Let $H$ be a complex Hilbert space and $\mathcal{L}(H)$ be the algebra of all bounded linear operators on $E$.

If $A,B\in \mathcal{L}(H)$, It is true that $\overline{\text{Im}(A)}\otimes \overline{\text{Im}(B)}\subset \overline{\text{Im}(A\otimes B)}$?

I try as follows:

Let $z\in \overline{\text{Im}(A)}\otimes \overline{\text{Im}(B)}$, then $z=\displaystyle\sum_{i=1}^dx_i\otimes y_i$, where $x_i\in\overline{\text{Im}(A)},\;y_i\in\overline{\text{Im}(B)}$ and $d\in\mathbb{N}$. So, there exists $(\alpha_i(n))_{n\in\mathbb{N}}\subset \text{Im}(A)$ and $(\beta_i(n))_{n\in\mathbb{N}}\subset \text{Im}(B)$ such that $x_i=\lim_{n\rightarrow\infty}\alpha_i(n)$ and $y_i=\displaystyle\lim_{n\rightarrow\infty}\beta_i(n)$. Moreover, $\alpha_i(n)=A\tilde{\alpha}_i(n),\;\beta_i(n)=B\tilde{\beta}_i(n)$ and $$\alpha_i(n)\otimes\beta_i(n)=(A\otimes B)(\tilde{\alpha}_i(n)\otimes \tilde{\beta}_i(n))\subset \text{Im}(A\otimes B),$$ where $(\tilde{\alpha}_i(n))_{n\in\mathbb{N}}\subset \mathcal{H}$ and $(\tilde{\beta}_i(n))_{n\in\mathbb{N}}\subset \mathcal{H}$. Let $$z_n=\displaystyle\sum_{i=1}^d\alpha_i(n)\otimes \beta_i(n)\in \text{Im}(A\otimes B).$$ Since $\|x\otimes y\|=\|x\|\|y\|$, then the map $(x,y)\longmapsto x\otimes y$ is continuous. So $z_n\longrightarrow z$ as $n\longrightarrow \infty$, this implies that $z\in \overline{\text{Im}(A\otimes B)}$. As a consequence, $\overline{\text{Im}(A)}\otimes \overline{\text{Im}(B)}\subset \overline{\text{Im}(A\otimes B)}$.