Hyponormal measurable operators, affiliated to a semifinite von Neumann algebra

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