One of my professors mentioned that since a matrix A is positive semi definite and B is hermitian, hence the inner product $<A,B>$ is real. Is this an if and only if condition? So if we know that B is hermitian and $<A,B>$ is real then does it imply that A is hermitian as well?
Update 1: From the answer it seems like my answer is no. Could you also then explain, if we are given a linear map T which takes hermitian operators to hermitian operators, then why is the adjoint map hermitian as well?
No. Take any matrix A, and define the matrix $$\hat{A} = \overline{\langle A,B\rangle } A$$
Then
$$\langle \hat{A}, B \rangle = \langle \overline{\langle A,B\rangle } A, B\rangle = \overline{\langle A,B\rangle }\langle A, B\rangle \in \Bbb R$$
And $\hat{A}$ is generally not hermitian