On clarification on representation of operators on direct sum of Hilbert spaces as matrices

Question

Suppose $G$ is a countable group, consider $\mathcal{H}=\oplus\{\mathcal{H}_{g}:g\in G\}$, Suppose $T$ is an operator in $\mathcal{H}$, then how to give a isomorphism from $B(\mathcal{H})$ into matrices with entries are operators on $\mathcal{H}_{g}$.

Answer 1

If you write $P_g$ for the projection $P_g:\mathcal H\to\mathcal H_g$, then the entries of your matrix will be $$ T_{hg}=P_hT|_{\mathcal H_g}. $$ If this is not obvious to you, do it explicitly when $\mathcal H=\mathbb C^2$ and it should be clear.