I wonder if someone could help me to understand the following sentence in the second last paragraph below: "Then $0\leq q\leq 1.$"
This extract is taken from the book of E. Christopher Lance on Hilbert modules.
The context is that $t$ is a densely defined regular operator between Hilbert modules $E$ and $F$. We know that the extension of the inverse $(1+t^*t)^{-1}$, which is referred to as $q$, is a positive element, which explains the first half of that sentence. But I'm not sure why $q\leq 1$, since $1+t^*t$ itself is not an element of $\mathcal{L}(E)$ so you can't just invert.
Many thanks for your help.
From the second sentence of the second paragraph, $(1+t^*t)^{-1}$ is norm-reducing (on its domain). Hence its bounded extension is norm-reducing, and so is this extension's square root, which we call $q$. In other words, $\|q\|\leq1$, which implies $q\leq1$.