at z ~ 0.6-1.0 favored by the shift of the 7.7-micron PAH band into the selection 15 micron band, while AGN spread over the 0.5 < z < 3.1 range. Star formation rates (SFR) are estimated for individual sources from the luminosity of the PAH features. An estimate of the average PAH luminosity in QSOs and obscured AGN is obtained from the composite spectrum of all sources with reliable redshifts. The estimated mean SFR in the QSOs is 50-100 Mo yr^-1, but the implied FIR luminosity is 3-10 times lower than that obtained from stacking analysis of the FIR photometry, suggesting destruction of the PAH carriers by energetic photons from the AGN. The SFR estimated in obscured AGN is 2-3 times higher than in QSOs of similar MIR luminosity. This discrepancy might not be due to luminosity effects or selection bias alone, but could instead indicate a connection between obscuration and star formation. However, the observed correlation between silicate absorption and the slope of the near- to mid-infrared spectrum is compatible with the obscuration of the AGN emission in these sources being produced in a dust torus.
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