  We present a study of the thermal structure of the hot X-ray emitting
atmospheres for a sample of 49 nearby X-ray and optically bright elliptical
galaxies using {\it Chandra} X-ray data. We focus on the connection between the
properties of the hot X-ray emitting gas and the cooler H$\alpha$+[NII]
emitting phase, and the possible role of the latter in the AGN (Active Galactic
Nuclei) feedback cycle. We do not find evident correlations between the
H$\alpha$+[NII] emission and global properties such as X-ray luminosity, mass
of hot gas, and gas mass fraction. We find that the presence of H$\alpha$+[NII]
emission is more likely in systems with higher densities, lower entropies,
shorter cooling times, shallower entropy profiles, lower values of min($t_{\rm
cool}/t_{\rm ff}$), and disturbed X-ray morphologies (linked to turbulent
motions). However, we see no clear separations in the observables obtained for
galaxies with and without optical emission line nebulae. The AGN jet powers of
the galaxies with X-ray cavities show hint of a possible weak positive
correlation with their H$\alpha$+[NII] luminosities. This correlation and the
observed trends in the thermodynamic properties may result from chaotic cold
accretion (CCA) powering AGN jets, as seen in some high-resolution hydrodynamic
simulations.
