unitary,close to it self defintion question

there is A matrices order n over C.we define $M_{nxn}^{c}$ a transformation T as T(B)=AB for every B $\in M_{nxn}^{c}$
A.prove that if A unitary if and only if T is unitary $TT^{*}=T^{*}T=I$?
B.prove that if A normal if and only if T is normal $(TT^{*}=T^{*}T)$?
C.prove that if A close to itselfs if and only if T is close to itself ( $T=-T^{*}$)?
i wrote the definition near each question ,also i know this $T*=\overline{T^{t}}$.
for A.
$T(B)T(B)*=AB(AB)^{*}=AB\overline{AB^{t}}$
what next?
the prof uses (T(x1),x2)=(x1, $T^{*}$(x1))
i dont know this definition and how it linked to the transformation we are asked to do
?