Some condition for unitary operator to be identity.

Question

• $U$ and $T$ are positive self adjoint operators.
• $K$ is unitary/orthogonal.
• $U = T \circ K$

Proof that $K=I$.
I tried to show that $\forall x <x-Kx,x-Kx>=0$, but got stuck. The only equation I managed to show is $<Kx,Ty>=<Tx,Ky> \forall x,y$. Can you give me some hints?

Answer 1

Suppose $U=T= \left( \begin{matrix} 1 &0\\ 0&0 \end{matrix} \right)$ and $K= \left( \begin{matrix} 1 &0\\ 0&-1 \end{matrix} \right)$ then $U=TK$ but $K \neq I$.