Uniqueness of polar decomposition of normal operator

Question

Let $H$ be a Hilbert space and $T \in \mathcal{L}(H)$, a bounded linear operator. It is well-known that $T$ is normal if and only if there exist positive and unitary operators $P$ and $U$ such that $$ T=UP=PU. $$ It is also true that if $T$ is invertible, then the decomposition is unique (Rudin, "Functional Analysis", Th 12.35). I would like to ask: when is unique the positive operator $P$? And the unitary operator $U$? Thanks in advance.

Answer 1

If $T=UP$ where $P \ge 0$ and $U$ is unitary, then $T^*T=P^2$; so $P$ is the unique positive square root of $T^*T$. $U$ is not determined on the orthogonal complement of the range of $P$.