$T^p$ increases to $T$ in strongly operator topology or not.

Question

In a Hilbert space $H$, let $T$ be a positive operator on $H$ with $\|T\|_\infty\le 1$. Then, obviously, $T^p$ is increasing as $p$ decreases to 1. But I am not sure whether $T^p$ increases to $T$ in strongly operator topology or not.

Answer 1

Yes, this follows from the monotone convergence theorem for nets (see Theorem IV.15 of Reed and Simon's Functional Analysis, for instance)

Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$ and let $(f_{\alpha})$ be an increasing net of continuous functions converging pointwise to $f$. If $\sup \|f_{\alpha}\|_1 < \infty$, then $\lim_{\alpha} \|f_{\alpha} - f\|_1 = 0$.

Now, if $X = \sigma(T) \subset [0,1]$ as in your case, and $x \in H$, then there is a regular Borel measure $\mu_x$ on $X$ such that $$ \langle f(T)x,x\rangle = \int_X fd\mu_x $$ So if we take $f_p(x) = x^p-x$ for $p \in (1,\infty)$, then $(f_p^2)$ satisfies the conditions of the above theorem with $f=0$. Hence, using the fact that $$ \|(T^p-T)x\|^2 = \|f_p(T)x\|^2 = \langle f_p(T)x,f_p(T)x\rangle = \langle f_p^2(T)x,x\rangle = \int_X f_p^2d\mu_x $$ we see that $\|(T^p - T)x\| \to 0$ as $p\to 1$