# Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators

Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the separable Hilbert space of Hilbert Schmidt operators. Then, e.g. $(\phi_{ij})$ is a ONB of $\mathcal{S_H}$ with $\phi_{ij}(e_k) = e_i\delta_{jk},\,\forall k,$ where $\delta_{jk}$ denotes the Kronecker-Delta.

Moreover, for $p \in \mathbb{N},$ let's consider the $p$-dimensional case. $(\mathcal{S}^p_{\mathcal{H}},\langle\cdot, \cdot\rangle_{\mathcal{S}^p_{\mathcal{H}}})$ can be considered as the space of vectors where each component can be considered as a Hilbert Schmidt operater from $\mathcal{H} \rightarrow \mathcal{H}$ (in other words every component is an element of $\mathcal{S_H})$ and for $\boldsymbol{s} \colon\!= (s_1, ..., s_p)^T,$ $\boldsymbol{t} \colon\!=(t_1, ..., t_p)^T \in \!\mathcal{S}^p_{\mathcal{H}}\colon \langle\boldsymbol{s}, \boldsymbol{t}\rangle_{\mathcal{S}^p_{\mathcal{H}}} \colon\!= \sum_{i=1}^p\langle s_i, t_i\rangle_{\mathcal{S_H}}.$ Also, $(E_j)_j = ((E^{(1)}_j\!\!\!, ..., E^{(p)}_j)^T)$ is a ONB of $(\mathcal{H}^p, \langle\cdot, \cdot\rangle_{\mathcal{H}^p})$ where $\langle\boldsymbol{x}, \boldsymbol{y}\rangle_{\mathcal{H}^p} \colon\!= \sum_{i=1}^p\langle x_i, y_i\rangle_{\mathcal{H}}, \boldsymbol{x} \colon\!= (x_1, ..., x_p)^T \in\mathcal{H}^p$ and $\boldsymbol{y} \colon\!=(y_1, ..., y_p)^T \in \!\mathcal{H}^p.$

Question: Am I able (and how?) to construct a ONB of $\mathcal{S}^p_{\mathcal{H}}$ given a ONB $(E_j)$ of $\mathcal{H}^p$ ?

Thank you in advance!

If $(E_j)$ is a ONB of $\mathcal{H}^p,$ then I can choose a subsequence $(j_k)_k \subseteq \mathbb{N}$ such that $(E^{(1)}_{j_k})_k$ is maximal linear independent. Then the orthonormalised version of $(E^{(1)}_{j_k})_k,$ denoted as $(e_k),$ is a ONB of $\mathcal{H}$ which leads to a ONB $(\phi_{ij})$ of $\mathcal{S_H}$ defined by $\phi_{ij}(e_k) = e_i\delta_{jk},\,\forall i,j,k.$
Answer A ONB of $\mathcal{S}^p_{\mathcal{H}}$ given $(E_j)$ could be: $(\Phi_{ijl}) \colon\!= \big((0, \dots, \phi_{ij}, 0, ..., 0)^T\big),$ where $\phi_{ij}$ is located at the $l$-th position, since $\forall i, i', j, j' \in \mathbb{N}$ and $\forall l, l' \in \{1, ..., p\}\colon$ $$\left\langle \Phi_{ijl}, \Phi_{i'j'l'}\right\rangle_{\mathcal{S}^p_{\mathcal{H}}} = \sum_{k=1}^p \left\langle \Phi^{(k)}_{ijl}, \Phi^{(k)}_{i'j'l'}\right\rangle_{\mathcal{S_H}} \!\!\!= \delta_{ll'} \left\langle\Phi^{(l)}_{ijl}, \Phi^{(l)}_{i'j'l}\right\rangle_{\mathcal{S_H}} \!\!\!= \delta_{ll'}\left\langle \phi_{ij}, \phi_{i'j'}\right\rangle_{\mathcal{S_H}} = \delta_{ll'}\delta_{ii'}\delta_{jj'},$$ where the last step follows from the fact that $(\phi_{ij})$ is a ONB.