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Let $H$ be a separable Hilbert space and $\mathcal{L}(H)$ the set of bounded operators on $H$. If $D \in \mathcal{L}(H)$ is a positive operator, $D \geq 0$, is true that the set $$\{T \in \mathcal{L}(H)\mid T^*T \leq D\}$$ is closed in the weak operator topology (WOT) on $\mathcal{L}(H)$?

Note that the mapping $T \mapsto T^*T$ is not continuous in WOT, like the counterexample $H = \ell^2$, $T_n = S^n$ ($S$ - unilateral (right) shift) shows.

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    $\begingroup$ Do you already know that the norm on a Hilbert space is weakly lower semicontinuous? $\endgroup$
    – MaoWao
    Commented Jul 23, 2020 at 16:27
  • $\begingroup$ @MaoWao Yes, this is well known $\endgroup$
    – Andrei Kh
    Commented Jul 23, 2020 at 16:59

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Suppose $T_\alpha \to T$ in WOT and $T_\alpha^*T_\alpha≤D$. What we would like to have is that $T^*T^\mathstrut≤D$, ie $$\langle x , (D-T^*T^{\mathstrut})x\rangle=\langle x,Dx\rangle - \|Tx\|^2\overset!≥0$$ for all $x$. Now we may note that $\|Tx\|=\sup_{\|y\|≤1}|\langle y, Tx\rangle|$, while $$|\langle y, Tx\rangle |= \lim_\alpha |\langle y, T_\alpha x\rangle|≤\liminf_\alpha \|T_\alpha x\|$$ if $\|y\|≤1$, giving $\|Tx\|^2≤ \liminf_\alpha\|T_\alpha x\|^2$. Now using $\langle x ,(D-T^*_\alpha T_\alpha)x\rangle ≥0$ we find: $$\langle x, Dx\rangle -\|Tx\|^2 ≥ \langle x, Dx\rangle -\liminf_\alpha \|T_\alpha x\|^2 =\limsup_\alpha \langle x, (D-T^*_\alpha T_\alpha)x\rangle$$ but for every $\alpha$ the term on the right is $≥0$, giving us the conclusion we wanted.

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  • $\begingroup$ I don't understand how $T \mapsto T^*T$ is continuous here: If $S$ is the right shift on $\ell^2$ and $T_n = S^n$, then certainly $T_n$ is bounded, as $||T_n|| \leq 1$ and $T_n \to 0$ in WOT, but $T_n^*T_n = I$, so clearly $T_n^* T_n \not \to 0$ here $\endgroup$
    – Andrei Kh
    Commented Jul 23, 2020 at 18:25
  • $\begingroup$ You are right, I confused myself. Multiplication is not jointly WOT continuous on bounded sets, rather it is WOT x SOT -> WOT continuous on bounded sets. I'll edit the answer. $\endgroup$
    – s.harp
    Commented Jul 23, 2020 at 18:30

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