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?