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Let

  • $U$, $H$ be separable $\mathbb R$-Hilbert spaces
  • $\mathfrak L(A,B)$ denote the space of bounded linear operators from $A$ to $B$
  • $Q\in\mathfrak L(U,U)$ be nonnegative and self-adjoint with finite trace, i.e. $$\operatorname{tr}Q\stackrel{\text{def}}=\sum_{n\in\mathbb N}\langle Qe_n,e_n\rangle_U<\infty\tag 1$$
  • $(e^n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\in\mathbb N\tag 2$$ for some $(\lambda_n)_{n\in\mathbb N}\subseteq[0,\infty)$
  • $L\in\mathfrak L(U,\mathfrak L(U,H))$
  • $h\in H$ and $$B_h(u,v):=\langle(Lu)v,h\rangle_H\;\;\;\text{for }u,v\in U$$

It's easy to see that $B_h$ is a bilinear functional on $U\times U$ with $$\left|B_h(u,v)\right|\le\left\|L\right\|_{\mathfrak L(U,\:\mathfrak L(U,\:H))}\left\|u\right\|_U\left\|v\right\|_U\left\|h\right\|_H\;\;\;\text{for all }u,v\in U\tag 3$$ and hence there is a unique bounded linear operator $\tilde L_h\in\mathfrak L(U)$ with $$B_h(u,v)=\langle v,\tilde L_hu\rangle_U\;\;\;\text{for all }u,v\in U\tag 4$$ by Riesz' representation theorem. Note that $$\operatorname{tr}Q\tilde L_h=\sum_{n\in\mathbb N}\lambda_nB_h(e^n,e^n)\tag 5$$ and hence $$\left|\operatorname{tr}Q\tilde L_h\right|\le\left\|L\right\|_{\mathfrak L(U,\:\mathfrak L(U,\:H))}\left\|h\right\|_H\operatorname{tr}Q<\infty\;.\tag 6$$ Again, by Riesz' representation theorem, we obtain a unique $\tilde{\operatorname{tr}}\:QL\in H$ (be careful, I just use $\tilde{\operatorname{tr}}\:QL$ as a suggestive symbol) with $$\operatorname{tr}Q\tilde L_h=\langle h,\tilde{\operatorname{tr}}\:QL\rangle_H\;.\tag 7$$ Now, let $$x_N:=\sum_{n=1}^N(Le^n)(\lambda_ne^n)$$ for $N\in\mathbb N$. $(4)$, $(5)$ and $(7)$ imply that $$\langle h,x_N\rangle_H\xrightarrow{N\to\infty}\langle h,\tilde{\operatorname{tr}}\:QL\rangle_H\;,\tag 8$$ i.e. $$x_N\xrightarrow{N\to\infty}\tilde{\operatorname{tr}}\:QL\tag 9$$ $\color{red}{\text{weakly}}$ in $H$, for all $h$.

How can we show that the convergence in $(9)$ holds in the strong sense too?

From the general theory, I know that we only need to show that $$\limsup_{N\to\infty}\left\|x_N\right\|_H\le\left\|\tilde{\operatorname{tr}}\:QL\right\|_H\;,\tag{10}$$ but for some reason I'm not able to obtain this relation.

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