For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows :
$$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(x_1,\ldots, x_N)dx_1\cdots dx_N \right) \Psi,\quad \forall \Phi\in H^N.$$
Let $H^N=H_1\otimes H_2$ with $H_1:=H^k$ and $H_2:=H^{N-k}$ for some $1\le k\le N$. What are the partial traces on $H_1$ of
$$D,~~ \nabla D,~~ \Delta D,~~ V_{ij}D,~~ K_i^*K_iD,~~ K_iDK_i^*?$$
$\nabla$, $\Delta$ denote respectively the gradient and Laplacian operators. $V_{ij}$ and $K_i$ (with $K_i^*$ its adjoint operator) denote the multiplication operators identified by (bounded) functions $k:\mathbb R^3\to\mathbb C$ and $v:\mathbb R^3\times\mathbb R^3\to\mathbb C$ defined by
$$V_{ij}(\Phi):=V(x_i,x_j)\Phi(x_1,\ldots, x_N),\quad K_{i}(\Phi):=V(x_i)\Phi(x_1,\ldots, x_N),\quad \forall \Phi\in H^N.$$
I'm looking for references on the partial trace operator. Most books that I find are for physics and quantum mechanics and for finite dimension, e.g.
http://www.ueltschi.org/AZschool/notes/EricCarlen.pdf
https://web.math.princeton.edu/~lieb/cm-summerschool.pdf
The operator theory books don't include the computation of the partial trace. I appreciate very much the concrete calculus.
