Tensor products of Hilbert spaces

Show that $\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}}$ is exact in $H \otimes K$, and its biorthogonal system is $\left\{ a_{m} \otimes b_{n}\right\}_{m, n \in \mathbb {N}}$.

Prove the following relationships among the partial sum operators (where we let $S_{0}^{X}$ and $S_{0}^{Y}$ be the zero operators):

Show that $\left\| S_{N}^{Z}\right\| \leq 3 \mathcal{C}_{X} \mathcal{C}_{Y}$ for each $N \in \mathbb {N}$, and conclude that $\left\{ z_{k}\right\}$ is a basis for $H \otimes K$.

Let $\left\{ x_{n}\right\} ,\left\{ y_{n}\right\}$ be frames for Hilbert spaces $H, K$, respectively. Show that the tensor product sequence $\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}}$ is a frame for $H \otimes K = \mathcal{B}_{2}(H, K)$ (see Appendix B for definitions).

Let $E, F$ be measurable subsets of $\mathbb {R}$. Show that if $\left\{ f_{n}\right\}$ is a frame for $L^{2}(E)$ and $\left\{ g_{n}\right\}$ is a frame for $L^{2}(F)$, then $\left\{ \overline{f_{m}(x)} g_{n}(y)\right\}_{m, n \in \mathbb {N}}$ is a frame for $L^{2}(E \times F)$.

$T \in \mathcal{B}(H, K)$ has rank one if and only if $T = x \otimes y$ for some $x \in H, y \in K$ not both zero.

$T \in \mathcal{B}(H, K)$ has finite rank if and only if $T = \sum_{k = 1}^{N} x_{k} \otimes y_{k}$ for some $N \in \mathbb {N}$, $x_{k} \in H$, and $y_{k} \in K$.

$\left\{ x \otimes y: x \in H, y \in K\right\}$ is complete in $\mathcal{B}_{2}(H, K)$.

If $x_{1}, x_{2} \in H$ and $y_{1}, y_{2} \in K$, then $\left\langle x_{1} \otimes y_{1}, x_{2} \otimes y_{2} \right\rangle_{\mathrm{HS}} = \left\langle x_{2}, x_{1} \right\rangle_{H} \left\langle y_{1}, y_{2} \right\rangle_{K}$.

If $\left\{ e_{n}\right\}$ is an orthonormal basis for $H$ and $\left\{ f_{n}\right\}$ is an orthonormal basis for $K$, then $\left\{ e_{m} \otimes f_{n}\right\}_{m, n \in \mathbb {N}}$ is an orthonormal basis for $\mathcal{B}_{2}(H, K)$.

If $T \in \mathcal{B}(H, K)$, then $\left\langle T, e_{m} \otimes f_{n} \right\rangle = \left\langle T e_{m}, f_{n} \right\rangle$ for every $m, n \in \mathbb {N}$.

$\left\| x \otimes y\right\|_{\mathrm{HS}} = \left\| x\right\|_{H}\left\| y\right\|_{K}$.

$(a x) \otimes y = \bar{a}(x \otimes y)$ and $x \otimes (b y) = b(x \otimes y)$.

$(x+w) \otimes y = (x \otimes y)+(w \otimes y)$ and $x \otimes (y+z) = (x \otimes y)+(x \otimes z)$.

$\left\| x \otimes y-w \otimes z\right\|_{\mathrm{HS}} \leq \left\| x-w\right\|_{H}\left\| y\right\|_{K}+\left\| w\right\|_{H}\left\| y-z\right\|_{K}$.

Given $x_{k} \in H$ and $y_{k} \in K$, show that the adjoint of $T = \sum_{k = 1}^{M} x_{k} \otimes y_{k}$ is $T^{*} = \sum_{k = 1}^{M} y_{k} \otimes x_{k}$

Given $T \in \mathcal{B}(H, K)$, show that $T$ is compact if and only if $T^{*}$ is compact.

If $x \in H$ and $y \in K$ then $V(x \otimes y) U^{*} = U x \otimes V y$.

There exists a unique bounded operator $T \in \mathcal{B}(H \otimes K)$ such that $\left\| T\right\| \leq \left\| U\right\| \left\| V\right\|$ and $T(g \otimes h) = T g \otimes T h$ for all $g \in H$ and $h \in K$. We call $T$ the tensor product of $U$ and $V$, and write $T = U \otimes V$.

$\left(U_{1} \otimes V_{1}\right)\left(U_{2} \otimes V_{2}\right) = U_{1} U_{2} \otimes V_{1} V_{2}$.

If $U$ and $V$ are each topological isomorphisms, then so is $T = U \otimes V$.

Specialize to $H = L^{2}(E)$ and $K = L^{2}(F)$. In particular, show that if $U \in \mathcal{B}\left(L^{2}(E)\right)$ and $V \in \mathcal{B}\left(L^{2}(R)\right)$ then there exists a unique operator $U \otimes V \in \mathcal{B}\left(L^{2}(E \times F)\right)$ such that $(U \otimes V)(g \otimes h) = U g \otimes V h$, where $g \otimes h$ and $U g \otimes V h$ are the functions defined by equation (B.3).

Prove that if $A$ is a Hilbert--Schmidt operator, then the quantity $\left\| A\right\|_{2}=(\sum_{\beta \in B}\left\| Ax_\beta \right\| )^{1/2}$ does not depend on the choice of basis and defines in the space $\mathcal L_2(H)$ of Hilbert--Schmidt operators a norm which majorizes the usual operator norm.

Prove that the norm $\left\| \cdot \right\|_{2}$ is generated by the scalar product $\langle A,B\rangle_{\mathcal L_2(H)}=\sum_{\gamma \in \Gamma }(Ay_\gamma ,By_\gamma )_H$, where $\left\{ y_\gamma \right\}_{\gamma \in \Gamma }$ is any Hilbert basis in $H$.

Prove that every Hilbert--Schmidt operator is compact.

Construct an isomorphism between $\mathcal L_2(H)$ and the Hilbert tensor product of $H$ and $H'$.

Prove that every Hilbert--Schmidt operator on $L^{2}\left({X,\mu }\right)$ is an integral operator with kernel $K\inL^{2}\left({X\times X,\mu \times \mu }\right)$.

An orthonormal basis in $\mathbb {R}^{3}$ is given by

Use this orthonormal basis and the Kronecker product to find an orthonormal basis in $\mathbb {R}^{9}$.

A basis in the vector space $M_{2}(\mathbb {C})$ of $2 \times 2$ matrices over $\mathbb {C}$ is given by the Pauli spin matrices (including the $2 \times 2$ identity matrix) $\sigma_{0} = I_{2}$, $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$. Use this basis and the Kronecker product to find a basis in the vector space $M_{4}(\mathbb {C}^{4})$ of all $4 \times 4$ matrices over $\mathbb {C}$.