Hyponormal measurable operators, affiliated to a semifinite von Neumann algebra

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\) and \(\tau \) be a faithful normal semifinite trace on \(\mathcal {M}\), \(S(\mathcal {M}, \tau )\) be the \( ^*\)-algebra of all \(\tau \)-measurable operators. Assume that an operator \(T\in S(\mathcal {M}, \tau )\) is paranormal or \( ^*\)-paranormal. If \(T^n\) is \(\tau \)-compact for some \(n\in \mathbb {N}\) then T is \(\tau \)-compact; if \(T^n=0\) for some \(n\in \mathbb {N}\) then \(T=0\); if \(T^3=T\) then \(T=T^*\); if \(T^2\in L_1(\mathcal {M}, \tau )\) then \(T\in L_2(\mathcal {M}, \tau )\) and \(\Vert T\Vert _2^2=\Vert T^2\Vert _1\). If an operator \(T\in S(\mathcal {M}, \tau )\) is hyponormal and \(T^{*p}T^q\) is \(\tau \)-compact for some \(p, q \in \mathbb {N}\cup \{0\}\), \(p+q \ge 1\) then T is normal. If \(T\in S(\mathcal {M}, \tau )\) is p-hyponormal for some \(0<p\le 1\) then the operator \((T^*T)^p-(TT^*)^p\) cannot have the inverse in \( \mathcal {M}\). If an operator \(T\in S(\mathcal {M}, \tau )\) is hyponormal (or cohyponormal) and the operator \(T^2\) is Hermitian then T is normal.