We say that a bounded linear operator $T$ on a Hilbert space
$\mathcal{H}$ belongs to the class $\mathcal{F}$ if $T$ satisfies
the following Fuglede's property that, for a given isometry
$W$ on $\mathcal{H}$, $SW^*=TS$ for some bounded linear operator
$S$ on $\mathcal{H}$ always implies $SW=T^*S$. Such class
is wider than the class of paranormal contractions, the class
of dominant operators and the class $\mathcal{Y}$ which was
introduced in [4]. In this paper, we prove that, for the class
$\mathcal{F}$ contraction $T$ on $\mathcal{H}$, the positive
square root $A_{T^*}$ of the strong limit of $T^nT^{*n}$ is
the projection from $\mathcal{H}$ onto $\mathcal{H}_T^{(u)}$
on which the unitary part of $T$ acts.