Consider the Banach algebra $B\_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$ with Hilbert Schmidt norm. We know that $B\_2(H)$ is a Hilbert space as well with $\left<A,B\right>=tr(B^\*A)$.I am looking for an example of pair of sequences $A\_i,\tilde A\_j$ and $T\_i,\tilde T\_j$ in the closed unit ball of $B\_2(H)$ and $B(B\_2(H))$ respectively and a $D\in B\_2(H)$ such that both the iterated limits of $\left<T\_i\tilde T\_j(A\_i\tilde A\_j),D\right>$ exists but $$\lim\_i\lim\_j \left<T\_i\tilde T\_j(A\_i\tilde A\_j),D\right>\neq 
\lim\_j\lim\_i \left<T\_i\tilde T\_j(A\_i\tilde A\_j),D\right>$$
 Otherwise prove that if iterated limits exists then they are equal.



Iterated limits equal?
For $\xi,\eta\in H$ let $\theta\_{\xi,\eta}$ be the rank-one operator $\theta\_{\xi,\eta}(\gamma) = (\gamma|\eta) \xi$ for $\gamma\in H$.


Let $(e\_i)$ be an orthonormal sequence in $H$, set $S\_i = \theta\_{e\_1, e\_i} \in B(H)$ and let $R\_j$ be the projection onto the span of $\{e\_1,e\_2,\cdots,e\_j\}$. Then
$$ \lim\_i \lim\_j (S\_iR\_j(e\_i)|e\_1) = \lim\_i (S\_i(e\_i)|e\_1) = \lim\_i (e\_1|e\_1) = 1, $$
while
$$ \lim\_j \lim\_i (S\_iR\_j(e\_i)|e\_1) = \lim\_j \lim\_i (R\_j(e\_i)|e\_i) (e\_1|e\_1) = 0. $$


Now we "embed" this into your example. Let $\tilde A\_j = \theta\_{e\_1,e\_1}$ for all $j$, and set $A\_i = \theta\_{e\_i,e\_1}$ so $A\_i \tilde A\_j = A\_i$. Define $\tilde T\_j(a) = \theta\_{R\_j(a(e\_1)), e\_1}$ for $a\in B\_2(H)$; a simple estimate shows that $\tilde T\_j$ is bounded. Define $T\_i(a) = \theta\_{S\_i(a(e\_1)),e\_1}$. Then
$$
\tilde T\_j (a) (e\_1) = \theta\_{R\_j(a(e\_1)), e\_1} (e\_1)
= R\_j(a(e\_1)) \qquad (a\in B\_2(H)). $$
Thus
$$ T\_i \tilde T\_j(a) = \theta\_{S\_i R\_j(a(e\_1)), e\_1} \implies
T\_i \tilde T\_j (A\_i \tilde A\_j) = T\_i \tilde T\_j(\theta\_{e\_i,e\_1})
= \theta\_{S\_iR\_j(e\_i),e\_1}. $$
If we now let $D=\theta\_{e\_1,e\_1}$ then
$$ \Big\langle T\_i \tilde T\_j (A\_i \tilde A\_j), D \Big\rangle
= \operatorname{Tr}\Big( \theta\_{S\_i R\_j(e\_i),e\_1} \Big)
= (S\_i R\_j(e\_i) | e\_1). $$
As above, this has different iterated limits.


