If $T$ is normal and given $T$ is invertible, then is $T$ unitary? and Given two unitary operators $U$ and $R$ prove $U^{-1}R$ is also unitary.

Question

Question 1. If $T$ is normal that is $T^{*}T = TT^{*}$, and given $T$ is invertible, then is $T$ unitary?

I am familiar with the result which states the converse. That if T is unitary, T is invertible and $T^{-1} = T^{*}$. Not sure if they are the same. If they are can you please explain.

Question 2. Given two unitary operators $U$ and $R$ prove $U^{-1}R$ is also unitary. My solution: $(U^{-1}R)(U^{-1}R)^{*} \to U^{-1}RR^{*}(U^{-1})^* \to U^{-1}U \to I$ where the second last implication follows from $U^{-1} = U^{*}$ and $U^{**} = U$. Is this reasoning correct?

Thank you for the help.

Answer 1

For question 1, note that every complex diagonal matrix is normal, and most of them are invertible but not unitary. In particular, if $|c|\ne0$ or $1$, $cI$ is invertible but not unitary.

Your reasoning in question 2 is correct.