Adjoint Operator and Inverse

Another way to solve the Question 1 is:

We have $ (T^*)^{-1}T^* = 1 $ then $ ((T^*)^{-1}T^*)^* = 1^*$

Since $1^* =1 $ and $(AB) = B^*A^*$ for every operators A and B then $ T^{**}((T^*)^{-1})^* = 1 $

But $T^{**} = T$ so $((T^*)^{-1})^* = T^{-1}$ and then $(T^*)^{-1}=(T^{-1})^*$

By definition of adjoint, considering $T^*T$ as a product of operators: $$(T^*Tu,v) = (Tu,Tv) = (u,T^*Tv).$$

On other hand, if we take them as a whole: $$(T^*Tu,v) = (u,(T^*T)^*v).$$

Can you conclude now?