  We present a multi-wavelength study of the molecular gas properties of a
sample of local Seyfert 2 galaxies to assess if, and to what extent, the
presence of an active galactic nucleus (AGN) can affect the Interstellar Medium
(ISM) properties in a sample of 33 local Seyfert 2 galaxies. We compare the
molecular gas content (MH2), derived from new and archival low-J CO line
measurements of a sample of AGN and a control sample of star-forming galaxies
(SFGs). Both the AGN and the control sample are characterised in terms of
host-galaxy properties (e.g., stellar and dust masses, Mstar and Mdust,
respectively; and star formation rate, SFR). We also investigate the effect of
AGN activity onto the emission of polycyclic aromatic hydrocarbon (PAH)
molecules in the mid-infrared (MIR), a waveband where the dust-reprocessed
emission from the obscured AGN contributes the most. The AGN hosted in less
massive galaxies (i.e., log(Mstar/Msun)<10.5; log(Mdust/Msun)<7.5) show larger
molecular gas content with respect to SFGs matched in stellar and dust mass.
When comparing their depletion time(tdep~MH2/SFR), AGN show tdep~0.3-1.0 Gyr,
similar to those observed in the control sample of SFGs. Seyfert 2 galaxies
show fainter PAH luminosity at increasingly larger dominance of the nuclear
activity in the MIR. We find no clear evidence for a systematic reduction of
the molecular gas reservoir at galactic scale in Seyfert galaxies with respect
to SFGs. This is in agreement with recent studies showing that molecular gas
content only is reduced in regions of sub-kpc size, where the emission from the
accreting supermassive black hole dominates. Nonetheless, we show that the
impact of AGN activity on the ISM is clearly visible as suppression of the PAH
luminosity.
