Let $\{e_n\mid n \in \mathbb{N}\}$ be an orthonormal basis for the Hilbert space $H$ and define for each $T \in B(H)$ the doubly infinite matrix $A = \{\alpha_{n,m}\}$ by letting $\alpha_{n,m} = (T e_m\mid e_n)$.
- Show that every row and every colomn in $A$ is square summable (i.e. belongs to $\ell^2$ ).
- Use this to prove that the matrix product $AB = C$, where $C = \{\gamma_{n,m}\}$, where $\gamma_{n,m} = \sum_k \alpha_{n,k} \beta_{k,m}$ is well defined when matrices $A$ and $B$ correspond to operator $T$ and $S$ in $B(H)$.
- Show that the matrix $C$ is corresponding to the operator $TS$.
- Also find the matrix corresponding to $T + S$ and $T^*$.