Let T be a linear operator on an inner product space V. Let $U_1=T+T^{*}$ and $U_2=T T^{*}$. Prove $U_1=U_1^{*}$ and $U_2=U_2^{*}$

Question

Let T be a linear operator on an inner product space V. Let $U_1=T+T^{*}$ and $U_2=T T^{*}$. Prove $U_1=U_1^{*}$ and $U_2=U_2^{*}$

How am I supposed to prove this? This doesn't tell it's finite-dimensional.

Answer 1

Use the following: $(A+B)^{*}=A^{*}+B^{^*}$, $(AB)^{*}=B^{*}A^{*}$ and $A^{**}=A$.