Tail of increasing convergent net of self-adjoint operators is bounded

Question

Let $H$ be a Hilbert space and $(T_\alpha)$ an increasing net of self-adjoint operators that converges (in some topology) to an operator $T$. Then $(T_\alpha)$ is not necessarily norm-bounded I think (is this correct?). Assuming that the index set is non-empty, for any $\alpha_0$ we can consider the net $(T_\alpha)_{\alpha\geq\alpha_0}$. I want to prove that this net is actually norm-bounded. I think that $T_{\alpha_0}\leq T_\alpha\leq T$ and that $$-\|T_{\alpha_0}\|\leq\|T_\alpha\|\leq\|T\|$$ for all $\alpha\geq\alpha_0$, but I don't know how to make this precise or mathematically rigorous. In particular, I just assumed $T_\alpha\leq T$, since this is natural for increasing convergent sequences of real numbers. I'm not really familiar with nets. Any help would be greatly appreciated!

Answer 1

"Some topology" is pretty vague, and for sure one can come up with pathologic topologies where $T$ is not even normal. But since the tag is "strong convergence", that's what I'll assume.

Indeed the net needs not be bounded. This is already true in dimension one: take $(T_\alpha)_{\alpha\in\mathbb Z}$, with $$ T_\alpha=\begin{cases} \alpha,&\ \alpha\leq 0\\[0.3cm] 1-\tfrac1\alpha,&\ \alpha>0\end{cases}. $$ Then $T_\alpha\to1$, but the net is not bounded.

Regarding the tail, indeed from $T_{\alpha_0}\leq T_\alpha\leq\|T\|$ you can deduce that the tail of the net is bounded. For this you may use that $$\tag1 \|T_\alpha\|=\sup\{\langle T_\alpha x,x\rangle:\ \|x\|=1\}. $$ From $(1)$ you see that $\|T_\alpha\|\leq\max\{\|T\|,\|T_{\alpha_0}\|\}$.