Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $H$ be a complex Hilbert space and let $m\in\mathbb{N}$. Let $H^{m}$ be the set of $m$-tuples $\zeta=(\zeta_{1},\dots,\zeta_{m})$ in $H$, with componentwise addition and multiplication by complex scalars and with

the finite sum in $\mathbb{C}$. For $k\in[m]$ let $\iota_{k}:H\to H^{m}$ send $v$ to the tuple whose $k$-th entry is $v$ and whose other entries are $0$. For $A\in\mathcal{L}(H)$ let $A^{(m)}:H^{m}\to H^{m}$ send $\zeta$ to $(A\zeta_{1},\dots,A\zeta_{m})$. Sums of elements of $\mathcal{L}(H)$ and of $\mathcal{L}(H^{m})$ are finite sums in these complex vector spaces.

1. (Hilbert structure)¶ $H^{m}$ with these operations and $\langle\cdot,\cdot\rangle_{H^{m}}$ is a complex Hilbert space, and $\lVert\zeta\rVert_{H^{m}}^{2}=\sum_{k=1}^{m}\lVert\zeta_{k}\rVert_{H}^{2}$ for every $\zeta\in H^{m}$.

2. (Coordinate inclusions)¶ For $k,l\in[m]$: $\iota_{k}\in\mathcal{L}(H,H^{m})$; $\iota_{k}^{*}\zeta=\zeta_{k}$ for every $\zeta\in H^{m}$; $\iota_{k}^{*}\iota_{l}=I_{H}$ if $k=l$ and $\iota_{k}^{*}\iota_{l}=0$ if $k\ne l$; and $\sum_{k=1}^{m}\iota_{k}\iota_{k}^{*}=I_{H^{m}}$.

3. (Block entries)¶ For $T\in\mathcal{L}(H^{m})$ and $k,l\in[m]$ let $T_{kl}=\iota_{k}^{*}T\iota_{l}\in\mathcal{L}(H)$. Then, for all $S,T\in\mathcal{L}(H^{m})$, $\zeta,\zeta'\in H^{m}$ and $k,l\in[m]$,

Conversely, for every family $(A_{kl})_{k,l\in[m]}$ in $\mathcal{L}(H)$ there is exactly one $T\in\mathcal{L}(H^{m})$ with $T_{kl}=A_{kl}$ for all $k,l\in[m]$, namely $T=\sum_{k=1}^{m}\sum_{l=1}^{m}\iota_{k}A_{kl}\iota_{l}^{*}$.

4. (Diagonal operators)¶ Let $A\in\mathcal{L}(H)$. Then $A^{(m)}\in\mathcal{L}(H^{m})$, $(A^{(m)})_{kl}=A$ if $k=l$ and $(A^{(m)})_{kl}=0$ if $k\ne l$, and $(A^{(m)})^{*}=(A^{*})^{(m)}$. A map $T\in\mathcal{L}(H^{m})$ commutes with $A^{(m)}$ if and only if $T_{kl}$ commutes with $A$ for all $k,l\in[m]$.