  The polar factor of a bounded operator acting on a Hilbert space is the
unique partial isometry arising in the polar decomposition. It is well known
that the polar factor might not be a best approximant to its associated
operator in the set of all partial isometries, when the distance is measured in
the operator norm. We show that the polar factor of an arbitrary operator $T$
is a best approximant to $T$ in the set of all partial isometries $X$ such that
$\dim (\ker(X)\cap \ker(T)^\perp)\leq \dim (\ker(X)^\perp\cap \ker(T))$. We
also provide a characterization of best approximations. This work is motivated
by a recent conjecture by M. Mbekhta, which can be answered using our results.
