Vtyjkyuk

I am trying to get a better understanding of tensor products of Hilbert spaces. Suppose that $x,yinmathbb H$, where $mathbb H$ is a Hilbert space. As far as I understand, we can think about the tensor $xotimes y$ as a bounded linear operator from $mathbb H$ to $mathbb H$ defined by
$$
(xotimes y)(z)=langle z,yrangle x
$$
for each $xinmathbb H$ and the tensor product $mathbb Hotimes mathbb H$ 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 $xotimes y$ as an element in a Hilbert space $HS(mathbb H)$ and we can consider a tensor $xotimes ytildeotimes zotimes w$ with $x,y,z,winmathbb H$ as an element in the tensor product $HS(mathbb H)tildeotimes HS(mathbb H)$.

Does it make sense to define the tensor $xotimes ytildeotimes zotimes w$ by setting
$$
(xotimes ytildeotimes zotimes w)(varphi)=langlevarphi,zotimes wrangle_mathrmHS(xotimes y)
$$
for each $varphiin HS(mathbb H)$, where $langlecdot,cdotrangle$ is a Hilbert-Schmidt inner product? This would mean that we can think about $mathbb Hotimes mathbb Htildeotimes mathbb Hotimes mathbb H$ as the space $HS(HS(mathbb H))$ and, if $varphi=uotimes v$ for some $u,v
inmathbb H$, we would have
$$
(xotimes ytildeotimes zotimes w)(uotimes v)
=langle uotimes v,zotimes wrangle_mathrmHS (xotimes y)
=langle u,zranglelangle w,vrangle(xotimes y).
$$

References (textbooks, papers, etc.) are very welcome. Any help is much appreciated!