Let $T: H\to H$ be a bounded operator on Hilbert space $H$. $T(e_n) = a_n e_{n+1}$ where $\{e_n\}$ is orthonormal basis and $\{a_n\}$ is bounded sequence.
- What is the polar decomposition of $T$?
- For what sequences $T$ is Fredholm?
- For what sequences $T$ compact?