I am trying to get a better understanding of tensor products of Hilbert spaces. Suppose that $x,y\in\mathbb H$, where $\mathbb H$ is a Hilbert space. As far as I understand, we can think about the tensor $x\otimes y$ as a bounded linear operator from $\mathbb H$ to $\mathbb H$ defined by $$ (x\otimes y)(z)=\langle z,y\rangle x $$ for each $x\in\mathbb H$ and the tensor product $\mathbb H\otimes \mathbb H$ as the space of Hilbert-Schmidt operators from $\mathbb H$ to $\mathbb H$ (see here). Let us denote the space of Hilbert-Schmidt operators from $\mathbb H$ to $\mathbb H$ by $HS(\mathbb H)$.
It seems that we can think about $x\otimes y$ as an element in a Hilbert space $HS(\mathbb H)$ and we can consider a tensor $x\otimes y\tilde\otimes z\otimes w$ with $x,y,z,w\in\mathbb H$ as an element in the tensor product $HS(\mathbb H)\tilde\otimes HS(\mathbb H)$.
Does it make sense to define the tensor $x\otimes y\tilde\otimes z\otimes w$ by setting $$ (x\otimes y\tilde\otimes z\otimes w)(\varphi)=\langle\varphi,z\otimes w\rangle_{\mathrm{HS}}(x\otimes y) $$ for each $\varphi\in HS(\mathbb H)$, where $\langle\cdot,\cdot\rangle$ is a Hilbert-Schmidt inner product? This would mean that we can think about $\mathbb H\otimes \mathbb H\tilde\otimes \mathbb H\otimes \mathbb H$ as the space $HS(HS(\mathbb H))$ and, if $\varphi=u\otimes v$ for some $u,v \in\mathbb H$, we would have $$ (x\otimes y\tilde\otimes z\otimes w)(u\otimes v) =\langle u\otimes v,z\otimes w\rangle_{\mathrm{HS}} (x\otimes y) =\langle u,z\rangle\langle w,v\rangle(x\otimes y). $$
References (textbooks, papers, etc.) are very welcome. Any help is much appreciated!