The unitary part of class $\mathcal {F}$ contractions

Abstract

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.