  We use a sample of 47 homogeneous and high sensitivity CO images taken from
the Nobeyama and BIMA surveys to demonstrate that, contrary to common belief, a
significant number (~40%) of HI-deficient nearby spiral galaxies are also
depleted in molecular hydrogen. While HI-deficiency by itself is not a
sufficient condition for molecular gas depletion, we find that H2 reduction is
associated with the removal of HI inside the galaxy optical disk. Those
HI-deficient galaxies with normal H2 content have lost HI mainly from outside
their optical disks, where the H2 content is low in all galaxies. This finding
is consistent with theoretical models in which the molecular fraction in a
galaxy is determined primarily by its gas column density. Our result is
supported by indirect evidence that molecular deficient galaxies form stars at
a lower rate or have dimmer far infrared fluxes than gas rich galaxies, as
expected if the star formation rate is determined by the molecular hydrogen
content. Our result is consistent with a scenario in which, when the atomic gas
column density is lowered inside the optical disk below the critical value
required to form molecular hydrogen and stars, spirals become quiescent and
passive evolving systems. We speculate that this process would act on the
time-scale set by the gas depletion rate and might be a first step for the
transition between the blue and red sequence observed in the color-magnitude
diagram.
