unitary,close to it self defintion question

there is A matrices order n over C.we define $\displaystyle M_{nxn}^{c}$ a transformation T as T(B)=AB for every B$\displaystyle \in M_{nxn}^{c}$

A.prove that if A unitary if and only if T is unitary$\displaystyle TT^{*}=T^{*}T=I$?

B.prove that if A normal if and only if T is normal$\displaystyle (TT^{*}=T^{*}T)$?

C.prove that if A close to itselfs if and only if T is close to itself ($\displaystyle T=-T^{*}$)?

i wrote the definition near each question ,also i know this$\displaystyle T*=\overline{T^{t}}$.

for A.

$\displaystyle T(B)T(B)*=AB(AB)^{*}=AB\overline{AB^{t}}$

what next?

the prof uses (T(x1),x2)=(x1,$\displaystyle T^{*}$(x1))

i dont know this definition and how it linked to the transformation we are asked to do

?