  Band structure of metallic sodium cobaltate Na$_x$CoO$_2$ ($x$=0.33, 0.48,
0.61 0.72) has been investigated by local density approximation+Hubbard $U$
(LDA+$U$) method and within Gutzwiller approximation for the Co-$t_{2g}$
manifold. Correlation effects being taken into account results in suppression
of the $e'_g$ hole pockets at the Fermi surface in agreement with recent
angle-resolved photo-emission spectroscopy (ARPES) experiments. In the
Gutzwiller approximation the bilayer splitting is significantly reduced due to
the correlation effects. The formation of high spin (HS) state in Co $d$-shell
was shown to be very improbable.
