TITLE: Tensor product of operators
QUESTION [11 upvotes]: We know that if $T_1$ is a linear bounded operator on a Hilbert space $H_1$ and $T_2$ is a linear bounded operator on a Hilbert space $H_2$ there exists a unique linear bounded operator $T$ on $H_1 \otimes H_2$ such that
$$ T (x_1 \otimes x_2) = T_1x_1 \otimes T_2x_2$$ for all $x_1$ in $H_1$ and $x_2$ in $H_2$. This operator is called a tensor product of operators $T_1$ and $T_2$ and denoted by $T_1 \otimes T_2$.
We can extend this to the arbitrary finite tensor product of Hilbert spaces and even infinite one which works with respect to some stabilising sequence. 
How about the converse? Is it true that if $T$ is a linear bounded operator on a Hilbert space $H_1 \otimes H_2$ (here of course $H_1$, $H_2$ also Hilbert spaces) then we can find operators (unique?) $T_1$ and $T_2$ on $H_1$, $H_2$ respectively, such that $Tu= (T_1 \otimes T_2)u$ for every $u \in H_1 \otimes H_2$. If not I would be grateful for any counterexamples. If it is true, how about the arbitrary tensor product and what about inifinite one?
Thank you for the help. Any suggestions for good books are welcome.

REPLY [5 votes]: When dealing with infinite dimension we have to be careful with our notation since  although the sets do not coincide, the closures (with respecto to any of the topologies below) do coincide.
Denoting with $\otimes _a$ the algebraic tensor product and without the subindex the closure w.r.t. any of the weak, $\sigma$-weak, strong, $\sigma$-strong, strong*, $\sigma$-strong* topologies, it is immediate that $L(H) \otimes_a L(H) \subset L(H \otimes H) \Rightarrow  L(H) \otimes L(H) \subset L(H \otimes H)$.
Now, take a $X\in L(H \otimes H)$. We will show that there exists a sequence in $L(H) \otimes_a L(H)$ weakly convergent to$X$. Consider  $\{e_i \otimes e_j \}_{ij}$ an orthonormal basis of $H \otimes H$. For any $n$ define the trunked operator \begin{align*}X_n&=\sum_{i,j,i^\prime,j^\prime}^n |e_{i^\prime} \otimes e_{j^\prime} \rangle \langle  e_{i^\prime} \otimes e_{j^\prime}| X |e_{i} \otimes e_{j}\rangle \langle e_{i} \otimes e_{j}|\\ &=\sum_{i,j,i^\prime,j^\prime}^n \langle e_{i^\prime} \otimes e_{j^\prime}, Xe_{i} \otimes e_{j} \rangle |e_{i^\prime} \rangle \langle e_i | \otimes |e_{j^\prime} \rangle \langle e_j |\in L(H) \otimes_a L(H). \end{align*} 
For arbitrary simple tensors $u\otimes v,z\otimes w\in H \otimes H$,
\begin{align*}\langle u \otimes v ,X_n z \otimes w \rangle& =\sum_{i,j,i^\prime,j^\prime}^n \big\langle e_{i^\prime} \otimes e_{j^\prime},X e_i \otimes e_j \big\rangle \big\langle u\otimes v, |e_{i^\prime}\rangle\langle e_i |z \otimes | e_{j^\prime} \rangle \langle e_j | w \big\rangle \\
&= \bigg \langle\sum_{i^\prime,j^\prime}^n \langle e_{i^\prime} \otimes e_{j^\prime},u\otimes v \rangle e_{i^\prime} \otimes e_{j^\prime} ,  X \sum_{i,j}^n \langle e_{i} \otimes e_{j}, z\otimes w \rangle e_{i} \otimes e_{j}  \bigg \rangle, \end{align*} from where it follows that $\langle u \otimes v ,X_n z \otimes w \rangle \xrightarrow[n]{\ \ \ \ \ } \langle u \otimes v ,X z \otimes w \rangle $ since $X$ is bounded.
While for general tensors $s,t\in H \otimes H$, $s=\sum_{m^\prime l^\prime} s_{m^\prime ,l^\prime}\ e_{m^\prime} \otimes e_{l^\prime}$, $t=\sum_{m,l} t_{m l}\ e_m\otimes e_l$, where $s_{m^\prime,l^\prime}$ y $t_{ml}$ are
complex numbers such that \begin{align*}\sum_{m^\prime, l^\prime} |s_{m^\prime, l^\prime}|^2<\infty,\ \ \ \  \sum_{m, l} |t_{m l}|^2<\infty,\end{align*}
\begin{align*}\langle s, X_n t \rangle &=\lim_{h^\prime,h\to\infty}\sum_{m^\prime, l^\prime, m,l}^{h^\prime,h} \overline{s}_{m^\prime l^\prime} t_{m l} \big\langle e_{m^\prime} \otimes e_{l^\prime}, X_n e_{m} \otimes e_{l} \big\rangle \\ 
&=\lim_{h^\prime,h\to\infty}\sum_{m^\prime, l^\prime, m,l}^{h^\prime,h} \overline{s}_{m^\prime l^\prime} t_{m l}\bigg \langle \sum_{i^\prime,j^\prime}^n \langle e_{i^\prime} \otimes e_{j^\prime}, e_{m^\prime} \otimes e_{l^\prime} \rangle e_{i^\prime} \otimes e_{j^\prime}, X \sum_{i,j}^n \langle e_i \otimes e_j, e_m \otimes e_l \rangle e_i \otimes e_j   \bigg \rangle \\
&=\sum_{i,j,i^\prime,j^\prime}^n \bigg \langle \langle e_{i^\prime} \otimes e_{j^\prime},s \rangle e_{i^\prime} \otimes e_{j^\prime}, X \langle e_i\otimes e_j,t \rangle e_i\otimes e_j \bigg \rangle \xrightarrow[n]{\ \ \ \ \ } \langle s, X t \rangle,  \end{align*} since $X$ is continuos. Hence $L(H\otimes H) \subset \overline{ L(H) \otimes_a L(H)}=L(H) \otimes L(H)$.
So not every operator $T$ can be written like $T_1 \otimes T_2$ , but as the limit of linear combination of simple tensors.