Hello,

I have a question about trace measurable operators and I think it's not a hard one. However, I'm quite confused because I cannot prove it.

Let $\mathcal{M}$ be a semi-finite von Neumann algebra with a faithful normal semi-finite trace $\tau$. Let $T$ be a $\tau$- measurable operator (densely defined closed (possibly unbounded) operator affiliated with $\mathcal{M}$ such that $$ \forall_{\varepsilon >0} \ \exists_{E - \text{a projection in} \text{M}} \ \mbox{Range}(E) \subset D(T) \ \& \ \tau(1-E) \leq \varepsilon.)$$

Let $E_{(s,\infty)}(|T|)$ be a spectral projection of $|T|$ corresponding to the interval $(s, \infty)$, $s \geq 0$.

How do we know that $\| |T|E_{(s,\infty)}(|T|) \| > s$ or $\| |T|E_{[0,s]}(|T|) \| \leq s$.

Thank you in advance for any help.