Relation between $A^{*}B=B^{*}A$ and $AB^{*}=BA^{*}$

Question

Let $A$ and $B$ be two matrices. Can we say $A^{*}B=B^{*}A$ implies $AB^{*}=BA^{*}$? how about when $A$ or $B$ are normal?

Any comments could be useful.

Thanks.

Answer 1

Suppose that $A^*B = B^*A$. That is, you know that $(A^*B)^* = (A^*B)$. You want to show that $(BA^*)^* = BA^*$.

This is certainly not generally true when $A,B$ are not both normal. As a counterexample, take $$ A = \pmatrix{0&1\\0&0}, B = \pmatrix{1&1\\1&0} $$ Notice that $AB^*$ is symmetric, but $B^*A$ is not.

The implication does hold if both $A$ and $B$ are Hermitian.