$\textbf {Polar decomposition}$ $:$ Let $\mathcal H$ and $\mathcal K$ be Hilbert spaces. Then any operator $T \in B(\mathcal H, \mathcal K)$ admits a decomposition $T = UA,$ where $U \in B(\mathcal H, \mathcal K)$ is a partial isometry and $A \in B (\mathcal H)$ is a positive operator. The proo...